E LE
M E N T A RY A EG E B E E
GE
OR G E W M Y E R S
.
THE
U N IV E R S IT Y O F
C H ICA G O
A ND
GE
OR G E E
N E VV B U RG H ,
.
A T W OO D
NE
W
YO R K
S C O TT , F O RE S M A N A N D C O M PA NY
C H IC A GO
NE W
YO R K
,
C OPY R IG H T
1916
BY
S C O TT
.
F
OR E S M A N
A ND C
O M P A NY
;
E D UC A T l O N D E PT
P RE FA C E
ma ke
ol o gy for o ffer i n g a n o t h er a lg eb ra
In i n fl ue n t i a l pl a c es a lg e b r a h as b e e n
t o t h e s c h oo l pub li c
Is
c h a l l e n g e d as a s ui t a b l e s u b j e c t for h ig h s c h oo l pu pil s
it n ot t he part of w i s do m b e fo r e e li m i n a t i n g a s u b j e c t of s o
l o n g an d u n di s put e d st a n di n g as a lge b r a t o t ry r e c o n st r u c t
in g a nd i m pr ovi n g it s fo rm an d e v e n s o m e o f it s s ub st a n c e ?
T h e a ut h o rs b e li e v e t h at t his t ext h as a c c o m pli s h e d m u c h
in bot h o f t h ese pa rt i c ul ars
T h i s b o o k is n o t writ t en h o we v er w i t h t h e t h oug h t o f
d e fe n di n g a n u nw o rt h y c l ai m a n t t o a pl a c e in t h e c u rr i c ul um
T h e t ru e V i ew is t h at t h e hi gh e du c a t i o n a l m e r i t o f s c h o o l
a lg e b r a m a y b e r ai s e d e v e n hi g h e r b y a t r ea t m e n t w h ose
l a n gu a ge an d m od e o f expo s it i o n a re in a c c or d wi t h t h e poss i
bil it ies a n d a ppr ec i at io n s o f yo u t h a n d w h o s e s c i e n t ifi c
s o u n d n ess is a t t h e s am e t i m e n o t ser i o u s l y c o m pr o m i s e d
It is t h e aut h o rs c o n vi c t i o n t h at r igh t l y t augh t a lg e b r a is o f
g re at e du cat i o n a l v a lu e a nd t h at t o m o st hi gh s c h oo l s t u d e nt s
it is n o t di s t as te ful
In c a r ry in g o ut t h e i r vi e ws o n t h i s li ne t h e a ut h o rs h a v e
a tt e m pt e d se v e ra l s pec ifi c t hi n g s
So m e o f t h e s e s t a t e d
b ri efl y a re as follows
1 T o pres e n t t h e m at e ri a l in a l a n g u a g e a n d m o d e t h a t a r e
s i m pl e a n d a t t h e sa m e t i m e m a t h e m a t i c a ll y s o u n d w i t h o u t
r es o rt t o m at h e m a t i c a l t e c h ni c ali t i es
2 T o m o t iv at e t h e v a rio u s t o pi cs o f a lge b r a e i t h e r t h r o ug h
s pe c i a l pr obl em a t i c
s ituat i o ns
gr a du a ll y
o r t h r o ug h t h e
r i s i n g d e m a n d s of t h e e q u at i o n fo r p a rt i c u l a r ph ases o f a lg e
braic t ec hniq ue
As e xa m pl e s s e e pa ges 2 7 3 2 5 9 2 66 e t c
3 Pe rs i st e n t l y t o m a ke t h e fi rst st e ps i n t o t h e t r e at m e n t s
o f a lg e b r a i c s u b j e c t s t h r o ug h t h e a n a l o g o u s s ub j e c t s o f a r i t h
m et i c
2
2
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4
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1
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54 l 291
,
,
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,
.
P RE
iv
t h e pu pil
ome
FACE
ll y v a lu ab l e h e l p in l e ar n i n g
t o re a d t o c o m pr e h e n d a n d t o i nte rp r e t a lge b r a i c l a n gu a g e
a n d t o e xpr e s s m at h e m a t i c pr i n c i pl es a n d r u l es in t h i s l a n
gu a g e C h a pt e r X I II o n G e n er a l N um b ers F o rm ul as a nd
Typ e fo rm s m ay b e c it e d as a go o d illu st r at io n of t h i s
4
.
To
give
s
rea
,
,
,
,
.
,
-
t r eat m e n t
.
T o giv e a n e ar l y i n t r odu c t io n t o s i m u l t a n eo u s s i m pl e
e q u at io n s a n d t o c o m pl e t e t h e i r s t ud y b y r e c u rr e n t t r e a t m e n t s
as t h e c our se d e v e lo ps
6 T o m a k e e ar l y a n d fr e q u e n t u se o f t h e gr a ph fr ee d fro m
a n a l yt i c a l t e ch n i ca li t i e s a s a n a i d to the dev el op men t of a l g e
bra t h r oug h c l arifyi n g a n d vivifyi n g m ea ni n gs o f a lg e b r a i c
i
n
h
n
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e
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c
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ess
es
t
q
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g
g
p
a n d l e arn i n g t h e m
7 T o se e k di lig e n t l y fo r s u c h a n o r d er o f t r e a t m e n t o f t h e
s pe c i a l t o p i cs a s is di c t a te d b y t h e h ig h e s t e c o n o m y in t h e
m ast e ry o f t h e el e m e nt s of t h e sc i e n ce of a lg e b r a
B y this
m e a n s it is h o pe d t o giv e a s t ro n g er a n d a m o re h ig h l y e d u
c at iv e fi rs t ye ar c ourse in t h e c u s t o m a ry t i m e
( See T a bl e
o f C o n t e nt s )
8 C ar e full y t o gr a d e as t o diffi c u l t y a n d t o b a l a n c e as t o
n
n
m
n
h
k
u
li
u
a
t
i
t
e
r
b
l
e
s
a
d
exer
c
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es
of
t
b
s
a
t
a
d
t
o
e
oo
h
y
p
q
y
q
a g a i n wi t h a n e ye s i n gl e t o t h e u n fo ldi n g n e e d s of a lg e b r a
(S ee probl em li st s give n u n d er t h e di ff er e nt t o pi c s )
9 T o c o rr e l at e wi t h a r i t h m e t i c g e o m e t r y g e n era l sc i e n c e
a n d e v ery d a y lif e to a s g r e a t a d e g r ee as t h e be s t s c h o o l
i nt erest s of fi rst-yea r a lgeb ra r e q ui re
1 0 T o h e ig h t e n t h e w o rk a b ili t y o f t h e t e xt by a s yn o pt i c
ta b l e of c o n t e n t s a s u mm a ry of d e fi ni tio n s ( pag e
a nd
a go o d w o r ki n g i n d e x
A lit t l e of th e pe d agogi c a l b a c kgro un d of t h e o rg a niz a
t i o n of t h i s t ext m ay b e s t a te d h e r e
Th e a u t h o rs h old t h e
vi ew t h at t e a c he rs o f pr e se n t d ay se c o n d ary a lg e b r a s h o uld
re c og ni z e t h a t t he y a re u n de r t hr ee sig nifi c a nt pr ofe ss i o na l
o b lig at io n s t o t h e ir pu pil s viz
5
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-
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P RE
I
To
.
the
ra ti ona l ize
F ACE
a n a l ogou s
meti c
a ri th
f the
o
a l gebr a i c
topi cs ta u ght
.
o n a b l e t o expe c t o f b e gi nn e r s in se c o n d ary
e ve n
a lge b r a t h at t h e y re a ll y u nd er s ta nd t h e i r ar i t h m e t i c
St ill l es s m ay se c o n d ary t e a c h e rs r igh t fu ll y
as a rit hm e t i c
e x pe c t t h a t be gi n ni n g pu pil s h a v e g ras pe d t h e i r a ri t h m e t i c
Thi s
in s u c h fo rm t h at it c an b e m a d e t h e b as i s fo r a lg e b r a
is a m u c h m o r e diffi c ul t m at t e r b e c a us e a l t h ough bot h a r i t h
m e t i c an d a lge b ra a r e ab str ac t s c i e n c es a lge b r a i n v o lv es a
m u c h high er o r d er o f ab s t r a c t n ess t h a n a r i t h m e t i c
In vi e w o f t h e sc o pe an d c o m pl exi t y o f m o d e rn e l e m e n t a ry
o f t h e s lig h t e m ph as i s o f s c h oo l o ffi c i a l s
s c h oo l a r i t h m e t i c
e x a m i n e rs a n d s urv e yo r s a n d e v e n o f s c h oo l pr o g ra m s u po n
r a t i o n al i z i n g pr o ce s s e s it is w o rs e t h a n u se l ess t o e xpe c t l e t
t h e m o s t c o n sc i e n t iou s t e a c h er s t r iv e as h e m ay t h a t m o re b e
d o ne in t h e e l e m e nt a ry s c h o o l t h a n t o r a t i o n ali z e t h e m ost
n t a ry n ot i o n s an d pr o c esse s o f a r i t h m e t i c
In f a c t fo r
e l em e
se v e r a l yea r s e l e m e n t a ry t ea c h e rs h a v e b e e n u r g e d b y s o m e
a ut h o r i t i e s t o r e n o u n c e r at i o n a li z in g fo r m e re h a b i t u at i n g a n d
d rill pro c e d ures T h ese t hi n gs c o u pl e d wi t h t h e fa c t t h at
a r i t h m e t i c o f t h e so rt c o v e r e d in o ur g r a mm a r g r a d es is o n e
o f t h e m o s t di ffi c ul t o f al l m at h e m at i c a l b r a n c h e s a n d w i th
li m i t at io ns o f pro gr a m t i m e a n d i mm at uri t y o f pupil s h o pe
l e ss l y pre c lud e a n y a t t e m pt s a t t h o se fa r r e a c h i n g i n d u c t i o ns
a n d g e n e r a li z a t i o n s t h at a re e s se n t i a l a t t h e v ery b e gi nn i n g
o f r a t i o n a l a lg e b r a
T he re fo re t h i s fun d a m e nt a l w o rk fo r
t h e h ig h l y s pe c i a li z e d n ee d s o f t h e s e v er a l a lg e b r a i c t o pi c s
b e l o n gs pro pe rl y t o t h e a lg eb r a t e a c h e r
T hi s t e xt s uppli es
t h e i n i t i a t o ry a r i t h m e t i c a l r at i o n a li z i n g fo r t h e a lg e b r a i c
t o pi c s a n d s ub j e c t s at t h e pre c i s e pl a ce s w h ere it is n e e d e d
a n d of t h e s o r t t h at is a ppr o pri a t e
It is h ar dl y
re as
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-
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‘
.
II
.
To s how tha t ma n y
ri ca l l y,
i
di agr am s
T o see
.
,
,
e
.
,
by t h e
a id
a l g ebr ai c
of
t he
thin gs
be done geomet
s pa c e m a t e r i a l o f
ca n
c o n c re t e
f
n
c
r
es
o
a
i
t
u
p
y gr a ph i ca l h e l ps t o c l ea r t h i n ki n g
t o c a l c u l a t e a n d t o co m pr e h e n d is t h e t r ue o r d er o f
,
.
,
P RE
vi
in m ast e r i n g
FACE
lge b ra i c t as ks T h e c o nc ept s o f li n es
r ec t i l i n ea r figu r es
a n d so lid s ar e s o m u c h s pa c e m a t e ri a l
a l w a ys a n d e v e r yw h e r e a v a il a b l e fo r c o n cr et i n g vi s u a li z i n g
a n d vivifyi n g n u m b e r l a w s a n d re l a t i o n s
at n o g re at c o s t
in m o n e y o r e ffo rt
T h e h ig h s c h oo l yo u t h h as liv e d l o n g
e n o u g h in t h i s w o rld o f s p a c e t o h a v e b ec o m e f a m ili a r w i t h it
a n d his s pa t i a l e x pe r i e n c e s n ee d o n l y t o b e d r a w n u p o n t o
e n a bl e him t o l a y fi rm h o ld o n t h e h ig h l
a b s t r a c t fun d a m e n
y
t a l s of b e gi nn i n g a lg e b r a
R eall y t o see t h at a lg eb r a m e re l y
g e n er ali z es m e n s u rat i o n l aws t h at a lg eb r a i c n u m b ers l a ws
a n d p r o b l e m s pi c t u r e i n t o vivid f o r m s a n d t o l ea r n t h e s e c r e t
o f l a yi n g b e f o r e h is e ye s di a g r a mm at i c a ll y t h e c o n dit i o n s o f
a lg e b r a i c p r o b l e m s as a n aid in fo r m ul a t i n g t h ese c o n di t i o n s
i nt o a lg eb r a i c l an gu a ge a n d t e c h ni q u e a r e of t h e h igh e st
i nt ere st a n d v a lu e t o t h e b e gi nn e r Th e pro fes si o n a l d ut y
o f e m pl o yi n g t h e c o n c r e t i n g a g e n c i es o f p i c t u r e s di a g r a m s
g e o m et r i c a l figu r es a n d g r a ph s t o vivify a n d vi t ali z e a lge b r a
will b e re a dil y a c c e pt e d b y t h e t ea c h e r w h o st riv e s t o re a li z e
in pr ac t i c e t h e e du ca t i o n a l m e r i t s o f w e ll t a u g h t a lg e b r a
N o c l um sy l a b o r at o ry e q ui pm en t o f e xt e n s iv e a n d e x pe n s iv e
a pp ar a t u s is r e q ui r e d t o e n ab l e t h e a lg e b r a t e a c h er t h r oug h
s p a c e m at e r i a l s t o s u ppl y g e n e t i c b a c k g r ou n d s fo r a lg e b r a i c
p r o b l e m s t r u t h s a n d l aw s
st e ps
a
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-
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-
,
To s how the pu pi l tha t a l gebra wil l en a bl e him to d o mu ch
tha n he ca n do wi th either a ri thme tic or g eometry, or both
III
m ore
.
,
.
f i
l d ut i es are re all y prel im i
n a ry t h r o u g h w h i c h m o t iv a t i n g a n d c l e a ri n g t h e w a y fo r
T h i s t hi r d dut y is pe c u
e ff e c t iv e a t t a c k a re a c c o m pli s h e d
l ia rl y d ue t o a lg e b ra
It is in f a c t d ue t o b o t h pu pil a n d s ub
f
h
r
t
e
h
n
r
b
m
as
t
e
o
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l
g
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u
d
a
e
c
t
a
t
t
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r
c
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r
a
s
b
e
sec
e
t
t
a
t
o
y
p
y
j
s u b j e c t m at t er s h a ll a pp e a r in t h e l ea r n i n g ac t s
F o r ex a m pl e t h e pu pil s h o uld s ee s u c h t h i n g s as t h at by
a r i t h m et i c h e c a nn o t s u b t r a c t if t h e s u b t r a h e n d h a ppe n s t o b e
g re at er t h a n t h e m i n ue n d ; t h at h e c a n no t s o lv e s o s i m pl e
=
n
i
n
3 ; b ut t h at if h e i n c lud e t h e n e g at iv e
as x + 9
a
e q ua t o
T he
fi rs t
a n d se c on
d
pr o e ss o n a
,
.
.
-
.
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,
,
P RE
n
u m b er s
am on
g
F AC E
V
n um b e r n o t o n s
his
he
i
c an
do
b oth
e as
ii
il y
.
He s h o u ld s ee t h at h e c a n s q ua r e a n d c ub e num b ers g eo
m e t r i ca ll y , b ut t h a t h e c a n g o n o fu rt h e r w i t h i n v o l ut i o n t h a n
I f , h o w e v e r , h e w ill l ea rn t h e s ym b o li sm o f a lge b ra h e
t hi s
m ay e as il y expr ess a n d w o rk w i t h 4 t h , 5t h , 6t h , e v e n wi t h n t h
h
n
r
H
h
h
n
h
h
c
a
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uld
il
s o lv e e q u a
w
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w
t
a
t
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e
o
p
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t i o n s in
o ne ,
a nd
two ,
gr aph i c a l pi c t u res
in t h r ee
r
h
a ps
e
p
u nkn o w n s
wit h
g e o m e t ri c a ll y t h e gr eat pow er h e g a i n s
b y m as t erin g t h e a lg eb r ai c w ay e n a b l es h im t o g o r ig h t o n
e as il y t o t h e s o l ut i o n of s i m ul t a n e ou s e q u at i o n s in 4
5 6
a n d e v e n n u nk n o w n s
H e sh o u ld b e m a d e t o f e e l t h a t w h il e
a ri t h m e t i c w o uld e n a b l e him
by a s l o w pro c e s s o f f ee li n g
a bou t t o fi n d one s o l u t i o n o f m a n y pro bl e m s a lg e b r a if he
w ill l ear n its l a n g u a ge a n d m e t h o d will l ea d him di r e c t l y n o t
t o o n e b ut t o a l l pos si bl e s o lu t i o n s
It will t h u s e n a b l e hi m
t o kn o w w h e n h e h as s o lv e d h is pr obl e m co mpl e tel y
Th e se
a n d s i mi l a r g a i n s of p o w er o v e r q u a n t i t a t iv e pr o b l e m s a re t h e
r ea l r e aso n s w h y t h e e du c a t e d m a n o f t o d ay c a nn o t a ffo r d
n o t t o kn o w a lg e b r a
Le t t ea c h e rs pe rfo rm t h i s pr of e ssio na l
du ty we ll a n d t h e fo es of alge b ra as a s c h oo l s ub j e c t will b e
c o n fi n e d t o t h o s e w h o a re ig n o r a n t of it
The o ne who h a s
l e arn e d t h e s ub j e c t w ill t he n r e g ar d it as t h e ema nci pa tor
of qua ntita ti ve thin king
It is d e si re d t o c a ll pa rt i c ul ar a tt e nt i o n t o t h e i n t r odu c t o ry
pa g e s o n R e as o ns fo r S tu d yi n g A lg e b r a a n d t o Sugg es t i o n s o n
P rob l em so l v i n g o n p age 1 1 3 a n d t o t h e c a r e ful t r eat m e nt o f
f ac t o ri n g Th e t re at m en t o f t h e fu n c t i o n n o t i o n o n pa g e s
50 5 6 will a ppe a l t o m a ny t e a c h er s
It w ill b e no t e d a l s o
t h a t t his e l em en t a ry c o u r s e is divid e d i nt o h a lf yea r u n i t s
T he probl em a n d e x e r c i se li s t s are full v ar i e d a n d c a re full y
c h o se n
Teac h e rs wh o e m pl o y suppl e m e n t a ry li st s of exer
c i s e s w i t h t he re gul a r t e x t s h o u ld n o t r e q ui r e
u
il
t
s
r
t
o
t
p p
y o
so lv e a l l t h e pr obl e m s a n d e xe r c i s es g iv e n h e r e
T h ese li st s
a re m a d e full a n d v a ri e d t o a ffo r d c h o i ce a n d r a n g e of m a t e ri a l
G r eat c a re h a s b ee n e x er c i se d t o c o v e r al l t h e st a n d ar d d iffi
,
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-
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-
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.
.
.
P RE F A C E
V iii
c ul t ie s o f
t as k t o
b o o k m a k es it s
fi rst yea r a lg e b r a , fo r t h i s
t eac h g oo d a lg e b r a
-
pr
im al
.
T h i s t e x t is t o b e fo llo w e d pr e s en t l y by a s e c o n d c o u rse o n
I nt e rm edi at e A lg e b r a
Th e t wo t og e t h e r will c o v e r t h e
s t a n d a r d r e q u i re m e n t s o f se c o n d ary a lg e b r a
T h e pl e as a n t t as k n ow r e m a i n s t o a c k n ow l e dg e t h e as s i st
a n ce t h e a u t h o rs h a v e r e c e iv e d f r o m M r J o h n D e Q B r igg s
o f St P a ul A c a d e m y St P a ul M i nn ; f r o m t h e M i s ses E ll e n
G o ld en a n d E s t e ll e F e nn o o f C e n t r a l H igh S c h o ol W as h i n g
t o n D C ; a n d fr o m Pr o f ess o r H C C ob b of Le wi s I n s t i t u t e
C h i c a g o a l l of w h o m r ea d a n d c ri t i c i z e d t h e p roo f s o f t h e
b o ok
T h e i r c ri t i c i sm s a n d su gg es t io n s h a v e r es ul t e d in
n u m e r o u s i m pr o v e m e n t s
M ay t h i s b oo k fi n d fr i e n d s a m o n g st t e a c h ers a n d pu pil s
a n d a d e s e r vi n g pl ac e a m o n g s t t h e i n flu e n c e s n ow m a ki n g fo r
t h e i m pr o v e m e n t o f t h e e du c a t i o n a l r es ul t s o f h ig h s c h oo l
a lg e b r a
TH E A U T H O R S
.
.
.
.
.
.
,
.
,
,
.
,
.
.
.
,
,
.
.
,
.
.
Chica go , S eptember , 1 91 6
.
C ON TE NTS
F IRS T H AL F Y E AR
-
C HA PT R
E
I N T RO D U C T I O N R E A SO N S F OR STU D YI N G A L G E B R A
N O T ATI O N IN A L G E B R A T H E E QU ATI O N
.
.
N o t a ti o n
Th e E
.
q ua ti on
D i r e c ti o n s fo r
Pr o bl e m s
M akin g
St a t em e n t s
an d
So l v i n g
.
P
OS I TIV
E
AND
P os iti v e
G ATIV
NE
and
E
N UM B E
RS
.
DE
FI N I TI O N S
N e g a ti v e N u m b e r s
De fin iti o n s
A DDI T I O N
Ad d iti o n o f M o n o m i a ls
Ad d i n g Si m ilar Te r m s
A dd in g D iss i m i l ar Te r m s
Add itio n o f P o lyn o mia l s
SU B TR A CTI O N S Y M B O LS O F A GGRE G AT I O N
S ub t r a c ti o n o f M o n o m i a ls
S ub tr a c ti n g Si m ila r T e rm s
S ub t ra c ti n g D i ss i m ilar T er m s
S ubt ra c ti o n of Po l yno m i a ls
Sym b o ls o f A ggr e g a ti o n
Ad d itio n o f T e rm s P a rtly Si m ila r
S ubt r a c ti o n o f T erm s P artl y Si m il ar
G RA P HI N G F UN CTI O N S S O LVI N G E QU ATI O N S IN
UN K N O W N G R A P H I C A LL Y
G raph i n g F un c ti o ns
So l v i n g E q ua ti o n s in O n e Unkn o w n G r a ph i c a ll y
.
.
ON E
'
.
E QU A T I
E
ON S
G E N E R A L R E VI E W
.
qua ti o n s
.
C le a r i n g E
q ua ti o n s
of
F c ti
ra
o ns
G en er a l Re v i ew
G RA P HI N G D AT A S O L VI N G S I M U LT AN E O U S E QU ATI O N S
G RA P HI C A LLY
G r a ph i n g D a t a
So lv i n g Sim ult an eo us E qua ti ons Gr aph i c a ll y
.
.
IX
C O N TE N T S
PAG
S I M U LT A N E O U S S I M P LE E QU AT I O N S E L IM I N A TI O N BY
ADDI TI O N OR SU B T R A C TI O N
Sim ult an eo us Si m p l e E q u a ti o n s
E l i m i n a ti o n b y A d d iti o n o r S ubtr a c ti on
M U LT I P L I C A T I O N
T h e Sign o f t h e P r o d u c t
T h e E xp o n e n t in t h e P r o d u c t
M ultiplyi n g On e M o n o mi al b y An o t h er
P ow er s o f M o n o m i al s
M ultiplyin g a P olynom ia l b y a M o n om ia l
M ulti p lyi n g a P o l yn o m i a l b y a P o l yn o mi a l
S I M P LE E QU AT I O N S
D IVI S I O N
D i v i d i n g a M o n o mi a l b y a M o n o i a l
D i v i d i n g a P o l yn o m i a l b y a M o n o m i a l
D i v i d i n g a P o l yn o m i a l b y a P o l yn om i a l
A PP L I C ATI O N S O F S I M P LE E QU ATI O N S E L I M I N AT I O N BY
SU B S TI TUTI O N
S ugges ti o ns o n Pro bl em S o lv i n g
E li m i n a ti o n b y S ub s tit u ti on
G E NE R AL NUM B E RS F ORM U L A S T Y PE —F ORM S
G en er a l N um b er s
F or m u l as
for m s of A lge b r a i c N um b ers
Fo rm s a n d Type—
F A C T OR I N G
M o n o m ia l F a c to rs ( Typ e form : a x + a y + az)
C o mm on C o m po un d F a c t or :
S quar e o f t h e Sum o f T w o N um b ers :
2
2
S qu ar e of t h e D i ff er en c e of T w o N um b ers ( a 2a b + b )
2
2 i
Tr i n om i a l S quar es (x 2xy + y )
f T w o N um b e r s
P ro d uc t of t h e Sum a n d D i ff er e n c e O
E
.
.
.
.
.
m
.
XII
.
.
-
XIII
.
.
.
.
XI V
.
-
.
( a + b) ( a
b)
— b2
D iffer e n c e o f T w o S quar es ( a
Pr od u c t o f T w o B i nom i a ls w ith
Z
a
)
.
C o mm on T er m
S pe c i a l Q uad r a ti c Tr i n om i a ls :
—
—
Th e G e n er a l Q ua d r a ti c T r i n om i a l : ( a mH bx l c )
4
2 2 — 4
I n co m ple t e Tr i n om i a l S quar es : (113 + x y l y )
3 — 3
D i ff er en c e o f t h e S a m e O dd P o w er s : ( x
y)
Sum o f t h e S a m e O dd P ow er s
Re v i e w
-
-
1 13
1 13
1 20
1 23
1 23
1 24
1 30
134
134
1 35
1 37
1 38
140
C O N TE N T S
C HA PTE R
H AL F YE AR
E X E RC I S S F OR RE VI W N
SE C O N D
E
Xi
-
E
QU ATI O N S
E
So lu tio n o f E qua ti ons b y Fa c t or in g
E xer c is es for R ev i ew an d P r a c ti c e
.
PA G
A
D
P
RA C TI C
E
.
.
.
H I G HE S T C O MM O N F A CT OR
H igh es t C o mm o n F a c t or
LO W E ST C OMM O N M U LT I P LE
.
H igh t C
es
F ct
n F ct
o m m on
a
o r of
M o n o m i a ls
.
H igh es t C o mm o
a
o r o f P o lyn o mi a ls b y F a c t or i n g
Lo w es t C o m m on M ulti ple
Lo wes t C o m m o n M ulti pl e o f M o n o m i a ls
L o wes t C o mm o n M ultipl e o f P o lyn o m i a ls b y Fa c
t or in g
XV II
.
F
RA C TI O N S
.
Red uc ti o n o f I m pr o pe r Fr a c ti o ns
Red uc ti o n o f M ixed E xpr e ss i ons
Lo w es t C o mm o n D en om i n a to r
Ad d iti o n a n d S ub tr a c ti o n o f Fr a c ti o n s
M ulti pli c a ti o n o f Fr a c ti o n s
D i v i s i o n of Fra c ti o ns
.
X V III
.
L I TE RA L
FRA C I O N AL E QU ATI O N S
FORM U L AS
L it l d F c ti l E q u ti
AND
T
.
S O LUTI O N
.
er a
an
on a
ra
o ns
a
S pe c i a l M e t h od s
G e ner al Pro b le m s
So l uti on of Form ul as
.
.
S I M U LT A N E O U S S I M P LE E QU AT I O N S
E l i m i n a ti o n b y C o m pa r iso n
P ro b l e m s in Si mult a n eo us S im
.
T hr ee or M or e Un kn o wn N um
P ROPORTI O N V A R IATI O N
Ra tio
.
P r o po rti o n
P ri n c i p l es
XXI
.
of
Pro
p ti
or
P O WE RS R OO TS
I nv o l uti o n
.
.
P o wer o f a Pr od u c t
P o wer o f a r a c ti o n
P o we rs o f B i n o m i a l s
F
on
le E
ers
q ua ti o n s
E
C O N TE N T S
PA G
TE R
I
P O W E RS R OO TS— C on tin u ed
E v o l u ti o n
Roo t o f a P ow er
R oo t o f a P r od u c t
Roo t of a Fr a c ti o n
N um b er o f R oo t s
I m a gi n ar y Ro o t s
Signs o f R ea l R oo t s
T o F in d t h e R ea l Roo t s o f M o n o mia l s
S quar e R oo t o f a P o lyn omia l
S qua r e R o o t o f N um b e rs
T o F i n d t h e S q u a re R oo t o f a D e c i m a l
T o F i n d t h e S q u ar e R o o t o f a C o m m o n
.
.
.
.
E X PO N E N TS
R ADI C A L S
.
F c ti
ra
on
.
E xp o n e n t s
R a d i c a ls
.
Sim p lifi c at i on
of
Ra d i c
T o R ed u c e a M ix ed N u m b e r t o a n E n tir e
Ad d iti on a nd Sub tr a c ti o n o f S ur d s
T o R ed u c e S ur d s t o t h e S a m e O r d er
u lti p li c a ti o n o f S ur d s
D i v i s i o n o f S ur d s
R a ti o n a l i z i n g S ur d s
S quar e Ro o t o f B i n o m i a l S ur d s
Appro x i m a t e Va l ues o f S ur d s
I rr a ti o n a l E qua ti ons in On e Un kn o wn
S ur d
M
.
.
.
.
Q U AD R A T I C
QU AT I O N S
T h e G r a phi ca l M e t h o d o f S o l uti o n
S o l v i n g Q ua d r a ti cs b y F a c t or i n g
S qu ar e R o o t M e th od o f S o l uti o n
T o C o m pl e t e t h e S q ua r e Wh en a is 1
T o C o m p l e t e t h e S quar e wh e n a is n o t 1
S ol uti o n b y Fo r m ul a
T o F i n d A p pr o x i m a t e V a l u es o f R o o t s o f Q ua d r a ti c
E q ua ti o n s
E q ua ti o n s In Q u ad r a ti c F or m
G r aph i c a l S o luti on o f Q uadr a ti c s
C h ar a c t er o f th e R o o t s o f Q ua d r a ti c E q ua ti o ns
T o F o r m a Q u ad r a ti c E q ua ti o n w it h G i v e n R oo t s
F a c tor i n g b y P ri n c i pl es o f Q uad ra ti cs
Pr o bl e m s in Q ua dr a ti c E q ua tion s
E
.
.
.
~
.
.
S I M U LT A N E O U S S Y S TE M S S O LV E D BY Q U AD R A TI CS
SUM M A R Y O F D E FI N I T I O N S
I NDE X
.
E
IN T R O D UC T ION
AS O NS
RE
F O R S TUD YI N G
AL G E B RA
ould b e c o m e c o n vi n c e d a s e a rl y
t h a t t h e r e are s t r o n g re a s o n s w h y h e S h o u ld
a s po s s i b l e
l ea r n alge b r a T h e ki n d o f wo rk t h e pu pil w ill d o a n d h is
t o him d e p e n d s o
c o n s e q ue n t se n se o f it s a c t u a l v a l ue
l a rg e l y o n t h e a ppr o v a l h e gives t o it s s t ud y t h a t it s ee m s
wo rt h w h il e e v e n b e f o r e b e gi nn i n g it t o c o n s id e r t h e rea s o n s
fo r st u d yi n g a lg e b r a
T h e h ig h
sc h oo
l
pu pil
sh
,
,
.
,
,
,
.
ALL T A S KS REGARDED A S P RO BL E MS
T O B E S O LVED
n
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w o r k o f life h e w ill s oo n l ea r n t h a t t h e b es t w a y t o d e a l w i t h
t h e q u es t i o n s a n d diffi c u l t i es t h a t a r i s e is t o re g a r d t h e m a s
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t o l ea r n h is l ess o n s t o w r i t e a c o m po s i t i o n t o d o a n e x pe ri
m e n t t o d e b a t e a q ues t i o n t o win in a c o n tes t t o d o a n y
t h i ng t h e fi rs t fe w t i m es a r e f a m ili a r pr obl ems t o t h e h ig h
s c h o ol pu pil
H o w to ea r n m o r e a n d w as te l es s t o m a n a g e a ff a i r s m o re
e c o n o m i c a ll y t o g e t m o r e o u t o f a n d t o p u t m o re i n t o lif e
h o w t o c o n d u c t h o u se h o ld a ff ai rs m o r e e c o n o m i c a ll y t o
l ea r n to a ppre c i ate a n d t o u n d e rs t a n d m o r e o f t h e re a ll y
go o d a n d t r ue in b oo ks a n d in life a re a c t ua l pr o bl em s t o
e v e ry r ig h t m i n d e d m a n a n d w o m a n
R igh t livi n g is li tt l e
m o r e t h a n s o lvi n g a c o nt i n uo u s c h a i n o f pro b l e m s
Th e
“
H ow fa r c a n
q u e s t i o n fo r e v e r y yo u n g pe r s o n s h ould b e
I a dv a n c e in t h e pro bl em b o o k o f t h e g re at w o r ld b e fo re t h e
”
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“E LE M E N TARY ALG E BRA
I MP OR TA N T T O A CQ UI RE P O WER A N D SKILL I N PRO BLE M W ORK
C l ea rl y t h e n , it
is
g r e at i m po rt a n c e t o l ear n w h a t it
m e a n s t o s o lv e a pr o b l e m a n d t o a c q u i r e w h a t e v e r Skill w e
m ay in t h e a r t o f pro bl e m so lvi n g a n d t h i s t o o n o t m e r e l y
b e c a u s e o u r t e a c h e r o r o u r par e n t s w a n t u s t o d o s o b ut
for o u r o w n s a k es p ur e l y
In a n e s pe c i a l s e n s e a lg e b ra
t e a c h es t h e ta cti cs a n d t h e techn ique o f pr o b l e m s o lvi n g
Th e t o o l s b y w h i c h b o t h t h e s c i e n c e a n d t h e art a re w r o u g h t
o ut a r e t h e a lg e b r a i c n u m b e r a n d t h e a lg e b ra i c e q u a t i o n
T o b e Wi t h o u t t h e a b ili t y t o u s e the e q u a t i o n s killf u ll y is t o
b e w i t h o u t t h e a b ili t y t o d o m u c h pro b l e m t h i n ki n g
Po w e r
t o u s e t h e e q u a t i o n w i t h s k ill a n d i n s ig h t is t h e m a i n pa rt
o f t h e e q u i pm e n t o f a n a c c u r a t e t h i n ke r
a n d a lg e b r a is
e s s e n t i a ll y t h e s c i e n c e a n d t h e a rt o f t h e e q u at io n
of
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.
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T W O REA S O N S WH Y ALGE B RA SH O UL D B E S T UD IE D B Y ALL
v e ry pe r s o n w h o h a s h is w a y t o m a ke in t h e wo r ld m u s t
s u c c e e d o r f a il in h is s t ru ggl e w i t h life s pr o b l e m s
The
w o r ld s pr o b l e m s a r e h ar d e r t h a n t h o s e o f a lg e b r a b ut t h e
b e st w a y t o a c q uIre a bili t y t o g r a pp l e w i t h h a r d e r pr o b l e m s
is fi rst t o g e t so m e s kill w i t h e as i er o n e s
A lge b r a st a rt s
w i t h c o m pa r a t iv e l y s i m pl e di ffi c u l t i e s t h a t g ra d u a ll y i n c r ea s e
in c o m pl exi t y a s o n e s s kill g ro w s t o diffi c ul t i es g re a t
e n o ug h t o t a x t h e p o w ers o f e v e n t h e b r ig h t es t pu pil s
F o r t w o reas o n s a t l e a st t h e pro bl em s o lvi n g o f a lg e bra
is e a s i e r t h a n t h a t o f e v e r y d a y life
In the fi rs t p l a ce t h e l a n gu ag e o f a lg e b r a m a k es reas o n i n g
e a si e r
t h a n d o es a n y o t h er l a n gu a g e m e n h a v e ye t
d e vi se d
B e fo r e a lg e br a i c l a n gu a g e w a s i n v e nt e d t h e a n c i e nt
m a t h e m at i c i a n s u se d o r di n a ry w o r d s a n d sen t e nc e s in t h e
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w a s l ar g el y s e n t e n c e m a k i ng is n o w c a ll e d r hetori ca l a lg e br a
It n e v er a m o u n t e d t o m u c h as a pro bl em s olvi n g i n st r u m e nt
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RE A S O N S F OR
S T UD Y IN
M at h e m at i c i a n s l at er m a d e us e o f a bb re v i a t e d w o r d s
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fo rm e d w h at is n o w c a ll e d a bb rev iat io n al o r s yn copa ted
T h i s w as a r e a l a dva n c e a n d a v er y fa i r s ort o f
a lg e b r a
and
a lg e b r a n ow d e v e l o pe d a s t h e n ee d fo r it c a m e a l o n g
B ut it w a s st ill c u m b e r so m e
m e n g r e w i n t e r est e d in it
a n d m e n co n t i n u e d t ryi n g t o i m pr o v e it in t h i s w a y a n d
th a t u n t il fi n a ll y a ft e r m a n y c e n t u r i es t h e y h it u po n t h e
m o d e r n f o r m o f w r i t i n g a lg eb r a i c n u m b e r s a n d r e l a t i o n s
F r o m t h i s t im e fo rt h s ymbol i c a lg e b r a a s w e n o w k n o w it
T h e a d v a n c e in m a t h e
s t e p b y s t e p b u t r a pidl y g r e w u p
m at i e s a n d m a t h e m a t i c a l s c i e n c e t h a t s o o n f o ll o w e d is
T h us t h e h i st o r y o f m a t h e m at i cs S h o w s
a l m o s t i n c r e di b l e
t w o t h i n g s v iz
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in m a thema ti ca l thou g ht depen d s g r ea tl y
l a n gu a g e em p l oyed , a n d
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v e ry c ivili z e d r a c e u ses t h i s l a n gu a ge t o d a y O f a l l
e x i s t i n g l a n gu a g es o f t h e w o r ld it is b e s t e n t i t l e d t o b e c a ll e d
t h e u n iv e r sa l l a n g u a g e o f m a n
In the s econd p l a ce a lg e b r a i c pr o b l e m s h a v e d e fi n i t e
a n s w e r s so t h a t t h e b e gi nn er m ay a l w a ys h a v e a c o m pl e t e
h is t h i n ki n g d ur i n g t h e a ppre n t i c es h i p pe r i o d
c heck
on
w h il e h e is n e c essa r il y s o m e w h a t do ub t f ul a b o ut it s r e li a
O n t h e o t h e r h a n d t h e pr ob l e m s of life h a v e n o
bil it y
a n s w e r s o r t h e a n s w ers a r e o f t h e g e n era l n a t u re o f s u c c ess
o r f a il ur e in o n e s e n t er pr i s e s
Wi t h t h e l at t er t h e r e is n o
c h a n c e t o g o b a c k a n d c o rr e c t e r r o r s b e f o r e t h e e rr o r s h a v e
re s u l t e d f a t a ll y
T h i s is a s t r o n g r ea s on wh y a lge br a is a
go o d ea r l y t r a i n i n g in pro b l e m st u d y a n d pro b l e m s t rat e gy
We c a n d o h ar d t h i n g s b y vi rt u e o f t h e power a n d s kill
a c q u i re d in d o i n g s i m il ar b ut e a s i er t h i n g s
E
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“E LE M E N TARY ALG E BRA
ALGE B RA N O T C REAT E D FOR A M ERE SCH OOL D I SC I P LI N E
4
It t h u s
t h at
lg e b r a wa s n ot c r ea t e d a s p u pil s
a r e s o m e t i m e s pr o n e t o t h i n k m e r e l y a s a s e v e r e di s c i pli n e
fo r sc h o o l b o ys a n d gi r l s
A lg e b ra w as fo rm e d t h ro u g h
t h e u n i t e d e ff o rt s o f a l o n g s u c c e s s i o n o f s c i e n t ifi c m e n t o
d e vi se a t o o l a n d t e c h n i q ue fo r s o lvi n g t h e pr o b l e m s o f
—
sc i e n c e t h a t a r o s e f r o m a g e t o a g e
l
r
b
m
h
n
o
e
s
t
a
t
o
p
kn o w n s u b j e c t o r d e v i c e c o u ld c o n q u e r
It w a s c r ea t e d a s a
n e c e s s i t y t o W in e v e n t h e li t t l e s c i e n t ifi c k n o w l e dg e t h e r a c e
A ft e r a lg e b r a h a d r e v e a l e d t h e
a c qui re d fr o m a g e t o a g e
d es i re d so l ut i o n s s o m et i m es a n o t h e r m a t h e m a t i c a l s ub j e c t
w as f o u n d c a pa b l e of yi e ldi n g a s o l ut i o n a l s o b ut a lg e b ra
wa s u s u a ll y t h e pi o n ee r a n d it is o n l y r a r e l y t h a t a n y s c i e n c e
fu rn i sh e s e a s i e r a n d m o r e r e li a b l e w ays of s o l v i n g pr o b l e m s
T o b e ig n o r a n t o f a lg eb r a is t o b e d e pr iv e d
t h a n a lg e b r a
of t h e m o st e ff e c t iv e pr o b l e m s o l v i n g e n gi ne ye t i n v e n t e d
Wh y n ot se i z e t h e o p po rt un i t y t o a c qui r e s om e m as t e ry
?
r
v
r
h
i
s
e
f
u
l
t
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l
T h e b e gi nn i n g s o f t h e s ub j e c t a r e
o e t
po w
ea s il y w i t h i n t h e c o m pr e h e n s i o n o f t h e t w e lv e ye a r o l d b o y
o r gi r l
a pp ea r s
a
,
,
.
.
.
,
,
,
.
-
.
-
.
ALGE B RA I S F UN D A M E N T A L
T O ALL M AT HE M ATI C AL SC IE NC E S
lg e b r a is t h at it is
fun d a m e n t a ll y n ec e ss a ry t o so m a n y fi e ld s o f h ig h e r s c ie n
A sid e fr o m a litt l e e l e m ent a ry g eo m e t ry al m ost
t ifi c w o r k
n o m at h e m a t i c s b e y o n d t h e s i m pl est a r i t h m e t i c is p o s s i b l e
T o at t e m pt t o g e t o n in
w i t h o ut a kn o w l e dg e o f a lg e b ra
“
m at h e m a t i cs w i t h o u t a lg e b ra is v e r il y t o t ry t o w a l k w i t h ou t
”
feet
P e r h aps t h e m o s t wid e l y u se fu l m at h e m at i c a l
s u b j e c t w i t h i n r e ac h o f h ig h s c h o o l s t u d e n t s is t r ig o n o m e t ry
T r ig o n o m e t ry is t h e sc i e n c e of t h e t ri a n gl e a n d is m a d e up
v e ry l a r g e l y o f c o m pa c t pra c t i c a l r ul es o r l a w s express e d in
The
t h e l a n g u a g e o f a lg e b ra i c f o r m ul as a n d e q ua t io n s
t r a n s fo r m at i o n s o f t h es e f o rm ul a s t h a t l e a d t o t h e m o s t
O ne o f t h e s t r o n g est
r e aso n s
fo r st u d yi n g
a
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re c e n t
ly :
0
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L E M E N TARY ALG E BRA
t h at
I kne w
eno
ug h
a
lg e b r a
to
ena
bl e m e
to
u n d er s t a n d t h e fo rm u l a s o f Ke n t s E n gin eer s P ocket B ook
t o b e a b l e t o m a ke pr o pe r s ub s t i t ut i o n s in t h es e f orm ul a s
”
a n d t o k n o w t h e m e a n i n g o f t h e r es u l t s "
It i s t h e w e a k
n e s s o f t h e i r p r o b l e m so l v i n g a b ili t y t h a t m e n o f pr a c t i c a l
a ff a i r s se e m m o st t o r e g re t
T h e se m en o ft en c o n t e n d t h a t
m u c h o f w h a t t h e y h a d t o s t u d y in t h e h ig h sc h oo l h a s
b ee n o f li t t l e o r n o u s e t o t h e m b ut t h a t t h e y c o u ld n o t h av e
b ee n giv e n t oo m u c h m at h e m at i c s fo r t h e w o r k t h e y h a v e
T h ey t e ll u s t h e l e a d e r s t o d ay a r e n o t t h e
si n c e h a d t o d o
g r e a t o ra t o rs a n d c h a r m i n g t a l ke r s o f a g e n er at i o n a g o b ut
It is t h e l a t t er t h e y t e ll us t h a t
t h e m a them a tized t h i n k e r s
a re c a rry i n g o ff t h e pr i z e s o f t h i s c o m m e r c i a l a n d i n d u s
t ri al ag e
Le t bo ys a n d gi r l s w h o h a v e n o t ye t l o st t h e o ppo rt u ni t y
t o pr o fi t b y s c h oo l w o r k in m at h e m at i cs m a ke t h i s s t udy
b y t a ki n g up t h e
a s pro fi t a bl e as p o s s i b l e t o t h e m s e lv es
fu n d a m en t a l s u b j e c t o f a lg eb ra w i t h e n e r gy a n d d e t e rm i n a
Di sm i s s t h e id ea if yo u h o ld it t h at yo u a re st ud yi n g
t i on
E m brac e
t h i s s u b j e c t a s a f a v o r t o yo u r t e a c h er o r pa r e n t s
a n d c h e r i sh t h e t r u e id e a t h a t y o u a r e s t u d yi n g it fo r yo u r
o wn b e n e fi t t o r a i s e y o u r o w n e ffi c i e n c y a n d t h a t yo u a r e
T h e c h a n c es
o n l y c h e a t i n g yo u rs e lf if yo u d o p o o r w o r k
a r e m a n y t o o n e t h at t h e t a sk s o f a f t e r lif e w ill b e f o u n d t o
m ak e s t ro n g er d e m a n d s o n yo u r pro b l e m so lvi n g ab ili t y t h a n
D o n o t f o r g e t t h at a lg e b r a is in a pe culi a r
a lg e b r a re q ui r es
se n s e t h e s u b j e c t w h i c h c a n b e s t d e v e l o p a n d pe r f e c t a b ili t y
T h er e f o r e t a ke up t h e w o rk vig o ro u sl y t h e
o f t h i s t y pe
fi rs t d a y n e v er r e l ax i n g yo u r e ffo rt s t o m ast e r t h e s u b j e c t
u nt il t h e l a s t l ess o n is l e ar ne d
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E LE M E N TA R Y A LG E B RA
F IR S T
HA
L F - YE
AR
CH AP TE R I
N OTATI O N IN
AL G E B RA
THE
.
Q UATI O N
E
N OTATIO N
1
it s
T he po w e r o f a lg e b ra is d u e m a i n l y t o it s l a ngu a g e a n d
Yo u h a v e a l re a d y m a d e so m e s t a rt wi t h t h i s
s ym b o l s
.
.
l a n gu a g e fo r e v e ryt h i n g yo u h a v e c o r r e c t l y l e a r n e d a b o ut
t h e l a n g u a g e a n d t h e s ym b o l s o f a r i t h m e t i c h o ld s g o o d a l s o
B ut b e c a u s e a lg e b r a is a s o r t o f gene r a l a r i t h
in a lg e b r a
m et i c it a dd s so m et h i n g t o t h e l a n g ua g e a n d s ym b o l s o f
a r i t h m et i c a n d e m p o ys t h e m m o r e gen er a l l y t h a n a r i t h
l
m etic d oes
P e rh a ps t h e m o s t i m po r t a n t t h i n g s fo r t h e
b e gi n n e r t o ke e p in m i n d f r o m t h e o ut se t are t h a t w h a t t h e
l a ng u a g e t a l ks a b o u t a n d w h a t t he a lge b ra i c
a lg e b ra i c
T h o ug h
s ym b o l s s t a n d fo r a r e n u m ber s a n d n um b e r r e l a ti o n s
t h e b o o k o r t h e t e a c h e r m a y t a l k a b o u t a lg e b r a i c e x p r e s s i o n s
o r qua n tities o r m o n o m i a l s o r po l y n o m i a l s it is i m port a n t
,
'
.
,
.
.
,
,
,
to
r e m e m b er
t hat
all
,
t h e se t e rm s ,
and
m any
o t h e rs ,
l y o t h e r n a m e s fo r n umb e r s
Alg eb ra li ke a ri t h m et i c t re a t s o f n u m b er s
m ul t i pli es a n d d ivid es n um b ers r a i s es
s ub t r a c t s
po w e r s a n d e x t r a c t s t h e i r r o o t s
on
a re
.
,
,
,
,
.
No ta tio n is t h e m et h o d
figu res
or
,
It
a
dd s
,
th em t o
.
,
2
.
l e t t ers
.
of
i g
e xpr e s s n
n
u m b ers
by
E
8
LE M E N TARY ALG E BRA
i t h m e t i c n u m b ers a re
c h a r a c t e r s c a ll e d d igits o r fi gu re s
In
ar
Th e Num b e r
re pr ese n t e
of
Th e wh o l e
.
d by t h e
Th us
.
d by
A ra bi c
ten
,
400
45 3
p art s
r e p r es e n t e
,
se
t he
n
u m b e r is t h e
v e r a l digi t s
su m o f
the
.
R e pr e s e n tin g N u m b e r s In a lg eb r a n um b ers are re pre
s e n t e d b y fig u r e s b y l e tt e r s a n d b y a c o m b i n a t i o n o f b o t h
.
,
,
3
Pr o d u c ts
.
W h e n l e tt er s
.
g e t h e r in a lg e b r a
T h us
If
a
t h os e
,
and
figu r es
t h e i r p r odu ct is i n di c a t e d
wr i t t e n t o
are
4a m e a n s 4
u m b er is
ti m e s
a , an d
7a b m ea n s
t h e pr o du c t of t w o o r m o r e
*
factor s o f t h e pro du ct
n
u m b ers
,
.
u m b e r s r epr e se n t e d b y t h e digi t a n d
t h e r e f o r e f ac t o r s o f t h e w h o l e n um b e r
l et t e rs in
the
n
ar e
.
.
n u m b e r s a re
Th e
7a b
n
,
,
,
7a b
.
T h e e x pr ess i o n , 5x yar d s ,
Us ing Al g e b rai c Lan gua g e
m e a n s 5 t i m e s t h e n u m b er o f ya r d s r e pr e s e n t e d by x
4
.
.
.
E x er c i s e 1
1
.
W h at is m ea n t b y t h e
3 n m il es ?
2
.
s en t ?
3
.
If
n
sq
4
.
c e nt s
5
8y
*
.
io n
,
91: q u a rt s ?
1 2g
c e nt s ?
u a r e f e et ?
i
r e p r ese n t s a c e r t a n n
um b e r
,
what
d oes
4n
re pre
9n ? bn ?
If
A
t h e pr i c e
1 2 x?
3: r e pr ese n t s
r e pr e s e n t ?
5x
e x pr ess
boy
did h e
8x?
b o u gh t
8
o r an g es
T h e w or d
at
a
b
ya r
d
of s
il k
i
c e n t s a p ec e
.
,
w h at
d oes
How m a n y
pa y fo r t h e m ?
W h a t is m ea n t b y t h e
bu sh e l s ?
of
sq
u are
ro
.
,
63: ya r d s ? 4 a
d o ll a rs ?
ds ?
fac tor m e a n s m a ker
ma ker s by mul ti pl ic atio n
sio n
e x pr es
.
Th e
f a c t ors
of a
num
b er
are
it s
N OTATIO N
6
If
.
in 8
e arn
7
If
.
d o es
a
HO
.
r ec t a n
a
d ays ?
In
square
r e pr ese n t ?
4x
8
m an w o r k s fo r
a
Wm
any
gl e
a:
a
d ay , h o w m u c h
d a ys ?
a:
is
d o ll a rs
n
9
i n c h es o n e a c h s id e w h a t
Wh a t d oes xx r e pr ese n t ?
3:
,
i n c h e s a r e t h e r e in
l o n g a n d y i n c h es
s q ua r e
i n c h es
w id e ?
9
If yo u
.
a re
c
wh o is t h r ee t i m es
10
.
5
.
n
.
i
c e rt a n n
a
an d
.
The
s
in
a
lg e b ra
b y pl a c i n g
I n d i catin g Divi s io n
t he
wr i t i n g
f ra c t i o n
is yo ur
fat h e r
divid e n d
o
a
.
ver
umber
,
w h at
r e pre se n t s
6
num be r ?
of a
ign s
divi si o nm e a n
In d ic atin g M ul ti pl i catio n
.
c at ed
7
S ign s
Al g e brai c
ol d
as o l d ?
r e pre e n t s
m u l t i pli c a t i o n ,
6
t o d ay , h ow
s
t h e n u m b e r? m t i m e s t h e
.
t i m es
If
ol d
ye a rs
.
t he
ddi t i o n
sam e as
,
in
Mul t i pli c at i o n
d o t be t w ee n t h e
i
s ub t r a c t o n ,
ar
it h m eti c
.
is
oft e n i n di
fa c t o rs T h us
.
,
Divi s i o n is of t e n i n di c a t e d b y
t h e divi so r in t h e fo rm o f a
.
E xe r c i s e 2
1
Of
.
33,
2
.
ag o ?
3
.
I n di c a t e t h e s um o f 8 a n d 7
O f 2a 3 b a nd 1 2
y an d z
.
,
A
m a n is
.
fee t ?
n
y e a rs
E ig ht e e n ye a rs
If
a n ot h e r ,
4
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ol d
t o d ay
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HO W
ol d
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a a nd
b
.
w as
he 7 ye ars
ag o ?
m a n h as p s h ee p in o n e fi e ld
h o w m a n y h a s h e in b o t h fi e l ds ?
Wh at is m ea n t b y t h e
d ays ?
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a
n+8
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73:
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sh e ep
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a
r e pr ese n t s
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wh at wil l fr e pr ese n t
,
su
m of t w o
sm a
I n di c a t e in
b an d y
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12
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c ent s
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for
can
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a:
r e p r e se n t s
fe e t lo n g
an d
t he
y
u m b e r of
w id e ?
n
fe et
u m b e r of
fi e ld L r o d s l o n g a n d W r od s Wid e ?
17
.
18
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Wh a t w ill d e n o t e t h e
A
m an
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ld
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at n
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How m u c h did t h e h o r se
19
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and
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r
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a re ct an
g a i ne d
c
io n
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gul a r
doll ars
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o st him ?
A m an b o ught 51: sh e e p at m doll ars a
doll a rs
x
d b c en t s a n d t h en
c e n t s did h e h a v e l e f t ?
A
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How m any
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an d
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b oy h as m q u a rt e rs a n d n di m e s
r e pr e se n t s t h e n u m b e r o f c e n t s h e h as ?
15
of
a nd
.
di ffer e n c e b e t wee n
W h e n b is g re at e r t h a n
A b oy h a d
l a rg er n u m b e r
rod s l o n g
x
t w o w a ys t h e pr o du c t
and b
O f n , x, and 3
t he
.
n
num b er ?
b o ug h t x c o ws at $ 3 5 a pi e c e
m o n ey did h e h a v e a t fi rs t ?
g reat e r t h a n b
14
the
a: a n d
m an
Ho w m u c h
13
b e rs is
g u l a r pi ec e of l a n d is
d oes 2113+ 2 y r e prese nt ?
w id e , w h at
.
ll er
n um
r e ct an
a
r e pr es ent
ll er e v en n u m b er ?
is y, w h at is t h e
8
w h at w ill
H ow m a n y s q u a r e i n c h es a r e t h er e in
r e c t a n gl e m ya r d s l o n g a n d n i n c h es w id e ?
a
r
n u m b er ,
n um b er ?
r e pr e se n t s
n e xt s m a
th e
odd
u m b er o f s q u a r e
r e c t a n gl e a: f ee t b y y i n c h es ?
W h at
i n c h es in
7
n r e pr e s e n t s a n
l arg er o d d
n e xt
6
LE M E N TARY ALG E BRA
W h a t did a l l
c ost
hea d
h im ?
and
y
l am b s
E Q UATIO N
T HE
20
A
.
th e m
at
b o y b o u gh t
y
i
l
n
c e n t s a p ec e
at
a pp e s
cents
90
I f he ga ine d
.
11
and
i
ap ec e
w h a t w as his
,
so
ld
gai n ?
-
Si n c e
ti m es
5
a n y n u m b er
+4
ti m es
If
num b er ,
a n y n um
9b — 3 b
6b ,
b er — 3 tim es
what
23
A
.
t wi c e
24
If
.
and a
‘
25
m a ny
as
a se t of
h o rse 5 3;
If
.
as
a
u m b e rs
A
c o s t s 3:
ha s
ho w
E
Si gn of e qual i ty is
b e t wee n w h i c h it is pl a c e d are
i
T he
91:
an d
5 is t h e
.
s a m e as
the
e qua t
.
d o ll ars
,
l e ft
t he
of
r
the
igh t
.
s
ign
and
C has
i ge
a c arr a
3 a:
d o ll a rs
m
su
,
a
i n di c a t e s t h a t t h e
e q ua l
of
n u m b e rs
.
tha t
2x
and
expr es si on
9
f
o
t h e d i ff e r e n
c e b et ween
.
equal i ty
b e t wee n t wo
,
The fi rst m emb e r
.
n um b e r s ?
th e tw o
It
8x + 6 = 3x + 3 6
10
is 3 s ,
a l l c ost ?
m ea n s
io n is a n
T h us
e q u a l n u m b e rs
9 An
ll e r
Q UATIO N
9x
e x pres s o n
sm a
d o ll a rs a n d s pe n d s an d o ll a rs fo r
m a ny d o ll a rs h a s h e l e f t ?
The
.
the
5y
T HE
8
and
.
,
l
is 8 s
,
d o ll a rs W h at d o
o f c o t h es ,
it
su
n
.
h arn es s
m an
8 b — 3 b — b = 4b
B h a s t w i c e as m a ny a s A
Ho w m a ny h a v e a l l ?
and B
n s h ee p
h as
fo r
b er = 6 ti m es t h a t
5x
differ e n c e b e t w ee n
t he
r e pr e s e n t s
s a m e num
th e
d o ll ars
4 30
l
I f t h e l a rg e r o f t w o
.
and
a so
l oa — 3 d = 7 a
22
4n + 2 n + n = 7 n
ge t s a: d o ll a r s fo r c o rn
m u c h d o e s h e g e t fo r b o t h ?
ti m es
9
4a + a = 5 d
m an
a
w h e at , h ow
Si n c e
ti m e s t h a t
9
a so
9x + 4x = 1 3x
.
b er
l
n um b e r ,
21
s a m e n um
the
.
Th e
4n + 2 n = 5 4
u at i o n is t he n u m b er o n t h e
s e c o n d m e mb e r is t h e n u m b e r o n
of an e q
E
12
LE M E N TARY ALG E BRA
u at i o n e x pr es ses ba l a n ce of v a l ue s j u st as t h e
h o r i z o n t a l pos i t i o n o f t h e b a r o f t h e
b a l a n c es s h o ws ba l a n ce of w eig hts
To put
betw ee n t wo n u m b er
ex pr e ss io n s is t o s a y t h a t
if t h e
n u m b e r s w e r e w e ig h t s a n d t h e e x
n
r
ss
s in t h e t w o m e m b ers w e r e
e
i
o
p
r e pr e s e n t e d b y pro per w e ig h t s o n e in e a c h p a n t h e b a l a n c e
b ar w o uld st a n d h o r i z o nt a l
~
11
Th e
.
eq
.
.
,
,
.
12
n um
Th e
.
b er
or
valu e o f
n u m b e rs
l e t t e r in
a ny
w h i c h it
of
3
,
lb
eac h
.
of
L ea vi n g
w e ig h t s
.
ea c h
in t h e
in
i
e x pr e ss o n
is t h e
.
o ne
pan
are
bal a nc ed
pa n , w e m a y
o t h er
s ay
the
id e o n t h e
n
n
m
n
h
a
d
r
vi
g
r
e
a
s
t
e
o
e
st
p
t h e b a r w ill rem a i n h o r i z o n t a l
:
T ha t
o r w e m a y say a
6 lb
a
is t h e b a r c a n b e h o r i z o n t a l
w h e n t h e pa n s a re l o a d e d o n e
w i t h 3 33 lb a n d t h e o t h e r w i t h
18
3x
1 8 lb o n l y if x = 6 l b
B ut w i t h o ut t r o ubli n g w i t h t h e b a l a n c e b y m e r e l y a ppl y
in g t h e divi s i o n p r i n c i pl e t h a t e q u a l n u m b e r s divid e d b y
18
t h e sa m e n u m b e r giv e e q u a l n u m b ers t o t h e e q u a t i o n 3 x
In t h i s w a y a lg e b ra m a kes r ea s on in g t a k e t h e
we find x = 6
f
h
w
i
i
n
a
a
ra
t
u
s
l
ac
e
o
t
e
e
h
p
g
g pp
on
each
um ber
r e pr ese n t s
I f 3 u n k n o w n w e ig h t s o f a: l b
b y 6 w e ig h t s
a n
s
,
,
,
.
,
xi s
,
1 8 LB
.
.
:
.
.
,
,
,
.
.
In t h e
e
q ua l
on
ly
In t h e
e
q ual
on
e
e
ly
q ua ti o n
w h en
i c
s n e
t
3: r e pres e n s
q ua ti o n
wh en
,
3x = 1 8 ,
,
6
t
ti m es
x,
32:
and
18
a re
35
ar e
.
3 n + 2u = 3 5,
n r e pr es e n s
33: m ea n s 3
7;
i c 3 n + 2u is
o r t h i n k t h us
s n e
5 n , 3 n + 2n
and
‘
E
14
LE M E N TARY ALG E BRA
lg e b ra wh e n t h e r e a so n fo r a c h a n g e in a n e q u at i o n is
a s k e d t h e pu p il is e x pe c t e d t o q u o t e o r t o c i t e a n a x i o m t h a t
j us t ifi e s t h e c h a n g e
In
a
,
16
G iv e t h e
.
re aso n
fo r t h e
l
i
c on c us o n
in
e a ch
of
t he
fo ll o w i n g
x
=7
and
=4
y
x+y=
h
n
t
e
;
—
d=l 5
c
a n d d = 6 ; t h en
a
=r
e
= 2n
a n d b = 3 ; t h en
11
—
a
—
=
b x 3
= 20n
h
n
l
0
t
e
0
;
d = 32 ; th en
x
=7
and
y
=7
n
h
t
e
;
x
=
y
3y = 2 7 ; t h e n y = 9
If
n
= 5 t h en 8 n = 40
;
‘
If m = 9
10
.
a nd n
I f m = 28
=4
and n
= 36
h
n
m
n
t
e
;
=4
h
n
e
t
;
fl=7
n
11
.
12
.
13
.
H it
t h en
5 ; t hen
If
a
x
=8
= 80
t h en m = 7
I f mn = 7 n ;
E x e rc i s e 3
1
.
—
So lv e 8 x
—
3 x + 2x x
—
8r
30
3x +
.
—
2x
m= 3 0
6x
B y the
divi si o n
C h e c ki n g
,
i
x
A l w a ys ch e c k
,
5
40
30
or
f o un d
ax om ,
30
30
t es t t h e v a lue o f t h e un k no wn n u m b e r a ft e r it is
b y sub s tit uti n g it fo r t h e un kn o wn n um b er in t h e gi v en e q ua ti o n
or
.
T HE
E Q UATIO N
15
3
—
—
5n 2n + 4 n n = 4 8
a nd c hec k
So lv e
2
—
6x 2x + 3 x =
.
4
—
5 s + 6s
.
49
3 s = 48
5
.
—
—
2a + a = 45
9a 3 a
.
.
—
—
=
4
2
2
4
e 8y
y
y
7
.
8
—
7x + 2x 3x =
.
S olvin g
54
9
8 b+ 7 b
.
—
4n
.
-
~
—
b 4 b = 55
—
3 n + 6u n
= 72
m
i
h
f
n
n
ro
b
l
e
s
t
e
r
c
es
s
o
f
i
di
g
h
v
l
e
f
o
e
a
u
s
o
t
p
p
t h e u n kn o w n n u m b e rs i n v o lv e d in t h e pr o b l e m
In a r i t h m e t i c t h e u n k n o w n n u m b e rs a r e fo u n d b y o n e or
m o re o f t h e f un d a m e nt a l pr o c esses
In a lg e b r a t h e u n k n o w n n u m b e rs a r e r e p rese n t e d b y l e t t ers
a n d t h e i r v a lu e s a r e f o u n d b y t h e u se o f e q u a t i o n s
So lvi n g a pro b l e m in alg e b r a i n v o lv es t h r e e s t e ps : n ota tion
17
.
a
.
.
.
,
s ta temen t, s o l v in g a n equ a tion
E xer c i s e 4
1
.
s um o f
Th e
two
is 6 t i m es t h e s m a ll e r
s
t
r e p re s e n s
a nd
su
m
25 2
of
s
of
t he
T he
pr o b l e
n
= the
ar e
t h e t wo
S ta te m e nt ,
Th e
two
u n kn o w n
n um
s ta te m e n t
min o ne
the
or
l arge r
n um b e r
e qua ti n ,
n u m b er ;
i
i
u at i o ns
63
ea
ch
of
wh i c h
.
in
t h e pr o b l e m
3
.
e xpress o ns ,
e x pr ess o n o f
eq
l a rg e r n u m b e r
.
n um b e r
i
is t h e
o
t he
252
6s
73
S olvin g th e
ll e r
re pr es e n t a t o n
b e rs in
m o re
sma
n um b e rs
8
ota tion is
P r ob l e m s
,
h e n 6s = t h e
+ 6s
the
.
S o l v i ng
u m b ers is 2 5 2 a n d
Fi n d t h e n u m b e rs
Le t
N o ta ti o n ,
H e n ce
n
.
.
25 2
36
2 16
a
lg e b ra i c
s ym b o s
l
.
the
con
di t i o ns
of
t he
LE M E N TARY ALG E BRA
E
6
To check,
su
s ta tem en t
b s t it u t e in t h e
2 52 ,
E
c as e
t t
s u b s tit u t e
th e
v en
s a em e n
t it s e lf
Thus
.
or
252
m a y b e wr o n g
.
,
2 52
t es t w h e t h er t h i s
pr o b l e m it s e lf
To
is
c o n d iti o n s o f t h e
2 T h e s u m o f t w o nu m b e r s is 84 6 a n d t h e l a r g er n u m b e r is
Fi n d t h e n u m b e r s
t i m e s t h e s m a ll e r
th e
,
in t h e
.
,
.
8
.
Se v e n t i m e s
.
a c e r t a n n u m b er
l
Is 6 t i m es t h e
p I
u m b er
Fi n d t h e n u m b er
m i n u s 8 t i m e s t h e n u m b e r eq u a l s 1 7 5
To obt a i n t h e
1 8 O b tain in g S ta te m e n ts of P rob l e m s
s t a t e m e n t in a pr o b l e m is t o t r a n s l a t e t h e c o n di t i o n s o f t h e
n
n
m
a
n
u
a
t
l
i
e
i
b
e
t
o
o
r
o
q
p
3
.
i
n
.
.
.
.
.
DI RE C TI O NS FOR M AKI N G S TAT E M E N T S
1
Let
.
n u m ber s
2
3
ber
4
fi
es
to be
foun d
F rom the
.
s a me
the
a n y a p p r opr ia te
a nd
.
S O LVI N G PRO BLE MS
f
r epr es en t on e o
the u n kn own
.
c on di tion s o
f
l etter , the other
Fin d two
.
l e tter
AN D
the pr obl em
u n kn own n u mbers
n u m ber exp r ess ion s
u
a
l
r
i
f
o
l
a
ce
t
e
m
e
m
n
h
g
p
q
,
tha t
i n ter m s
expr ess ,
f
o
.
the
r ep r es en t
a n equ a ti on
s a me n u m
.
S ol v e the equ a tion a n d d etermin e whether the
the con di ti on s of the probl em
r esu l t s a tis
.
E xe rc i s e 5
1
.
On e
n
u m b er is
b et w ee n t h e m is 4 8
N o tatio n ,
5 t i m es
a n o t h er ,
t he
n um b ers
Fi n d
.
Le t
s
= the
sm a
ll e r
an d
the
di ff ere n c e
.
n um b er ;
t h e n 5s = t he
l a rg e r n um b e r
—
H e n c e 5 s s a n d 4 8 a r e t w o n u m b e r e x pr e s s i o n s
w h i c h r e pre s e n t s t h e d ifi er en ee b e t we e n t h e n u m b ers
.
,
.
S ta t e m e n t ,
S ol v in g th is
Ch e ckin g ,
5s
ion ,
e qua t
60
s
48
s
12
and
or
5s
60
,
each of
E Q UATIO N
T HE
2
A
.
t h e ir
3
as
a
l ess
B
Fi n d
’
.
th e
and
s ag e
di ff ere nc e
.
e a rn e
A
fi v e t i m es
d
h ow m u c h
,
as
did
m uc h
B
as
o f a re c t a n
i m et e r
is 2 24
i n c h es
wi
ce as
If B
.
b o t h t o g e th e r
l e ng t h
b e t w ee n
.
8 55 pupil s t h e r e a r e t
H o w m a n y gi rl s a r e t h e r e ?
th an
.
B
,
of
l
The
5
per
A
.
as o l d as
is 7 5 ye a rs
a s ch o o
In
.
t im e s
S ix
ges
b o ys
4
is
17
m any
e a rn e
gi rl s
d $64 8
?
r
n
ea
gl e is 3 t i m es it s wid t h
Fi n d t h e di m e n s i o n s
,
an d
the
.
.
In s o lvi n g pr o b l e m s
L e tte r s R e pr e s e n t N u m b e r s
It m u st n o t
a l w a ys l e t t h e l e t t e r r e p r ese n t so m e n u m ber
re pr e se n t m o n e y b ut a n u m be r o f d o ll a r s o r c e n t s ; n o t t im e
b ut a n u m ber o f d a ys o r h o u rs ; n o t we ig h t b ut a n u m ber o f
n
n
n
s
a
b
n
m
r
d
es
o
t
d
i
t
c
e
u
t
a
u
b
e
o
f
m
il
d
n
r
r
o
u
c
es
o
s
u
s
o
o
;
p
o r o t h e r u ni t s o f m e a s ure
19
.
.
,
.
,
,
,
,
,
.
E xerc i s e 6
1
.
c o st
as
A
h o rse ,
3 t i m es
the
i ge
c arr a
as
i ge
c a rr a
m uc h
,
as
a nd
P r ob l e m s
h a r n ess
c o st
$4 5 0
.
Th e
t h e h a r n ess , t h e h o rse t wi c e
Fi n d t h e
.
—
c o st o f e a c h
i ge
m u ch
c a rr a
as
.
c os t ;
t h en 3 n = t h e
o f d o ll a rs t h e c a rr i a g e c os t ;
an d
6n = t h e n um b e r o f d o ll a rs t h e h or s e c o s t
H e n c e n + 3 n + 6n an d 4 5 0 a re t w o n um ber e xpressi o n s
e ac h o f wh i c h re pr e s e n t s t h e c o s t o f a l l
Le t
n
= t he
ber
n um b e r
n um
of
d o ll ar s t h e h a rn es s
.
,
.
n + 3 n + 6n =
2
.
One
n
um b e r
is 9 t i m es
b et w ee n t h e m is 624
.
Fi n d
4 50
a no t he r ,
t he
n
um b e rs
an d t h e di ff ere n c e
.
E
18
3
A
.
m a ny
4
as
as
A
.
h as t w i c e
C
.
as
m a n y s h ee p
B h as 4 t i m es
I f a l l h a v e 665 h o w m a ny h as B ?
as
C,
as
a nd
as
,
l ot
and
h ou se
m uc h
L E M E N TARY ALG E BRA
the l ot
o st 37 2 50 t h e h o u se
Fi n d t h e c o st o f e a c h
.
c
,
i g
c ost n
4 times
.
c
o
f ho us e
st o
7 2 50
5
t he
6
I f twic e
’
.
s um
.
a nu
is 1 9 2
The
mb e r is a dd e d t o six t i m e s th e s a m e n u m b e r
Fi n d t h e n u m b e r
.
s um o f
the
a
the
.
f a th e r a n d so n is 9 6
t h e i r a g e s is t w i ce t h e
of
g es
,
di ff e re n c e b e t w ee n
W h a t is t h e f at h er s a g e ?
ye a rs ,
’
son s
and
ag e
.
’
7
.
A rec t a ngl e fo rm e d b y pl a c i n g t w o
id e b y s id e h a s a pe ri m
e t e r o f 2 7 0 f ee t
Fi n d t h e s id e o f e a c h
s q u a r e a n d t h e a r e a of t h e r e c t a n gl e
eq
u al
s q u a re s s
.
.
gl e s o f t h e s a m e w id t h a n d
a s w id e a re p l a c e d e n d
t o end the
w
i
f
n
r
m
e
r
m
t
er
h
r
e
s
i
gl
f
d
e
o
t
e
ec
t
a
e
o
p
1 8 0 i n c h es
Fi n d t h e ir di m e ns io n s
8
If
.
tw o
t wi c e
r ec t a n
as
l ong
,
.
.
9
.
On e
di ffer e n c e
10
.
A
t i m es
did
u m b e r is 4 t i m es a no t h er
is 5 7 6
Fi n d t h e n u m b e rs
n
m an
as
ld
a
h o rse
r i ge ?
ca r a
i g e fo r $ 340 re ce ivi ng 3
Ho w m u c h
fo r t h e c a rr i a g e
a n d c a rr a
m u c h fo r t h e h o rs e
h e g e t fo r t h e
4 t i m es t h e i r
.
.
so
,
a nd
as
,
.
E Q UATIO N
T HE
11
s um o f
The
.
5 t i m es t h e s m a ll e r
12
A B
.
,
m a ny
B
as
,
H o w m a ny d o
13
t he ir
14
A
.
a
g e s is
as
t w i ce
ds
7 5 years
as
C t o get he r
and
m i xt ur e
a
B
’
s
and
of
’
A
.
m a ny
.
A
as
3 t i m es
a nd
B
as
t o g e t h er
.
s age
228 b u s h e l s
’
s ag e , an d
the
s um
of
.
of
c orn and
o f c o r n a s o f o at s
.
o at s
t h ere
a re
Ho w m a ny b u s h e l s
t h e re in t h e m i x t u re ?
5 t i m es i t se lf is 65 0
A
di ffere n ce is
o wns
A n um b e r i nc re a se d b y 3 t i m es i t se lf
.
t hei r
o wn ?
b y 3 ti m es B
Fi n d A
.
m an y b us h e l s
o f o at s a r e
15
A
s h ee p
840
ow n
ow ns
s ag e exc e e
In
.
twice
’
C
a nd
is 3 2 2 ,
Fi n d t h e l a rg er n um b e r
.
C
and
,
n um b e rs
two
19
Fi n d
.
t he
n um b e r
,
4 t i m e s i t s e lf ,
a nd
.
l a m b s a t $3 a h e a d a n d t h re e t i m e s
as m a n y s h ee p a t $ 5 a h ea d re c e ivi n g $ 3 24 fo r a l l o f t h e m
Ho w m a ny o f e a c h did h e s ell ?
16
.
m an
so
ld
so m e
.
,
17
r
e
p
i m et e r is
18
A B
.
,
t im es
as
l e n gt h of
Th e
.
as
2 80 yar d s
and
,
m any
C
a c re s
,
for t h e
ot h e r
.
80¢
1 96
.
i nc hes
.
ac r e s
of
the
.
,
a
$ 50 fo r t w o pi ec es
ya r
d fo r
one
pi e c e
H o w m a ny ya r d s w e re in
of s
il k o f
and
e ac h
a
u al
ya r d
eq
pi ec e ?
gl e s w h o s e l e n g t h is 3 t i m es t h e w id t h
e nd
fo rm a re c t a ngl e wh o se peri m e t e r is
Fi nd t h e l engt h o f e a c h re c t a ngl e
T w o e q u al
pl a c e d e n d t o
20
if
i g
n
a
p y
an d
l a n d B o wn s 3
a s A a n d C o wn s h a lf as m a ny a c re s
Ho w m a ny ac r es h av e B an d C ?
600
own
.
v a lu e
,
.
.
A a nd B t o g e t h e r
1 9 A m e r c h a n t p a id
.
gl e is 4 t i m es it s wid t h
Fi nd t h e dim e n si o ns
a re c t a n
re c t an
,
,
.
CH AP TE R I I
N E G AT IVE
AND
P O S I T IVE
N UM B E R S
.
D E FI N I TI O N S
P O S ITIVE
AND N E G ATIV E
N u mb e r s of Ari th m e ti c
T he
N UM B E R S
l y r e l a t i o n o f n u m b ers
c o n s id ere d in a r i t h m e t i c ist h e r e l a t i o n o f s i z e
A b oy st a rt s fr o m 0 t a kes 1 2 st eps t ow a r d t h e right t h e n
H o w fa r is h e t h e n
t ur n s a n d t a k e s 7 s t e ps t o w a r d t h e l eft
f ro m t h e s ta r tin g p l a ce 0 ?
In a ri t h m et i c w e w o u ld so lv e t h i s pr o b l e m t h u s
20
.
.
on
.
,
,
.
,
-
,
7
12
B ut
s u pp o s e a
b ac k he
,
ft e r
t aki n g 1 2
h a d t a ke n 2 0
h e t h e n b e wi t h
re
gar d
s t e ps
t o t he
An
We
5
st e p s
to the
t ow ar d t h e
r
igh t
l e ft
s t a r t i n g po i n t ?
.
and
t ur n i n g
W h e re w o uld
-
A lg eb r a i c S c a l e
i t h m e t i c w e c a nn o t s u b t r ac t 2 0
f ro m 1 2 St ill b y u s i n g t h e a lg e b ra i c s c a l e a b o v e w e c a n
e a s il y s o lv e t h e pr o b l e m
a n d l e a r n t h a t t h e b o y Will b e 8
0
I f w e a g r ee t h a t
s t e ps t o t h e l eft o f t h e s t a r t i n g p o i n t
“
i n s t e a d o f m ea n i n g s ub t ra c t s h a ll m ea n go
t h e Sig n
l eftwa r d
w e m a y wr i t e
kn ow
in
t h at
ar
,
.
,
-
,
.
,
—
12
It w ill b e m o r e
i nst e a d
of
m ay m e a n
d l
,
v
8
.
l
l t o a g re e t h at t h e s ig n
”
“
a l w a ys m e a n i n g
ad d
a s it did in a r it h m e t l c
“
al s o
h e n c e we w ri t e
g o r ig htwa r d
,
,
w h i ch m e ans
ea
20
c o m p et e a so
+ 12
war
—
12
es o ne
—
20
8,
ig h t wa r d fo ll o w e d b y 2 0
s t e p s l eft o f t h e s t a r t i n g po i n t
s t e ps r
8
—
s t e ps
,
”
-
20
.
l e ft
,
E
22
L E M E NTARY ALG E BRA
u se d in a lg eb r a and t h e q u a lit y o f a n u m b e r is d e n o t e d
b y t h e s ig n
or
b e fo r e a n u m b e r d e n o t es t h a t it is posi tiv e a n d
T h e Sig n
—
t h a t it is n ega tive a s + 5
6
t h e Sig n
a re
,
,
,
24
olute valu e o f a n um b e r is
i n d e pe n d e n t o f t h e i r qu a li t y
Th e
.
in it ,
ab s
The
a b s o ut e
l
v a l ue
ab s o l ut e v a lu e
E
Le t
1
us c
and
.
If
a
.
If
a
m il es t h e
3
.
If
a
m il e s t h e
4
.
If
a
m il e s t h e
5
.
If
a
m il e s t h e
6
.
n um
If
a
m il es t h e
n o rt h
so
ut h
b e r of
un
its
n ext
s o ut h
fr o m
.
i
a c e rt a n
i
n
o
t as
p
s
o
p i
.
r es u
re s u
o ne
d ay
a n d n o rt h
13
lt ?
o ne
d ay
a n d so ut h
on e
d ay
and so
o ne
d ay an d
on e
o ne
'
lO
lt ?
res u
res u
resu
ut h
-
10
lt ?
uth
15
d ay
a n d no rt h
11
d ay
an d n
o rt h
17
so
lt ?
1 4 m il e s
lt ?
1 0 m il es
d a y , w h a t is t h e
8 is 8
.
7
1 0 m il es
d a y , w h a t is t h e
m a n w a l ks
of
—
1 4 m il es
d ay , w h at is t h e
m a n w a l ks
n ext
n ort h
+ 9 is 9
1 1 m il e s
d a y , w h a t is t h e
m a n w a l ks
n ext
s o ut h
of
1 2 m il e s
d ay , w h at is t h e
m a n w a l ks
n e xt
n or t h
d ay , w h a t is t h e
m a n w a l ks
n e xt
n or th
s ou th as n eg a tiv e
m a n w a l ks
n ext
i
x er c s e
o n s id e r di s t a n c e
di st a n ce
m il e s t h e
2
the
.
The
tive
.
,
re s
ul t ?
t h es e s ix q ues ti o n s as f o ll o ws : He is
s t a rti n g p o i n t ; 2 1 m il es s o u t h ; 4 m il es n o r t h ;
5 m il e s s o u t h ; 3 m il e s s o ut h ; 7 m il es n o r t h _
H er e a re t h e a lgeb r a i c so lu ti o n s of t h e s ix probl em s T el l h o w e ac h
r e s ult is o bt a i n e d a n d w h a t it r e p r e s e n t s
Y o u h a v e pro b a bl y
2 5 m il e s n or t h o f t h e
a n s w er e d
-
.
.
.
+ 12
+ 13
+ 25
— 11
— 10
+ 14
21
4
-
-
10
The
25
GATIV E
ul t s o f u n i t i n g t h e s e
fo ll o w i n g pr i n c i pl e s
p os i t i v e
res
t he
s h ow
P O S ITIV E A N D N E
The
.
sum o
f
a bso l u te v a l u es
the ir
26
The
.
sum o
f
n um ber s
a bs o l u te va l u es
E
A ppl yi n g
exa m p e s ,
l
ea c h
wi th the
fixed
cis e
—
RS
3
g a t iv e
s ign s
n um b e r s
i s the
p r efi x ed
u n l i ke s i g n s
re
p
it s pr o pe r
+ 33
14
+ 23
+ 15
xe r
s ig n
su
m of
.
i s the difi er
f
s ig n o
the
n u m be r
.
8
s um s
l es , w r i t e t h e
t h es e prIn p
gi v i n g
mo n
with
a bs o l u te v a l u e
ha vin g the gr ea ter
and ne
wi th l i ke
co m
w ith the
two
between their
e n ce
n u m ber s
two
N UM B E
in t h e
fo ll o w i n g
Sig n
31
— 41
16
+ 17
Doub l e M e an in g of
an d
T h us it a ppe a rs t h a t
a nd
a r e u s e d in a lg e b r a t o d e n o t e qua l ity o f
t h e s ig n s
n u m b e r s a s w e ll a s t o d e n o t e opera ti on s
27
.
.
E xe rc i s e 9
A s sig n
so
lve
1
.
them
A
—
P rob l e m s w ith P o s iti v e
n
r
i
n
u
li
h
a
t
e
u
m
b
e
t
s
t
o
q
y
a lg e b r a i c a ll y a n d i t e r p r e t t h e
n
m an
,
’
s
p r o pe r t y
Fi n d
2
.
n ex t
3
the
.
A
If
h is
n et
a m o u nt s
d ebt
or
a
n e xt
.
res u
pr o pe r t y
l ms
r
b
e
o
p
,
l ts
a nd
g a i n s 32 3 65 o n e ye ar
Fi n d t h e n e t g a i n o r l o s s
m an t ra v e l s
t h ese
to
m er c h a nt
r
ea
y
N e g ati v e N um b e r s
and
h is
d e bts
to
.
and
l ose s $ 1 7 90
t he
.
eas t
d ay , w h a t is t h e
5 8 m il es
n et res
o ne
ul t ?
day
and
we st 7 3 m il es
E
24
LE M E N TARY ALG E BRA
A m a n s an nu a l i n c o m e is
’
4
.
Ho w m u c h d o es h e
5
If
.
m il es t h e
6
A
.
sh
a
next
rea
o n a not he r
7
z er o
th e
10
°
il s
ve
a n n ua
d a y , w h a t is t h e
h is
o ne
d ay
n e t r es u
.
with
d e a l e r g a i n s $ 1 465 o n o n e
Fi n d t h e r es u l t o f b o t h sa l es
a
li n e
i g
r e prese n t n
an d
u th
39
l
sa e an d
$ 23 7 5
a
t h e rm o m e t e r
l
sc a e
; m ark the
th e
of a
w e ig h t
is
s to n e
t h e w e ig h t
a s pe c u a t o r
of a
g ar d e d
b a ll oo n ?
re
as
l
If
a sto n e
a
fo rc e
w e ig h s 3 4 po u n d s ,
and
Wha t
l o se s 328 7 5
b a ll o o n pull s u pwa r d
t h e c o m bi n e d w e ig h t o f
of
DE
po s i t iv e ,
and a
8 po u n d s , W h a t is
t h ey are f a s t e n e d t o g e t h e r ?
A
so
W h a t is t h e di ff er e n c e b e tw e e n
an d
If
,
and
.
r e pr e s e n t
b ot h if
.
lt ?
m a k e s $ 2 7 65 o n e m o n t h
n e x t m o n t h , w h a t is h is n e t g a i n o r l o s s ?
.
32 395
e x pe n s e s
ll y ?
5 3 m il e s
n o rt h
and
e st at e
.
If the
.
w o uld
9
sa
,
t h e + 24
8
l
D ra w
po i n t
.
ip
sa
$ 3 67 5
F I NITIO N S
o tatio n is a s ys te m of sym b o l s by m e a n s
ee n t h e m
a nd t he
o f w h i c h n um b e rs
t h e rel a t i o ns b e tw
o pe r a t i o n s t o b e pe r f o rm e d u po n t h e m c a n b e m o r e c o n c i se l y
e xpr ess e d t h a n b y t h e u s e o f w o r d s
28
.
s ys te m of n
“
,
,
.
29
n
Al g e b r a i c
o tatio n is t h e m et h od
b y figu re s a n d l e t t e r s
um ber s
30
n
.
.
31
.
r
A
s e pa a t e
of
i g
e x pr ess n
.
lg e b rai c
in a lg e b r a i c
An
um b er
n
a
te rm is
a
r
e xp e s s
i
io n is t h e
notat o n
n
u m b er
d b y t h e Sig n
2a >< 4b,
or
3 ab,
r e pr es e n t a t o n o f a n y
i
.
i
e x p ress o n
w h o s e pa rt s
thus,
my,
5 ax ,
and
a re
no t
F I N ITIO N S
DE
32
n
A
.
o mial
m on omia l is
is
an
25
o f / on e
e x pr e s s i o n
a n e x p r es s o n o f
i
two
t e rm
mo r e t e r m s ,
or
A poly
.
as,
2a —l- 4l2 — 3c — 5 d
The
b e t w ee n t h e t e rm s o f a po l yno m i a l
ig n s
an d
r e g a r d e d as s ig n s o f o per a ti on o r o f qu a l i ty
s
m ay b e
.
Wh en m o n o m i a l s a n d t h e fir s t t erm o f a po lyno m i al
w it ho u t a n y S ign b e f or e t h e m t h ey a r e posi ti v e /
ar e
tt
n
e
w
ri
I
,
33
is
a
A b in omial
.
pol yn o m i a l
34
A
.
is
of
a
po
l yn o m i a l
thr ee t e r m s
-
h o w m a n y t i m es t h e
sh ows
a
dd e n d
T h us
.
C oe ffi c i en ts
t h ey
m A
s
tri n o m i a l
.
.
t e r m is
c o e fi c ie n t o f a
f
two t e r
o
fa c t o r o f t h e t e rm
o t h e r f a c t o r is t a ke n
which
a ny
an
as
,
d is ti n g uis h ed
ar e
a r e e xpr e ss ed
in
figur es
or
n umerical
as
l e tt ers
l i ter a l ,
or
a
cc o r d i n g
.
te r ms a bo v e 4 is t h e n um er i c a l c o e ffi c i e n t
A n y o t h er fa c t or o f 4a a: m a y b e re g a r d e d a s t he c o e ffi c i e n t
pr o d uc t o f t h e r e m a i n i n g f a c to rs
In t h e t wo
as
.
,
th e
of
.
O b se r v e t h a t
T h i s s h o w s t h at w he n n o n u m e r i c a l co e ffi c i e nt is
t h e n u m e r i c a l c o e ffi c i e n t is c o ns ide re d t o b e 1
ex
pre s s e d
,
.
35
S imil ar te rm s
.
differ
on
l y in
5 xy, my,
36
.
t heir
and
Di s s im il ar
4a b,
37
.
a re
sa
fa ctor
Thus
bazy,
t e rm s w h i c h d o
n um e r c a
i
8xy;
3a b
te rm s
ax ,
l fa c t o rs
and
3 bc ;
,
and
ac
e,
4cx y
id
5a b;
3a e , 4x y ;
t o b e par tl y
no t
di ff e r
or
4a r ,
2x y,
a m,
a re
xe,
an d
no t
si
mi l a r ,
whi c h
or
si
r es pe
ct
s
7a r
.
i m il a r
,
as
3 yz
T e r m s t h a t ha ve
.
or
,
as,
,
t e rm s t h a t
a re
P a rt ly S im il a r T e rm s
fa c t o r
to tha t
a re
a
com m o n
mi l a r with
res pe c t
.
4x ,
and
a re s
bx
i m ila r
ar e
s
w it h
i m il a r
r es pe
w it h
ct
t o my
.
to
x;
a nd
5xy,
a xy,
38
L E M E N TARY ALG E BRA
E
26
.
Th e
lg e b r ai c e xpr e s s ion is t h e n u m b er it
n
i
r
i
u
l
ar
v
a
l
u
e
s
ass
ig
e d to eac h
a
t
c
p
val u e
of an a
re pr ese n t s wh e n so m e
l e t t er in t h e e x pr e ssi o n
S ub s t i t u t e 1 fo r a 2 fo r b 3 fo r
e x pr ess io n a n d s i m plify t h e r e s ul t
.
,
,
3 bc
2a b
c,
4 fo r d , in t h e
5 cd
foll o wi n g
4 ta
—
4
32
—
50
E x erc i s e 1 0
the
Fi n d
a
v a lu e
o f ea ch o f
= 1 b= 2 c = 3 d = 4
,
,
,
,
1
3
5
7
9
11
13
15
.
.
.
.
.
.
.
.
—
bed
9n
—
bcd
5
—
bc
6n
—
5 e 4 m + 3 en
—
8 a + 6mn
2 b + bcd
—
5 cd
8 m + 9a
4 bc +
—
7d
cdm
=0
,
4 a + 6a m
—
6b a dm +
—
e
-
6c n
9n + 7 a b
+ 3 cn +
—
ab
9n + edm + de
7e
2a b
fo ll o wi n g
l
2
)
n
e x pre s s
io n s
wh e n
1
3
__
6a m + 9a +
—
4ad
—
8b
8ad
—
2 bc
—
6n + 2 dm 2 b
4 a m + 9bn
—
3n
7
—
e
7 a + 9bd
~
-
2a
bm + 6d n
8 m + 9a n
—
5 bd + a e 4 bm + 6n
6bn + 5
—
e
6m +8 a d
—
cd
a cn + 8 m + 3 em
C H A PTE R I I I
ADD ITI O N
O F M O N O M IAL S
AD D ITIO N
39
i nt o
Ad d itio n is t h e pro c e s s o f u n i t i n g t wo
o ne n
40
the
.
.
n
41
.
um ber obt a i n e d
n um b e r s
t he
by
a
ddi t i o n
To Ad d S imil ar
i th m e t i c
t hen
,
m o re
t h e two
to be
a
dd e d ;
the
5 6
a nd
s um
is
.
T e rm s In
pr o d uc t s w h i c h
a
.
ddi n g
are
,
30
a nd
dd e d
Si n ce 5 t i m e s 6 pl us 3 t i m es 6 is 8 t i m e s 6
a dd e d a l so b y a ddi n g t h e c o e ffi c i e n t s o f 6 t h us
a
n u m b e rs
.
a d d e n d s a re
The
ar
and
u m b er
or
18,
3 6 in
a re
fo u n d
.
,
t h ey m ay b e
,
6
5
42
.
-
Ad d in g Ind i c ate d P rodu c ts
A lg e b ra i c
.
i n di c a te d pro d uc t s c a n b e u ni t e d i n to
l atter m e t h o d F o r e xa m pl e :
'
a re
,
the
o ne
t e rm s , w h i c h
t e rm
on
l y by
.
A sch ool
h a ll is l ya r d s l o n g
I g o t h ro ug h it 6 t i m e s o n
M o n d a y a n d 1 4 t i m e s o n T u e s d a y H o w m a n y ya r d s d o I
t r a v e l t h ro ug h t h e h a ll o n b o t h d a ys ?
1
.
.
.
M o n d ay
T ues d ay
,
6l ya r d s
1 4l ya r d s
B o t h d a ys
,
2 0 l ya r d s
,
2
.
T h e t i c ke t s fo r
e n te r t a n m e n t
i
an
w ere t
ld 3 4 a nd M a ry 2 8 t i c ke ts Fi n d t h e
fro m t h e s al e s o f G eo rg e a n d M a ry
G e o r g e 34 t c e n t s
M a ry
28 t c e nts
G e o rg e
so
.
.
,
,
B ot h ,
62 t
27
ce n t s
c en ts
to t a l
ea c h
re c e
.
i pt s
E
8
L E M E N TARY ALG E BRA
A DDI N G S I M ILAR
43
The
.
c ien ts
su m o
f
com mon
with the
W h e t h er t h e
th e
c
two
‘
s imil a r
l etters
te rm s h a ve
ter ms i s the
x
d
e
i
fi
a
li ke
o e ffi c i e nt s is f o u n d by
T ER MS
25
sum o
f their
ffi
c oe
.
u nlik e Sig ns
or
and
26
t he
,
s um
of
.
E x e rc i s e 1 1
G iv e
1
at s
igh t t h e
4 3
-
.
2
4a
.
3
12
.
the
f o ll o wi n g
8r
.
4
.
— ”b
— 3x
5a
53
11
s um o f ea c h o f
b
.
7 7
-
-
7x
25
44
.
to tha t
R ul e
Fin d the
.
r es u l t a
fi
x
the
m of the
a l g e b a ic s u
r
mon l etters
co m
.
.
t
i
n
s
c
e
i
,
fi
coe
4c
a nd
E
30
L E M E N TARY ALG E BRA
AD DI N G
DI SS I M ILAR T ER MS
Di s s im il ar te rm s c a nn o t b e u n i t e d i n t o o n e t e rm
Th e a ddi t i o n c a n o n l y b e i n di c a t e d b y w ri t i n g t h e m in s u c
c e s s i o n In a n y o r d e r e a c h pr ec e d e d b y it s o w n s ig n as h e r e
48
.
.
,
,
s how n :
5a
—
3 bc
be
2b
—
4a o
2 bd
— 3a
3a o
—
3a o
b e + 2 bd
2a
2b
— 2b
-
We w r it e a pos iti v e t er m fi r s t if t h er e is on e
n e g a ti v e a n y o n e o f t h e m m a y b e w r itt e n fi r s t
,
,
.
If
.
E x e rc i s e 1 3
G iv e
1
.
at
3a
2 1:
Sig h t
the
2
su m o f e a c h o f
b
.
—
2c
3
—
.
th e
2x
y
fo llo w i ng
4a c
al l
-
th e
3 bc
-
t er m s
2b
a re
A D D ITIO N O F
M O N O M IAL S
31
E
32
Si m plify t h e
51
.
53
55
57
59
61
.
.
LE M E N TARY ALG E BRA
fo ll o w i n g
4a + 2a + a + 5 a
2 x + zr
—
5c
-
~
5b
7x + 8x
6c + c + 4 c
— 3a —
6v
— 2x
.
7a
.
—
8x
—
4x
—
3x
.6y+
—
8x
—
9y 5 x
a
.
A ddi t i on
m ono m ial s ,
Th e
as
—
-
6b+ b + 9b
y + 9v
7n + 5 n
-
—
7 y+ 5 v
—
4n + 3a
n
—
—
4 b + 9b+ 7 b sb b
y
—
6a
AD D ITIO N O F
49
—2 b
—
7b
4 a + 3 b+ a
P OLYN O M IA L S
l yn o m i a l s pro c e e d s m u c h a s a ddi t i o n o f
t h e t w o f o ll o w i n g illu st r a t i o n s sh o w
of
o
p
S c h oo l h as 3
fligh t s of a b a n d
0 s t e ps r e s pe c t iv e l y
A b oy go es u p a n d d ow n t h e st a i rw ay
3 t i m e s o n M o n d a y 5 t i m e s o n T u es d ay 4 t i m e s o n We d
n e s d ay
6 t i m es o n T h u rs d ay a n d 4 t i m es o n F r id ay
Ho w m a n y s t e ps d o es h e t a k e o n t h e st a i r s du r i n g t h e w e e k ?
1
.
i
of
st a r w a y
a
,
,
,
,
,
M
o n d a y,
T ues d ay
6a + 6b + 60
1 0a + 1 0b + 1 0 0
,
W e dn esd a y,
Th ur sd ay
8 a + 8 b + 8c
1 2a
1 2 b 1 20
,
Fr i d a y,
8a
Sum ,
.
At
,
.
,
2
,
a
At
80
4 4a + 44 b + 4 40
m o ney c h an g er
-
tim e ,
8b
’
s a re o f er e
f
5 2 m a r ks , 3 5
t
s t e ps
s t e ps
s e ps
s te ps
t
s t e ps
s e ps
d fo r
ex c h an
ge
n
o
u
ds ;
p
n
o
u
ds ;
p
n
o
u
ds
p
fr a n c s 1 2
1 8 m ar ks 2 6 f r a n cs 24
At a n o t h e r
At a n ot h e r
2 2 m a r ks 1 5 f r a n cs 1 8
T h e e x c h a n g e v a lu e o f a m a r k b e i n g m c e n t s o f a f r a n c
n
fi
n
h
x
n
n
n
l
t
s
d
t
t
a
e
c
h
a
of
ou
d
c
o
l
g e v a l ue
c
e
t
s
a
n
d
a
e
e
t
p
f
o f t h e f o r e ig n c u rr e n c y in c e n t s
F i rs t ti m e 5 2m + 35f + 1 2l c en ts
S e c o nd ti m e 1 8m + 2 6f + 24l c e n t s
Th i rd ti m e 2 2m + 1 5f + 1 8 l c e n ts
9 2m + 7 6f + 5 4l c e n t s
S um
o ne
,
,
,
,
,
,
,
.
,
,
,
.
,
,
,
,
O F P OLY N O M IALS
A D D ITIO N
50
To
.
m
s
a
n
ol
o
i
l
,
p y
a dd
a n d add ea ch co l u m n ,
beg in n ing
at
s i mi l a r
5 bd
ac
ter ms i n
a
col u
mn
the kft
.
2 bc + 3bd + 5 xy
5 a b+ 3 a e
2a b
wr i te
33
7 x2
2 xy
be
5a o
3 bc
8ab
51
A
.
Ch e c k in g
.
lg e b ra i c w o r k is a n o t h e r
fi rs t r e s ul t c o rr ec t
ch e ck on a
t e n d s t o pr o v e t h e
52
7a c
Ad d ition
i
o pe ra t o n
wh ic h
.
b y S ub s ti tu tio n
A ddi ti o n m ay b e
.
bs t i t ut i n g a n y n um b e r in pl ac e o f t h e l e t t e r s
a n d d e t e r m i n i n g w h e t h er t h e su m of the va l ues o f t h e a dd e n d s
e q u a l s t h e va l ue of the s u m
c h e c ke
d
by
su
.
fo ll o wi n g s h o w s h o w a d di t i o n o f po l yno m i a l s m a y b e
c h ec ke d b y s ub s t i t u t i n g 1 fo r e a c h l e t te r
Th e
.
Th e
th e
s um o f
s um o f
t he
v a l ues o f t he a dd e n d s is
n
ol
p y o mi al s i s a l so 8
th e
8,
a nd
the
v a l ue o f
.
O b s e r v e t h at w h e n 1 is s u b s t i t ut e d fo r e a c h l e t t er
v a lu e o f e a c h t e rm is t h e n um e ri c a l c o e ffi c i e n t
,
t he
.
i g o r v e r ifyi ng a lge b ra i c pr o c e sse s a ny n um be r
T o a v o id l a r g e n um
m ay b e sub sti t ute d fo r e a c h l e tt er
b ers it is w e ll t o s u b s t i t ut e s m a ll n u m b e r s ; b ut s ub s t it u
t i n g 1 checks on l y the coefficie n ts a nd S h o u ld n o t i n gen era l
b e d on e
In
c h ec k n
,
.
,
,
.
,
L E M E N TARY ALG E BRA
E
34
E x er c i s e 1 4
1
.
A dd
—
—
n
l
x
2
.
an d
3
.
a nd
4
.
and
5
a nd
A dd
5 b+ 3
2 c + d + 4 b,
A dd
—
2b
—
4a
—
d
A dd
—
7x
A dd
3a r
A dd
-
—
c
—
c
6d ,
5 c,
5 y+
—
8 5z
3y
—
2r
4 y + 3z ,
4d
—8 a
+ 7 b, 3 c + 4 a + 3 d ,
— 4
y + 6z
,
-
5 x , 6y
—
—
2x
82 ,
.
y,
—
—
2ar
5 by
and chec k
by,
—
—
4 d 5 c + 3 b, 6d 4 c — 7 b,
.
—
a n d c he c k
+ 4 by
-
—
—
2 c 2 b+ d ,
—2 x
2 a + 6n + 6,
.
an d ch e ck
3z ,
7a
.
2 b+ 3 d ,
and check
3 b+
7 c,
7 xz + xy
.
—
—
+ 2x, 5n 4 x + 5 ,
—
—
4 , an d 5 a 4 n 3x + 4
4 e + 3 y+ 8 ,
.
6
—
4a 3n
3 xy
—
9by,
.
5z
+y
—
6x,
3 y+ z + 5x ,
—
8 y+ 4 x
an d 7y + 3 z
7
.
A dd
and an
8
.
—
—
7 a c a n + 3 n x,
—
5 a x nx
A dd
5a
,
+ 6b
5a x
an d c h e ck
-
—
4c
7 c,
+
—
4a n
6n x ,
—
—
2 n x 3 a n 5 a c,
.
3 b+ 5 ,
— 2c
—
+ 5b
8a ,
—
4c
—
7b
9,
an d
9
.
and
10
.
A dd
5ax +
—
6br
8 cx + a x
A dd
—
8ab
—
3 bx
—
3 dx 4 a zr
2 ex,
+ 5 cx ,
4 cx
—
7 bx
3 d zr ,
.
—
5 ad
6bo+ 4 a c ,
—
5ab
7 a c,
—
4 bc a d + 5 a c ,
—
—
a n d 3 bc
3 ad 3 a b
.
11
.
4 bn
an d
12
A dd
.
ac
A dd
7 bn + 5 a b, 4 bn
4a n
+ 4 a b,
—
6x
7y+ 5z ,
—
—
+ 4u x , a n d
13
14
—
.
.
A dd
A dd
5 + c,
15
a nd
.
6z +
—
4a b 2a c
—
4a
—
2 :e
+ 4 bc,
—
7b
5 c,
3 a n + 6a c
9a b,
.
—
—
4y u
3u
6a n ,
3 z,
— 2u
—
+ 6y
5x,
—
4z
5y
.
—
—
5 a o 2 a b+ 6bc ,
—
—
3c 7a d ,
—
and
—
2a o
ab
—
5 b+ 3 d 2 c ,
.
—
8b 2d
and
A dd
—
a n d c he c k
7a c
5xy
—
4xz + 3 yz ,
—
—
2 xz 2 xy 7 yz ,
—
8 yz + 6xy :ez , a n d c h e c k
.
—
3 xz x
y+ 9yz ,
C HAPTE
S UB TRA CTI O N
53
b e rs
54
t he
a
S YM B OL S
.
AG G RE G ATI O N
OF
n um
S u b tr ac tio n is t h e pr o c ess o f fi n di n g o n e o f t w o
w h e n t h e i r s u m a n d t h e o t h e r n um b e r a r e k n o w n
.
.
T h e in in ue n d is t h e
.
s
55
R IV
ub tr ah e nd
is
o n e of
dd e d
to the
the
a
t h at
dd e n d s
of
r e pr e se n t s
the
t he m inuen d
r e m ai nd e r , is t h e
s ub t r a h e n d gi v e s t h e m i n u e n d
Th e d ifi e re n c e ,
.
n um b e r
num
or
s um
;
.
w hi c h
b er
.
S UB TRA CTIO N O F M O N O M IAL S
1
.
A
t h e rm o m e t e r
d
di r e c t i o n
it
re a
fo u r h o u rs pr e vi o us l y
h o w m a ny d e g re e s a n d in w h a t
t h e m e r c ur y c h a n g e d m e a n wh il e ?
T h r o ug h
re a
h ad t he t op
Pr es en t r ea d in g
P r e v i o us r ea d i n g
T h e ch an ge
,
of
+ 13
°
7
°
,
a nd
ds
t i
o b a n ed
,
by
t c ti n g
s ub r a
— 7°
fr o m
—
r
an
st a
di n g a b oy g o e s u p 1 7 st e ps
a n d d r o ps his p e n c il w h i c h r oll s d o w n t o t h e l a n di n g a c ro s s
t h e l a n di n g a n d o n d o w n t o t h e 6t h s t e p b e l o w t h e l a n di n g
W h ere it s t o ps T h e s te ps a re a i n c h e s h ig h Ho w fa r a n d
in wh a t di r e c t io n m us t t h e b o y g o t o g e t t o t h e s t e p w he re
t h e pe n c il li es ?
2
.
St a rt i
n
g
f ro m
a
i l
,
,
,
,
,
.
.
C a lli n g
u p w ard
an d
i i g at
s t ar ti n g fr o m
go e s
d o wn w ar d
6a
arr v n
In t h e se
Le t
us
c ase s
n ow
n um b ers
.
+ l 7a
— 2 3a
,
m ea n i n g 23 a
we have bee n
l e a rn
t h e b oy
the
g e n era l
i n c h es
.
i g s ig ne d n u m b e rs
o f s u b t ra c t i n g s u c h
sub t ra c t n
pl a n
d o wn w a r d
.
E
36
LE M E N TARY ALG E BRA
SUBT RA C TI N G S I M ILAR T ER M S
56
.
w it h
fo ll o w i n g ex a m pl e s re pre se n t a l l c a ses in a d d i tio n
re fe re n c e t o s ig n s a n d re l a t iv e v a lu e s o f a dd e n d s :
The
5a
3a
3a
5a
8a
8a
exam p es
Wr i t e
l
m i nuends
in
a nd o ne a
su btr a c tion ,
dd e n d
as s
usi n g
the
ub t rah e n d
,
as
8a
8a
8a
2a
2a
3a
5a
5a
3a
3a
5a
3a
3a
5a
5a
ve s u m s
fo ll ow s :
abo
as
b t rac t i o n t h e di ff ere n c e o r i e m a i n
d e r in ea c h c a s e m ust b e t h e o t h er a dd e n d
S h o w t h at t h e c o rre c t r es ul t m ig h t h a v e b e e n o b t a i n e d in
e a c h c a s e by cha n gin g the s ign of the su btr a hen d a n d a ddi ng
B y the
de fi nition of
su
,
.
.
57
Prin c ipl e
.
S u btr a ctin g
.
a ddin g a n u m ber o
f
R ul e
58
to
from
ng
f S ign
ch a
e o
E
G iv e
1
.
re
9a
4a
si
f
n
o
g
and
to
or
T he
i
l d a l w ays
15 —
.
4x
6x
bu t opposi te qu a l i ty
the s u btra hen d
b e m a d e m e n t a l ly
cha n ged
.
S ub trac ti ng Sim il ar T erm s
m a i n d ers in t h e f o ll ow i n g
2
equ i va l en t
is
a s i n a d di tion
d
cee
r
o
p
sh o u
xe r c s e
—
n u m ber
equ a l a bsol u te va l u e
Con cei v e the
.
a ny
3
—
.
o ra
3b
8b
ll y
.
to
.
from
E
38
L E M E N TARY ALG E BRA
S ub tr ac tin g M onomi a l s
E xe rc i s e 1 6
G iv e
1
.
rem a i nd e rs in t he foll o wi n g
3a
b
2
.
— 4 a:
7x
3
.
4a
— 2n
o ra
ll y
S UBTRA C TIO N O F
P OLY N O M IALS
39
S UB TRA CTIO N OF P OLYN O M IAL S
60
1
.
Sub tra c t 7
.
d o ll a rs
1 2 di m e s
di m es fro m
3 q u a rt e rs , 8
,
16
d o ll a rs 7 q u a rt e rs
L ett i n g c b e t h e n um b e r o f c en t s in a doll ar q t h e n u m b e r
o f c e n t s in a q u a rt e r a n d d t h e n u m b e r of c e n t s in a di m e
we wri te
,
,
.
,
,
F
2
,
.
ro m
1 6c + 7q + 1 2d
T a ke
7c + 3q + 8d
D iff er e n c e ,
9c + 4q + 4d
—
Fr o m 5 a b 4 ao + 3 be b ush e l s
.
bu she l s
we r e
so
ld
,
,
Rul e
.
mn
col u
Write
.
B egin nin g
.
4a b
re m a ne
i
4a b
6ac
6a c + 2 ed
d?
ab
2a o
Zed
3bc
a
i
l
n
o
m
l s,
o
p y
the
at
,
5ab — 4ac + 3be
nuen d ,
Subtr a h en d
D iff er en ce
61
g ra i n
How m a ny b u sh e l s
.
Mi
-
of
f
the l e t,
2c d
s im il a r
ter ms
in
wi th mo n omi a l s
s ubtr a ct a s
a
.
S ubtr a c ti o n is c h e c ked by d e t erm i n i n g w h e th er t h e d i ff ere n c e
be t w ee n t h e v a l ues o f m i nuen d a n d sub tr ah e nd is e q u a l t o t h e v a l ue
O b s erv e t h e w o r k b e l ow
o f t h e r e m a i n d er
.
W ork
—
5a b 4 c e
Ch e c k
+ 3 be
—
4 ab 6a c
ab
Th e
bov e
s ub s t i t u t i n g 1
2 cd
-
exam p e
l
fo r
a,
16
—
8 18
+ 2 cd
+ 2 a c + 3 he
a
10
— 24 = 2
2+
in
s ub t r a c t
2 fo r b, 3 fo r
io n
h as b e e n
c, a nd
4 fo r d
c h e c ke
.
pl a i n t h at s u b t r ac t i n g is fi n di n g wha t
mus t be a d ded to the su btr a hend to give the min uend
It is
now
.
a no t h e r
a nd
go od
differen ce
c h ec k o n s
a n d s ee
ub t ra c t i o n is
if the
su m
to
a dd
d by
the
is the min uen d
.
n u mber
H en c e
,
s ubtr a hen d
LE M E N TARY ALG E BRA
E
40
S ub trac tin g
E x erc is e 1 7
So lv e t h e
1
.
a nd c he c k
fo llow i n g
Fro m
8ab
5 c + 4d
8
s
the
t
F r o m 4 cx + 7 by
1
0
0
Sub t ra c t 4 a r
4
Fr o m
0
Sub t r a c t 3 a b
3
0
9
10
f s u bt ra c t
Sub t r a c t 4 a c + 3 bd
.
.
F rom
Fro m
.
2 bc
9
xy
s
6a m
the
s
am
ax
4 a n + 4a r
7 rs
3a +
s
—
2b 30
+8
ub tr a c t
1 0 be
and
.
xy
.
8a b
.
7 + ar
7a y
4a b
2 d + 2a
.
am
.
1 2a n
.
6a o
6a m
12
.
.
3 xy
4 xy
.
5b
1 0 +3 ae
fro m
+d
z
4 a o+ 2 bd
2ac
ubtr ac t
6c
5 bc
10
ub tr a c t 7 by
7ac
6
sum o f
8d e
fro m
8 be
12
4 a az+ 6a y
7a b fro m 4 d x
4 xy
7a y+ 3 xy
ax
i
n ne
fro m
14
5 b+ 8 de
6bc
fi rst
u b t rac t 4 d + 3 ab
Sub t r a c t 5 a y
Fr o m
P olyn omial s
— 4b
s ub
t r ac t 3 b
11
.
12
F ro m
.
Sub t r ac t t h e
2u + 6
su m
of
.
.
— 2d
of
3z +
.
2y +
—
—
2 b 5x 3 a
F ro m
m
su
3d
f
—
S ub tr act 2 b
and
4 b+
—
3d
—
—
2 d 56 + 2f
of 3 c
—
a nd 3x
6b+ 3 y
-
—
—
—
86 4d 3f 6c
an d
.
2c +
—
d 2a
—
3a 3 c
fr o m t h e
—
2a 5b
+2 c
—
4x
a nd 3 z
10
s um
of
.
.
F r o m t h e s um of 3 x + 2y
—
5 y 4u s ub t r a c t 2z 3 5x 3u
15
sum
.
th e
—
5c
s ub t r a c t 5 e
14
-
the
—
2x 6
—
—
fr o m 2 a 3 b 2x + 4 y
+ 4a
13
—
3 y + 2z u s ub t r a c t
— 2z — x
+y
a nd
4g
—
4x
.
.
16
.
—
4a b
17
s
.
Fr o m
5 a b+
—
2 bd 4 bc
Fro m
ub t ra ct
,
—
2 ac
a nd
t he
4u + 4 y
—
bd
s um
52
3 bc
u b t r ac t
.
the
s um o f
2bc + 3 a c + bd ,
—
4 a c ab
.
of
5
s
2g
4x + y
3x
.
— 2z
a nd
4u + 3y
— 7x — 2 z
S Y M BOL S O F AGGR E GATIO N
18
0?
.
Wh at
To
19
.
—
3 ab
20
the
.
n u m b e r gm us t
8a + 4 b
giv e
an d
bc ,
Fr o m
sum o f
a
—
—
4 a + 6b 8 0 t o
to
dd e d
bd 7
sum
t he
giv e
4c?
—
4 a b 3 a c + 2 bc sub t r a c t t h e s um o f 3 bc + bd
Fr om
—
3 bd
—
be
41
-
-
a o,
—
2 ac
ab
.
of
3a
— 2 :c
—
n
+ 5 a d 4 x + 2 y 4 s ub t ra c t
—
—
—
3 x 2a + 3 an d y 2 x 2
.
—
—
—
=
=
2 b 3c,
5 a 3 b+ 4 c , y 3 a
2 1 If x
—
t h e v a l ue o f x y z
find
.
w
.
22
.
gi v
to
23
.
W h at
e m u st b e
n um b r
—
—
5 be + 2 a b 3 bd ?
e ac
—
2a
S ub t r a c t
—
2 b+ 2
a
fro m
s u b t r ac t e
To
—
2b 3a + 3
7,
t h e t h re e
z e r o , a n d a dd
d fr o m
giv e 0 ?
f ro m
3 b+ 4
—
—
2 a b 3 a c 5 bc
r es u
lts
f ro m u n i t y
*
,
.
SYM B OL S O F A G G RE G ATIO N
T h e pr o du c t 8 X 1 4
wh i c h m e a n s
62
.
T his us e
in l e a r n i n g
of
c an
be
s h o wn
s ym b
t he
c a ll e d
ol
r a pid m e n t a l c a l c ul a t i o n
a
thus :
r
i
e
n
t
h
e
s
s
a
,
p
so
.
.
A m an w a l ks n o r th 5 m il e s a n h o u r fo r 2 h ou rs a nd t h e n
ut h al ong
the
fa r is h e t h e n
T he
a ns w er
sa
me
fr o m
to
t his
ro a
t he
d
3 m il es
i g
s t a rt n
-
an
h o u r fo r 2 h o ur s
po i n t ?
pr o bl e m m ay b e wr itt e n
—3
t h us
)
Sh ow t h at t h e pe ri m e t er o f a
a n d y l o n g m a y b e W r i tt e n :
or
w+ y + r + ya n d t h a t
*
o f a id
t hus :
,
etc
63
is
Uni ty m e an s 1
.
re c t a n
gl e a: w id e
or
2 x + 2 y,
= 2x + 2
y
.
I
.
Ho w
LE M E N TARY AL G E BRA
E
42
64
and
f o u r o per a t io n s t h e m ul t i pl i c at io ns
T hus
b e pe r fo r m e d fi r st
a ser e s of
In
.
t he
i
divi s i o n s
are
to
,
.
,
8 + 7 X3
—6
+
8
t h e ter ms
a re
8+ 21
In
su
ch
i
s e r es
a
Th e
an d
i
s i m p lifi e d
c
t h an
der
t o use
s o m e s ymbol o
.
s ym b
j
t h e bra ce
,
s
ign s
Wh en
.
shou
ld
be
su
ch
fi t
rs
t o pe r fo rm t h e o pe r a t io n s o f a se r i e s in
t h e o n e m e n t i o n e d a b ov e , it is n e c e s s a ry
f
The
b y th e
.
a ny o r
65
t
s e p ar a ed
,
W h e n it is d e si r e d
ot h er
t h e p ar t s
= 17
4
l c o n t a i n s s ev e n t er m s
s i m p lifi ed o r r e d u c e d
ea c h te rm
to be
or r ed u e d
15
ex a m p e
a b ov e
e xpr es s o n s a r e
+5
ol s
a ggr ega tion
.
of a gg e g at
r
t h e bra cket
io n
H and
,
the
a re
t h e pa r en thes is
vin cu l u
m
T h ese m ea n t h a t t h e o per a t i o n s i n di c a t e d within them
a r e t o b e p e r f o r m e d b e f o r e t h e o p e r a t i o n s u pon them ; in
o t h e r w o r d s t h a t t h e e x pr e s s i o n s w i t h i n t h e m a r e in e a c h
E v ery p a rt w i t h i n t h e
c ase t o b e r e g ar d e d a s o n e n u m b e r
s ym b o l is a ff e c t e d b y t h e o p e r a t io n i n di c a t e d u p o n t h e
s ym bol
O b s er v e t h e follo w i n g
,
.
.
18 — 9
1 5 X 12 —8
= l 72
18
—
2
l
8
15 x
= 60
2 16
— 3 6 -z
2 16
60
N ot ic e t he
1
—
113
2 , t he i r
th e
.
ll e r
1
19
1 37
of
f o ll o w i n g :
—
t w o n um b e r s is 513
—
di ffe re n c e is ( tr 2 )
a rect an
4 ) = 1 37
t h e p a re n t h es i s in t h e
-
.
7
the
a nd
l ar g e r
(x
gl e is x + 8 in l o n g
o f t h e re c t a n gl e is
If
.
a re a
3
sm a
I f t he
.
2
use o f
— 35
a nd x
+ 3 in w id e , t h e
s q u a re
.
i n c hes
.
di s t a n c e b e t wee n t wo c i t i e s is x + 10 m il e s
w h o l e di st a n c e is 3 ( x + 1 0 ) o r ( x + 10 ) 3 m il e s
If
f
o
3
the
.
,
S Y M BOLS O F AGGR E GATIO N
43
E x e rc i s e 1 8
Re m o ve t h e
1
bol s
of a
—
—
—
z
4
24
5
5
5
6
)
(
.
7 64
3
.
—
23 8
4
.
ex
8 X9
-
(
s
i m plify
—
4 45
—
7 89
14 X 7 +
1 08
th en
an d
ggre g at i o n
24
1
44
9
X
+
(
4 65 + 67 X 8
.
2
s ym
—
84 5 8
—
54 0 9
)
1
2
7
8
X
+
(
O pe rati on s on Co m po un d E xpre s s io n s Sym b o l s o f
a gg r e g a t i o n a r e m u c h u s e d in a lg eb r a t o i n di c a t e o p e ra t i o n s
o n c o m po u n d e x pr e s s i o n s
T o i n di c a t e t h e s ub t r a c t i o n o r m ul t i pli c a t i o n o f a po l y
no m i a l a pa re n t h e s i s is n ec ess a ry
T h u s x ( a + b) re prese nt s t h e pro du c t o f a: a n d a + b a n d is
o
re a d a: t i m es a + b o r a + b t i m e s n
66
.
.
.
.
,
,
.
,
E x erc i s e 1 9
1
.
I n di c a t e t h e s u b t ra c t i
of
t h e pr o du c t
2
.
will
3
If
a
.
the
o f a sq
of r
.
n
the
u m b er
n
—5
h e h as
u mb e r
of
3x + 4
.
I n di c a t e
ll s
2 x + 35
of
t h em , w h a t
l eft ?
2)
-
fr o m
.
s h ee p a n d s e
W h at d o e s ( x + 5 ) ( se
1: r e pres e n t s
4
t w o b i n o mi a l s
m an has
de not e
on of x
fe e t
if
s id e
r e pr e se n t ,
o n e ac h
A.
a
u a re ?
W h a t d oes
( + 8)
x x
od s o n e a c h Sid e
re pr e s e n t ,
of a sq
if 1: s t a n d s
fo r t h e
n u m b er
u a re ?
R e pre s e n t in t w o f o rm s 4 t i m e s t h e s um o f a n y
n u m b ers
5 t i m es t h e d i ff e r e n ce o f a ny t wo n u m b e rs
5
.
.
.
R e pr ese nt t h e pro du c t
w h i c h is 8 g r e at e r t h a n T
6
tw o
.
of
t wo
eq
ual
n um
b e rs
ea c h
of
.
7
.
At 85 ¢
a re ct a n
a r o d , e xpr ess
gul a r f arm
a: r o
ds
in t wo w ays t h e
by y r o d s
.
l
i g
co st o f e nc o s n
E
44
8
I f x is
.
LE M E N TARY ALG E BRA
p o s i t iv e
a ny
—
g re at er o r l es s t h a n x
9
Wh a t
.
b e tw e e n
10
is t h e
—
513
9
11
12
If
.
is
x
a ny
5
.
di ff er e n c e
a?
a nd
12
T
—
x
8 is
n,
w h a t is
13
rd
of
.
of a
.
.
or
33?
10
A t $4 0
rd
.
long
an d
t wo
a re
d if t he re
,
a re
20
bi n o m i a l s divid e d by
s am e n
a re c t an
of
4
gul a r fi e ld
n
u m b e rs if
,
a ny
u m b er ?
3 t erm s ,
u n d f a c t o rs
in
.
t h e pr o du c t
d e r di ff e r by t h e
an a c re ,
of
t h e re
a cr es
r e pres e n t
c o m po
a n o r c h ar
.
eac h
t e rm
i i g
co n t a n n
.
v a lu e of 3 fa rm s c o n t a i n i n g
r e s pe c t iv e l y ?
w h at is t h e
—
113
5 a c r es ,
R epr e se n t t h e pro du c t
of e a c h n u m b e r b e i n g x
18
than
b
a
i
m ore
m+ 2 0 ,
g rea t er
ar
e
a r e ro w s ?
p r o du c t
a n e x pr e ss o n o f
Wr i t e
o ne or
17
an d x +
wid e
t h e m in
16
+ b and
W h at m ay
.
,
w h e n is
t h ere in
t h a n t h er e
a ro w
I n di c a t e h ow m a ny
.
15
are
Wri t e 3 a t i m es t h e
.
8
i t iv e i n te g er
Ho w m a ny t ree s
.
t h e pr odu c t
14
o
s
p
l e s s t h a n £13?
as
m o r e t r ees in
at,
is
t han 5 ,
w h i c h t e ll s t h a t t h e
I f t h e di ffe r e n c e b e t wee n
.
W h en is
2
Sh o w W h y
u ati o n
—
4
x
g e r g rea t e r
—
is x
v a lu e o f n ?
the
—
x
and
eq
3?
i
nte
of
.
t wo
u ne q
ual
n
u m b e rs
,
rt
pa
.
19
.
b e i ng
e ac h
20
W h at is t h e
.
The
a re ct a n
21
ge ts
.
gl e
x
in
.
a re a o f
and
the
a re a o f a sq
x+ 6
in
.
d ay
ex c ee
eq
l e n gt h
u a re
a:
—
by x 4
H ow m u c h d oe s
pe r
3
a
d s t he
b oy
n
u al
6 in
r ec t a n
.
gl es
,
t h e w id t h
g re at er ?
lo n g is t h e sa m e
in
E xpr e ss a s a n
in
.
.
as
eq
t h at
of
u at i o n
.
if t h e n um b e r o f c e n t s h e
o f d ays h e w o rks by 2 0 ?
e a rn ,
um ber
of
ELE M E N TARY ALG E BRA
46
E x erc i s e 2 1
R em o v e t he
the
e x pre ss
1
.
—
3a
(b
.
—
5a
b
7
9
—
3a
.
v a lu e
+c)
—
x
.
2
—
4a + b c
-f-
—
a
3
(
—
—
b 4a
b)
8
b
10
—
3 b+ 4 c , y 3 a + 2 b
o f e a c h o f t h e f o ll o w i n g :
2a
12
— z
y
15
.
.
(
4 x + 3 y+
.
5x
.
—
2x
.
—
4n
z
—
3 x 4 y)
—2x — 4
—
y)
.
50 ,
—
3y
3x +
(
—
—
3 n 4x)
= 4a — 5 b — 3c fi n d
,
—
x+y
z
— x — — z
y
—
x
— x
— z
+y
y+ z
r em o
To
.
3y
—2x — 4
y)
v e t wo o r m o r e sym b ol s o f a gg r e g a t i o n
w i t h i n a n o t h e r b e gi n w i t h t h e ou t e r o n e
69
and
3x
.
6
x + y+ z
.
14
fe w
as
ggr e g a t i o n in th e fo ll owi ng
t e rm s a s po ss ibl e :
a
4
—
c
hen x
11
l t s in
—
2a + b c +
.
b o l s of
—
(a 2b
—
b
4a
5
the
res u
.
3
W
—
s ym
,
o ne
,
3a
— 3a
It
s h o ul
c u l um
d be
,n o t
t o t he b
.
R e m o vi n g
M
ig n s
t he
and
b — c to
s am e s
t
n o ed
tha t
Th e
(
1
{b j
—
—
a + 2b
a
—
2b
a
th e
Sign
o ut e r
—
+a+b
s
of
-
c
-
n
)
—
c
n
ig n b e fo r e
the b
b e lo n g s
t o t he
v in
t h e b is
s ym b o
t h ese t w o t e rm s
l
g es t he Sig n b e fo re
b r ought do wn w i t h t h e
chan
a re
.
t e a c h er s pr ef er t o b e gin w it h t h e in n erm os t s ym b o l o f a g gr e
r
m
es
a
e
r
it
r
a
b
c
o
eas
ft
a
littl
e
a c ti c e
h
n
E
e
w
e
i
o
a
t
y
p
y
g
I t is j us t a b out as ea sy a n d it is ev en qui ck er to r em o v e a l l sym b o ls
o f a ggr e g a ti o n a t o n c e b y b e gi nn i n g a t t h e l e ft a n d b r i n gi n g e a c h s u c
c ess iv e t e r m d o wn w it h it s o wn o r t h e o ppo s it e s i gn a cc o r d i n g as t h er e
is an even o r a n odd n u m b e r o f th e a n t e c ed e n t m i n u s s ign s a ff e cti n g
it An y o ne o f t h e t h r ee w a ys b e c o m es e asy an d r eli able w it h a little
pr a c ti c e
*
any
.
.
,
'
.
.
S Y M BOL S O F AGGR E GATIO N
47
E xerc i s e 22
R em o v e t h e s ym b o l s
Si m plify t h e r e s u lt s :
1
3
.
6a
.
—
2a
(
—
3b
15
4a +
.
2a
.
3b
—
(b a
+
-
—
4 x 3 y)
5r
4n
— 3 m 3 n — 6x
+
)
)
7x
—4 —
y
)
3 x + 2y
a
c
2b — c
3x
-
an d
— 2 — 4x — 3
y)
y
)
-
—
fo ll o w i n g
6x
—
a+ o
b
—
—
—
3a
(b 2a + b
13
in t h e
— 2
3
+
y
y
—
a+b
c)
—
—
2
b
a
b
5a
+
(
.
ggr e g at io n
2y
4 a — ( 2b —
11
of a
)
b)
-
-
5x )
3 x + 3 y)
—
2 x 4 y)
4n
2y
-
—
—
2x 3x
3 y)
fo ll ow s t h a t in o r d er t o e n c l o se t w o o r m o re t e rm s
o f a po l yn o m i a l in a s ym b o l o f a gg re g at i o n pre ce d e d b y t h e
s ig n
we m ust c h a n g e t h e s ig n s o f t h e t e r m s e n c l o se d
Th us
—
ah
a c + bc
b
d
ab
cd
a
o
c
c
+
)
(
70
It
.
,
.
,
-
E x e rc i s e 2 3
l as t t h re e t erm s o f e a c h o f t h e s e
n
r
h
e
r
n
s
i
d
d
i
u
Sig
a
e
t
s
e
c
e
e
b
m
n
a
s
:
p
p
y
E n c lo s e t h e
a
1
3
5
7
9
.
.
.
—
ac
ax
+ a b + bx
—
a b+ be
ac + ax
—
—
—
ax
bx bc by
+ a b+
.
an
.
—
—
ac
ac
c
—
ac
2
4
6
bc
8
bc + bx
10
.
.
.
.
.
po l y n o m i a l s
—
—
2 x + 2 y :cy xz + yz
ar
—a
— 2x
—
+ xy 2 y
y
—
3 a + 2 b + a az a b+ bc
—
—
2 a a b ar
+ bc
bc + 2 a + a c +
-
2c
—
2x a c
E
48
LE M E N TARY AL G EBRA
AD D ITIO N O F
TE RM S
P AR TLY
S IM ILAR
T e rm s th a t a re part l y s im il ar i e simil a r as
m ay b e u n i t e d i nt o o n e te rm
o f t h e l e tt e r s o n l y
n
i
e
f
c
e
h
a
f
i
i
t
T
u
o
l
n
o
m
l
c
o
s
p y
71
.
.
,
.
,
,
.
,
ay
by
the
ax
an
x
2n
'
( a + b) y
(a + l ) w
(
Wri te
the di s si m i l ar pa r ts i n
m
l
oe
c
i
e
n
t
o
t
h
e
s
i
m
i
a
a
i
n
o
i
c
l
r
a
o
l
r
t
y
fi
f
p
p
72
.
t o pa rt
wit h a
Ru l e
.
a
cl
—2
M
p a r en thesi s
as
.
”
“a
d : a plu s b t i m e s y ;
p l us
”
“
”
a m i nus 2 t im es n
1 t im e s x
an d
a s lig h t pa u se in t h e
r e a di n g o c c u rr i n g w h e r e t h e l a s t c u r v e of t h e p ar e n t h es i s
st an d s
ab o
Th e
ve
a n swe r s a r e r e a
,
,
,
,
.
E x e rc i s e 24
R ea d t h e sum s o f t h e fo ll owi n g
1
.
ax
bx
2
.
by
y
3
.
.
S UBTRA C TIO N O F TE R M S
S UB TRA CTIO N O F TE R M S
P ARTLY
S I M ILAR
P AR TLY
49
S I M ILAR
T e rm s part l y Sim il ar i e s i m il a r a s t o pa rt o f t h e
li t e r a l fa c t o rs m ay be s ub t ra c t e d b y in di c at i n g t h e su b
T h us
t r a c t i o n o f t h e di ss i m il ar p a rt s
73
.
,
.
.
,
,
.
,
by
— c
y
bx
(
74
.
—
a
b
( b+ c) y
)x
R ul e
p a r ts in
a
—
(1
Write
.
pa r en the s is
the in dica ted
as a
s u btra ction o
p ol yn omia l
f
r
a
p t
to
1
.
ay
cy
21
.
ha:
2
d t h e r e s ul t s o f t h e fo ll o wi n g
-
.
bx
ax
3
.
4a
4b
the di ss im il a r
.
E x e rc i s e 2 5
a n d re a
)n
fi cien t
coe
O bs e rv e t h a t t h e Sig n o f t h e di ss i m il a r
h e n d is c h a n g e d fro m
to
o r fr o m
Subt r ac t
a
4
.
ax
x
in t h e
s ub t r a
CH AP TE R V
G RAP H I N G FUN CTI O N S
Q UATI O NS
G RA P H I CALL Y
S OL VI NG E
.
IN O N E UNKN O WN
FUN CTIO N S
G RA P H I N G
75
Al g e b rai c
.
Numb e r s ,
Fu n c t io n s
or
.
F o r t h e pr es e n t it
by t h e a id o f o n e o r
v e n i e nt t o c a ll a n u m b e r e x press e d
m o r e l e t t e r s a n a l g e b r aic n u m b e r o r a fu n c tio n o f t he
2—
—
n
r
t
e
b
t
h
e
e
t
er
h
2
d y
l t s T u s x + 3 n 2n 8
n um be s d e o
—
r
n
e
c
a
r
a
lg
e
b
a
i
c
u
m
b
r
fu
n
i
n
t
e
e
s
r
c
a+b x
o
t
o
s
y
is
con
‘
.
The
n
2
,
in
m e an s 5 X 5
,
,
.
,
,
,
n
2
.
— 2n
is
and
— 8 m ean s
,
r ea d
5- s quare
n
>< u a n d
is
r ea d
j us t
n - s qu a r e ,
as
5
2
.
lg eb r a i c n u m b er or fu n c t i o n s u c h a s 3 x + 5
2
n
m
2
n
t
w
m
b
rs
u
s
e
h
h
o
u
ug
o
r
n
e
t
b
t
o
a
b
u
t
v
i
z
h
o
t
t
e
:
(
a l g e b r a i c n u m b e r o r f un c t io n i ts e lf a n d t h e n um b e r x o r n
2
—
T h e n u m b er n
2 n 8 t e ll s
t h a t it d e pe n d s o n fo r it s v a lu e
u s t o f o r m a c o m po u n d n u m b er b y s q u a r i n g s o m e s i m p l e
n u m b e r ( n ) s ub t ra c t i n g t w i c e t h e s i m p l e n u m b e r a n d t h e n
T h e t w o n u m b e rs t o b e t h o ug h t a b o u t a re
s ub t ra c t i n g 8
2—
—
2n 8 i t se lf a n d t h e v a lu e o f n a n d s o fo r
t h e v a lu e of n
T h e n u m b e r x n t o r y in t e rm s
o t h er c o m po u n d n u m b er s
o f w h i c h t h e c o m p o u n d n u m b e r ( t h e fu n c t i o n ) is e x pr esse d
m ay b e c a ll e d t h e in d e pe nd e nt n um b e r
In o t h e r w o r d s t h e v a l u e o f 3 x + 5 d e pe n d s o n w h a t x is
2—
—
2 n 8 d e pe n d s o n t h e v a lu e o f n
a n d t h e v a l u e of n
The
at a n d t h e n a r e t h e i n d e pe n d e n t n u m b e rs
F o r t h e r easo n s j us t s t at e d a n u m b e r e xpr e sse d in t e rm s o f
x s u c h a s 3 x + 5 is c a ll e d a fun c tio n of x a n d is w r i t t e n f ( x)
a n d r ea d : fu n cti on of x
2—
—
m
r
S i il a l y n
2 n 8 o r a n y o th e r n u m b e r e xpr es s e d in
t e rm s o f n m ay b e d e n o t e d b y f ( n ) a n d re a d : fu n ction of n
W it h
e
v e ry
a
,
.
,
,
,
.
’
,
,
.
,
,
.
,
,
,
,
,
.
.
,
.
.
,
,
,
,
.
,
.
,
50
GRA P H I N G F UN C TIO N S
func tio n
A
for
v a l ue
i ts
a n u mber
is
a n o t h e r n um b e r
—
—
n
2n 8 ,
other n u mber
on s o
.
a n um b e r
r i f un c tion is
2
me
tha t d epend s
al g e b a c
An
51
a
is
—
x
+ b,
d
e x pr es s e
y,
et c
in
w h ose
d e pe n d e n c e o n
a lg e b r a i c s ym b o l s a s 3 x + 5
,
,
“
In t h i s b o o k t h e w o r d fu n c t io n m e a n s a l gebr a i c f u n c t i o n
A fu n c t i on t h at d e pen d s o n t wo o t h e r n um b e rs as a + b
is d e n o t e d by f ( a b) a n d r e a d : fu n ction of a a n d b
Th us
—
n
i
n
b
x
d
o
d
s
e
t
e
a
d r e a d : fu n c tion of x a n d y
a l so x
y)
y
y f(
in t h e f un c ti o n s ym b o l d o es n o t m e an m u lti pli
T h e p ar e n t h e s i s
c a ti on b ut is a pa r t o f t h e sym b o l
.
.
.
,
,
.
,
,
,
.
,
.
,
is r e p l a ce d by a po s i t iv e o r
l ett e r w i t h i n t h e
n e g a t iv e a r i t h m e t i c a l n um b e r a s in f (
t h e m ea n i n g is
—
2 is t o b e s ub s t i t ut e d fo r t h e l e t t er in
t h a t t h e n um b e r
T h us
t h e fu n c t i o n
the
If
,
,
,
.
,
then
If
if f ( n )
—
—
2n
n
2
Fi nd f ( 3 ) if
—
Fi nd f ( 4 )
7 6 T wo
.
v e ry
I Knowin g the
.
va l ue o
f
f(
-
2—
=
h
n
2
5
5
8 , t e f ( 5)
—
=
x
x
8
3
f( )
i m po rt a n t
va l ue o
f
f
the u n c ti on ;
—
1,
and
10
.
pr o b l e m s
of
the in de pen de n t
a
lg e b ra
n u m ber ,
to
a re
fi
nd
the
and
’
II Kn owi n g the
.
va l ue o
f
the i nd epen den t nu mber
77
.
the
fu n ction
,
to
fi nd
the
v a l ue
f
o
.
We a l re a d y kn ow
ho w t o
so
lv e Pro b l e m I
.
l t o fi n d t h e v a l ue o f 3 x + 5 fo r x = 4 we h a ve
o n l y t o s ub s t i t u t e 4 fo r x in 3 x + 5 t h u s 3
R e du c i ng
we fi n d 3 x + 5 = 1 7 fo r x = 4 a n d so a l s o fo r a n y o t h e r v a lu e
2—
—
of x
T o fi n d t h e v a l ue o f n
2 n 8 fo r so m e v a l ue o f n
2—
a s 5 w e s ub st i t ut e 5 fo r n t h us 5
2 X5
t o s ee t h at
2—
—
n
2 n 8 7 fo r n 5 a nd s o fo r o t h er v a l ue s o f n
Fo r
exa m p e ,
,
,
,
,
.
,
,
,
,
.
E
52
Thus
ha v e
LE M E N TARY ALG E BRA
w e kn o w t h at to
on l y
fu n cti on
to subs ti tu te the
a nd
to
s ol v e
the fi r s t of the
a bove
ms , we
ro
bl
e
p
f the in dependen t n umber i n
va l ue o
the
fy
s imp l i
.
d pr o b l e m o c c u rs v e ry fr e q u ent l y in a lgeb r a
v iz : T o fi n d t h e v a lu e o f t h e i n d e p e n d e n t n u m b e r w h e n t h e
va l ue of the fun cti on i s known
T h i s is Pr o b l e m I I a b o v e
a n d it is t h e con v er se o f P r o b l e m I
F o r e x a m pl e it is o ft e n
n e c ess a r y t o s o lv e s u c h pr o b l e m s as
78
.
T he
sec o n
,
.
.
,
.
G iv e n
G iv e n
Su c h
3x
n
2
=
5
8 , t o fi nd the
+
l
t o find the
2n
-
v a ue o f x , o r
v al ue o f
n
.
2—
=
2n
3 x + 5 8 an d n
i
a re equa
t io n s , a n d t o s o lv e t h e m m e a n s t o fi n d w h a t v a l u e o r v a lu es
2—
—
f
r
f
m
k
x
e
o x o o n w ill
a e 3 +5
2n 8 e qu al t o
qu al t o 8 o r n
7
C o nse q u e n t l y t o s o lv e t h e se c o n d p r o b l e m s t at e d a b o v e
76 I I ) r e q ui r e s a kn o w l e dg e o f t h e w ays o f so lvi n g e q u a
tio ns
We s h a ll fi rs t Sh ow by m e a ns o f pi c t u res w h at it
m e a n s t o s olv e e qu a tion s
e x pres s o n s as
,
,
.
,
,
,
.
.
Le t it b e ke pt in m i n d t h at al g e
b raic e qu atio n s a r e m a d e u p of
a
lg e b r a i c
n um b e s
r
.
D e pe n d e n c e of an Al g e b raic
N um b e r , or Fu n c tio n
Le t u s fi rs t
t ry t o u n d e r s t a n d t h e r e l a t io n t h a t
e xi s t s b e t w e e n x a n d 3 x + 5
79
.
.
.
D r a w a v e rt i c a l a n d a h o ri z o n t a l
a lg e b r a i c s c a l e ( YY a n d XX ) s o t h a t
t h ey s h a ll b e a t r ig h t a n gl e s w i t h
t h e i r O poi n t s t og e t h e r as s h ow n in
T h i s is q ui c kl y do ne
t h e fig ur e
r
P
s
u
il
c r o ss li n e d
a
e
wit h
p
p p
s h o uld h a v e s o m e p ag e s o f c r o s s
li ne d pa pe r in t he ir n o t e b oo ks
'
’
,
-
,
.
-
.
-
.
G ra ph
of
3x + 5
LE M E N TARY ALG E BRA
E
If
sh ou
ld
fra c t i o n a l
yo u
w ho l e
or
s ub st
i t ut e
po sit i v
a ny
e or ne
ga
v a l ue in 3 x + 5 fo r x a n d l o c a t e
t h e p o i n t p i c t ur e o f t h e r e s u l t i n g
n u m b e r p a i r yo u w o u ld a l w a ys fi n d
t h at t h e po i n t f a ll s o n t h i s s a m e li n e
t iv e
-
-
,
.
1 4,
T ry
Th e
l
2
4
,
et c
.
is t h at 3 x + 5 c o n
i n t o n u m ber—pa irs ,
i
c o n c us o n
n u m b er s
n e et s
w h o se p i c t u r i n g p o i n t s
t h e s a me s tr a i g ht l i n e
l ie
al l
l
a on
g
.
n um ber
A ny
by 3 x + 5
I
n
v
e
g
are
i
is
ca
tio n 3 x + 5
80
—
—
2n
n
2
A
8
v a l ue s
doing
in t h i s
.
ll e d g r a ph in g
fun c
th e
G r a ph
.
P i c tu rin g
.
of
i
p a rs
w e h a v e b een
W h at
se c t o n
of
—
n
2
—
2n 8
Le t
.
m a ke
us n ow
of n
2
— 2n — 8
c
r
e
i
t
u
p
a
of
.
s s um e
1,
n
2,
3,
8,
— 5
,
4,
6,
5,
0,
— 1 —2 —3
,
,
,
t h e n c a l c ul a t e
71
2
Th e
(4
— 2n — 8
n u m b er
— 9 —
,
pai r s
are
—
—
1
8,
5,
+ 7 , + 6,
0,
(1
here
0, + 7, + 1 6
(3
(2
,
.
,
)
— 4
,
and
pa i r
the
n-
v a l ue b e i n g
the
fi rs t
n
um ber
of e ac h
.
Us i n g
c ro s s
-
a
li n e d
D raw
In t h i s
.
a
i
pa r
p a pe r ,
c a re
A , B , C,
a
r
a
b
o
l
a
p
gai n
of
pi c t u re
the
full y fr e e h a n d
and so on
c a se
the
to F
di c ul a r a lg e b r ai c sc a l es o n
n u m b e r pa i rs a s in t h e figu re
p e r pe n
a
and
-
.
s m oo t h
the n t o L,
n u m b e r- pa rs
T h e pa ra b o l a is
c
i
l ie
u rv e
a s sh o wn
l
a on
a n op en c u r
t h ro ug h po i n t s
ve
g
.
a
.
cu rv e ,
ca
ll e d
a
GRA P H I C ALLY
SO LV IN G E Q UATIO N S
m ig h t t a ke fo r
A ny
v a lu e yo u
w o ul d giv e a n u m b e r
t h i s s am e c u r v e
-
i
pa r
n , s ub s t
‘
w h o se po i n t -
i t ut e d
p i c t u re
55
—
in n
2n — 8 ,
2
w o u ld l ie
on
.
T ry
et c
n
.
fun c t io n
8 is t h e n a n um b er l a w
p i c t u res i n t o a par ab ol a
Wh a t w e h a v e j us t b e e n d o i n g in t h i s s e c ti o n Is
z—
—
=
r
hi
n
u
n
2
n
a
f
8
g p
g ( )
—
n
The
2
—
2n
-
,
wh i c h
.
ca
ll e d
.
To m a ke p i c t u r es o f f un c t i o n s w e m e r e l y a s s u me
v a l ue s for x , o r n , e t c , s u b s t i t ut e t h e as s um e d v a l ue s in t h e
81
.
.
—
n
2n
f un c t i o n s ( 3 x + 5 o r
and
c a l c ul a t e t h e s e c o n d
It t h e n r e m a i n s t o pi c t ure
n u m b e rs o f t h e n um b e r p a i rs
t h e n um b e r pa i r s o n a p a i r o f pe r pe n di c u l a r a lg e b r ai c s c a l es
as abo v e
2
-
.
-
,
.
n um b e r o f n u m b e r - p a r s a r e
A ny
gi v e n b y e i t h e r
i
—
—
2n
or n
2
8,
or
by
a ny o t h er
fu n c t i o n
3x + 5
.
f un c t i o n h as s o m e s t ra ig h t o r c ur ve d li n e
n
n
n
i
r
T
h
i
c
l
a
r
u
m
b
e
r
a
i
r
giv
b
f
n
i
n
u
s
c
t
u
e
e
a
r
t
e
a
u
c
t
o
p
p
p
y
y
a l w ay s pi c t u re i n t o po i n t s a l l o f w h i c h l ie o n t h e s a m e s t ra ig h t
o r c ur v e d li n e
H e n ce e v e ry fu n c t i o n h as it s o wn pa r tic u
l a r li n e pi c t u re
E
v e ry
s uc h
-
.
.
,
-
.
Th e
i i g
r s n
fa lli n g o f
fu n c t i o n t h a t
a nd
ge s in t h e
i n d e pe n d e n t n u m b e r
c han
S OLVIN G
82
we re
.
E
,
t he
a re
as x or n
Q UATIO N S
IN
lin e o r c ur v e pi c t ure
pr o d u c e d b y c h a n gi n g
the
.
O NE
UN KN O W N
G RA P H I CALLY
=
l
3
x
8
G
r
h
i
l
u
se
v
i
n
5
a
c
a
l
S
o
So
+
pp
,
g
y
p
r e q ui r e d t o s o lv e t h e e q u a t i o n 3 x + 5 = 8
We
the
.
n ow
t h a t we
.
w o uld
l
l
c a c u at e
so m e
n um b e r - p a rs
t h e pi c t u r i n g po i n t s ( s ee fig ur e in
t h e poi nt s t h e s t ra ig h t li n e
.
i
of
an d
l o c a te
th r o ug h
3x + 5 ,
d r aw
LE M E N TARY ALG E BRA
E
56
So s o o n a s
ra t h er
tw o
we
kn o w t h e
w id e l y
li n e pi c t u r e
Si n c e w e w a n t
to be
li
i u re
se p ar at e d po i n t s a r e
n e- p c t
ig h t li n e
t o giv e t h e
a s tra
s u fli c ie n t
,
-
.
to fin d the
v a l ue o f x
t h at m a kes
3x + 5 = 8 ,
w e m e as u re 8
u n i t s u p o n t h e v e rt i c a l s c a l e a nd d ra w a
T h e l e n gt h
h o r i z o nt a l o ut u n t il it c r o s s es t h e li n e o f 3 x + 5
o f t h i s li n e o r it s e q u a l m eas u r e d a l o n g t h e h o r i z o n t a l s c a l e
T h e l e n g t h is 1 a n d as it e x t e n d s
is t h e r e q ui r e d v a lu e o f x
=
n
i
h
r
z
h
s
t
a
s
c
a
h
T
i
o
i
o
l
ll
d
r
i
l
h
i
1
e
t
e
a
h
h
x
r
t
c
a
to t e
+
g
g
p
s olu t io n o f 3 x + 5 = 8
N o t i c e t h a t w h il e a ny n um b er of n u m b er pa i r s a re giv e n
b y 3 x + 5 o n l y on e o f t h ese n u m b e r p a i rs w ill m ake 3 x + 5 8
,
.
,
,
,
.
.
,
.
-
-
.
,
83
—
S olv in g n
2
.
Si mil a r l y, l et it
G raphi c al l y
2n
.
—
e n
2n
ui r e d t o s o lv
g r a ph i c a ll y
C a l c u l at e s o m e n u m b e r p a i rs b y s u b s t i t u t i n g v a lu e s o f n
a n d d r aw t h e p a r a bol a p i c t u re
fre eh a n d a s in
a s in § 8 0
be
2
re q
.
-
-
,
,
80
,
.
Si n c e w e
a re se e
ki n g
th e
n
2
v a lu e of
n
t h at m a kes
— 2n
d r aw a h o r i z o nt a l t h r o u g h a poi nt 7 u ni t s u p o n t h e
v e rt i c a l s c a l e a n d pr ol o n g t h e h o r i z o n t a l both wa ys u n t il it
c r o ss e s t h e p a r a b o l a
T h e li n e is KE in t h e fig ure o f 8 0
It w ill c ut t h e p a ra b o l a in two p oi n t s
Th e l e n gt h s o f t h e
p a rt s of t h e h o r i z o n t a l b e tw ee n t h e v e r t i c a l sc a l e a n d t h e
c u r v e a r e t h e two v a lu e s o f n t h a t w ill m a ke
we
,
.
.
.
—
n
2
Th e t w o
S
v a lu e s
'
ub s t it ut e e a c h
t h e y m a ke it
t h a t w ill
—
—
2n
n
2
eq
giv e
8
.
ar e n
+ 5,
and n
—
3
.
—
—
2n
t h e t w o v a l u e s In n
of
2
8
a n d se e
if
T h i s s h o w s t h at t h e r e a re t w o v a lu e s
o n e v a lu e 7 fo r t h e a lg e b r a i c n u m b e r
u al 7
th e
=
2n
.
,
S OLVI N G
E
N oti c e t he n t ha t
gi v e n by
n
2
8,
2n
W h il e a n y n um b e r o f n um b e r
o n l y two o f t h e se p a i r s m a ke
n
T h i s m ea n s
t h er e
GRA P H I C ALLY
Q UATIO N S
ar e
2
2n
on
8
ly
2—
—
—
2n
n
2n 8 w here n
figu re
We
of
80
-
p a i rs
are
.
two po i n t s
They
2
E in t h e
7
57
t he
on
a re
g r a ph
t he po i n t s K
of
and
.
h ow t o m a k e pi c t u r es o f n um b e r
2—
—
l aw s s u c h as 3 x + 5 a n d n 2 n 8 , a n d h a v e a l s o s h o w n h o w
h i c a ll y s u c h e q u a t i o n s a s 3 x + 5 = 8 a n d
t o so lv e g r a p
84
.
h ave
—
n
2n
n ow s h o w n
For
2
t io ns t hat
a ny o t h e r
i
c o nt a n on l y on e
n um b e r s
lg e b r a i c
or
e qua
l etter , t h e m e t h o d is t h e
s am e
a
E x er c i s e 2 6
D
1
.
4
.
7
10
13
.
.
li
t he
r aw
n e - p c t u re s o f
i
2x + 5
2
.
2x + 3
5
.
—
3x 1
x
2
x
.
2
8
— 3x — 1 0
— 6x
+8
11
14
t he
fo ll o w i n g f u nc t i o n s
+5
x+ 3
3x + 2
3x + l
x
.
—
2x
.
—
—
2x 3
x
.
—
x
1
2
x
+ 8x + 1 2
—
x
1
2
2
of x
2
6x +5
x
2
— 4x
E x e rc i s e 2 7
So lv e t h e
1
.
4
.
7
10
.
i
2
.
2x + 3
5
.
3x
2
x
— 3x
g ra ph i c all y
e q ua t o n s
2x + 5 = 7
‘
.
fo ll o w i n g
8
11
.
.
x+5
=9
3x + 2 = 8
—
2x 1
x
2
-
2x
:
3
.
6
.
9
12
.
.
x
=
3
5
+
3x + l = 7
2
+ 8 x + 12
2
1=8
x
x
.
L E M E N TARY ALG E BRA
E
58
SUMM ARY
85
of t h i s
Th e w o r k
.
h as t a ug h t t h e
c h a pt e r
fo llow i n g
fac ts
A lg eb r a i c n u m b e rs
fu n c t io n s r e q u i re us t o k e e p in
m i n d t w o n u m b e r s t h e fu n c t i o n i t s e lf a n d a l s o so m e o t h e r
n um b e r as x o r n t h a t it d e p e n d s o n fo r it s v a lu e
1
.
,
or
,
,
,
,
2
tion
3
An
.
pa i rs
Al g e b ra i c
n um b er o r
fun c t i o n is a sh o rt h an d d es c rip
t o c a l c u l a t e it s o wn v a lu e
n um b e r s a s so c i at e n u m b e r s i n t o n u m b e r
lg eb r a i c
t h e way
of
.
a
.
.
.
Th e poi n t -p i c t u r es of t h e n u m b e r-p ai rs o f a n
n u m b e r giv e t h e li n e p i c t u re s of t h e a lg e b r a i c
4
.
-
ca
ll e d t h e g ra ph s of t h e a lg eb r a i c n u m b ers
lg eb ra i c
n u m b e rs
a
,
.
v a lu e of a n a lg e b r a i c fu n c t i o n wh e n t h e v a lue
w e s ub st i t u t e t h e
o f t h e n u m b er it d e p e n d s o n is giv e n
giv en v a lu e an d s im plify
6 T o fi n d t h e v a lu e o f t h e i n d e pe n d e n t n u m b er w h e n t h e
w e m u st
a lg e b r a i c fu n c t i o n is giv e n e q u a l t o a n um b e r
so lv e a n e q u at i o n
7 An e q u at i o n is o n l y a s h o rt h an d wa y of sayi n g a
fun c t i o n is t o h a v e a c e rt a i n v a lu e
5
.
To fi nd t h e
,
.
.
,
.
.
.
lg eb r a i c fu n c t ion m ay furnish a g re at n um b e r
—
n
u
b
f
m
r
u s u a ll y o n l y o n e o r a fe w o f t h e s e p a i r s
o
e p a i rs
f urn i s h a s o l ut i o n of t h e e q u a t io n wh i c h giv es t h e a lg eb r a i c
fu n c t i o n a pa r ti cu l a r v a lu e
A l t h o ug h t h e g ra ph i c a l s o l ut io ns of e q u at i o ns m ake t h e
m ea n i n g o f s o l ut i o n s c l e ar a n d c om pre h en s ibl e e v e n in
m i n u t e d et a il s s t ill t h e y a re m o re t e d i o u s a n d c u m b erso m e
Wh e n it is o n l y t h e r esu l ts o f
t h a n t h e a lg e b r a i c s o lu t i o n s
s o l u t io n s t h a t a r e w a n t e d a n d a f t e r it is l e a r n e d t h a t a lg e
b r a ic so lu t io ns a r e s h o rt e r a n d e a s i e r w ays of r e a c h i n g t h es e
resu l t s we s h a ll us e a lg eb ra i c s o l ut io n s
A lge b r a i c s o lut i o ns a re t re at e d in t h e n e xt c h apt er
8
.
W h il e
an a
,
.
‘
,
,
.
,
,
.
.
C H A PTE R v 1
E
Q UATI O NS
G E NE RAL
.
N
ATIO
S
U
Q
E
u io n is
RE VIE W
t h e b a c kb o n e
of a
lg e b r a It s v a lu e
O t h er
c o n s i st s in it s po w e r as a t o o l fo r s o lvi n g p r o b l em s
a lg e b r a i c t o pi c s a r e n e e d e d t o giv e i n s ig h t i n t o a n d po w e r o v e r
A lg e b ra i c s kill m e a n s a n d a l w ays h as m ea n t
t h e e q u a t io n
In m a t h e
n e a rl y t h e s a m e as s kill in u s i n g t h e e q u at i o n
m a t ic a l h i s t o ry t h e e volut i o n o f t h e e qu at i o n m e a n s t h e
e v o lu t io n of a lg e b r a
8 6 Th e
.
e q at
.
.
.
.
.
w e r e t h e E gypt i a n s
h u n dr e d ye a rs a g o t h e y s a id su c h t h i n g s as ,
Fin d t h e q u a nt it y
it s h a lf a n d t h i r d m a k e 1 9
T h irt y fi v e
A q u a nt i t y
T h ey u se d
n o sym b ol s o r a bb r e vi a t i o n s but t h e l a n g ua g e of wor ds o n l y
A b o ut s i xt een h un d re d yea rs a g o Di oph an t us a G re e k
m a t h e m a t i c i a n w r o t e d o w n t h e i n i t i a l l e t t e r s o f t h e v e rb a l
It w as s i m pl y a s h o rte n e d s e n t e n c e
s en t e n c e as h is e q u a t i o n
Th e
e ar
li e st
a
lg eb ra i s t s
.
-
.
,
.
.
,
,
,
.
.
A
t h o u s a n d ye ars l a t er c a l c u l a t o rs wr o t e d o w n
c a l c u l a t i n g in s ym bol s , m u c h a s a po s t a l c l e r k o f
r
ul es fo r
o ur
d ay
d o w n rul e s fo r c a l c ul a t i n g t h e po st a g e o n parc e l s
F o r e x a m pl e if fo r z o n e 3 t h e po s t a l r u l e
fo r v a ri o u s z o n e s
”
is 6¢ fo r t h e fi rs t po u n d o r f r a c t i o n a n d 2 ¢ fo r e ac h a ddi t io n a l
k
h
f
h
n
h
t
e
a
c
a
e
h
o
d
e
ig
g
o
l
l
n
i
e
w
t
o
t
e
s
a
r
k
t
u
t
c
e
p
p
p
m ig h t w r i t e 2x + 4 in w h i c h x is t h e w eig h t in pou n d s as a
O n w e ig h i n g t h e pa c k a ge h e m igh t
s h o rt f o rm o f t h e r ul e
d o as 2x + 4 s ays i e doubl e t h e n u m b e r o f pou n d s a n d a dd 4
t o g et t h e n u m b e r o f c e n t s t o c h a r g e as p ost a g e
N ow if a t t h e ot h e r e n d o f t h e r o ut e t h e pe r so n s r e ce ivi n g
t h e pa c k a g e h a d n o s c a l e s a n d d es i r e d t o kn o w t h e w e ig h t
o f t h e p ac k a g e k n o w i n g t h e po st a g e t o b e
t h e y m ig h t
m igh t w r i t e
,
.
,
,
,
.
,
.
.
,
.
,
59
ELE M E N TARY ALG E BRA
60
d o wn 2 x + 4 = l 2 a n d fi n d wh at x is if t h e y c o uld
so lv e t h e e q u a t i o n
A g a i n if a m a n st art s 5 mi l es fro m h is h o m e an d w a l ks
a w a y fr o m it x m il es a n h ou r fo r 2 h ou r s t h e r
ul e fo r fi n di n g
h is di st a n c e f r o m h o m e w o uld b e 2 x + 5
S u ppo se h e did
n o t kn o w h is ra t e b ut did kn ow h o w fa r h e w a s f r o m h o m e
T o fi n d h is r a t e h e m ig h t w ri te 2 x + 5 = l 3
s a y 1 3 m il e s
a n d if h e k n e w h o w t o s o lv e t h e e q u a t io n h e c o uld fi n d h is
r a te x o f w a l kin g
At a l a t e r d at e m e n c a m e t o r e g a r d Su c h fo rm s as 2 x + 4
a n d 2 x + 5 n o t as s h o rt e n e d r ul e s b u t a s t h e r e s u l t s o f fo l
l o wi n g t h e rul es t e as nu mber s T h e n t h ey b e g a n t o
a pp l y t h e l aw s o f n u m b e r t o t h e m t h a t is t h e y b e g a n l e a rn i n g
h o w t o a d d s u b t r a c t m u l t i pl y a n d divid e t h e m a n d a lg e
b r a w a s a r ea li t y
8 7 E q u a t io n s e x pres s e d p a rt l y o r w h o ll y in l ett ers a re
e i t h er i de n tities o r con di tion a l equ a tion s
writ e
,
,
.
,
,
.
,
.
,
,
,
.
,
,
,
.
,
.
.
,
,
,
,
,
,
.
.
.
88
.
An i d e nti ty is
b ers wh i c h m ay b e
re
d uc e d
li ke m em b e rs
s a m e f o rm
wit h
i
a n e q u at on
to the
.
S i gn
Th e
of
n e ed
S ign
d
,
is iden tica l with,
qu a li ty m a y a l s o b e use d in an i d e n tity w h e n th er e is n o
d i s ti n gu i s h t h e n a t ur e o f t h e e qua lit y
to
of e
.
T h us
E 8a + 3 a , a n d
4
a
2
a
5
a
+
+
,
e v i d en t t h a t t h e y ar e t r u e fo r a n y
ax
+c
l
v a ue
+ a x ar e i d e n titi e s , a n d
o f e a c h l e tte r in t h e m
E
of
of
.
s a me v a l u e
c
it is
.
S ub s ti tu tion is t h e pr o c ess
s ym bol i n t o a n e xp r ess io n in p l a c e
90
rea
.
,
The
m em
.
i d e nti ty is
It is
o r i s i d en ti ca l l y equ a l to o r s i m pl y i s
89
or
,
putt in g
an
ot h er
one
n
um b e r
which has the
.
u at i o n is s a id t o b e
s a ti sfi e d b y a n y n u m b e r w h i c h w h e n s u b s t i t ut e d in pl a c e o f
t h e u n kn o wn n u m b er r e d u c es t h e e q u a t io n to an id e nt i t y
91
.
S ati s fyin g
an
E qu a tio n
An
.
eq
,
.
,
of
q ua ti on
The
e
9 fo r
x
,
5x + 3x = 7 2 , is
giv es t h e i d en tit y
,
sa
t is fi ed
45 + 2 7 5 7 2
.
by
x
= 9 fo r t h e
,
b s tit uti o n
su
E
62
LE M E N TARY ALG E BRA
E x e rc i s e 2 9
P e rfo rm
the
n
n
i
u
es
t
io
s
q
1
.
2
.
the
i n di c a t e d
fo ll o wi n g :
(
If
x
3
4
( 5x
.
a n sw e
r
th e
—
( 8x 2 0 y) +
ag e of a
m an , ho w
ol d
is
an
4
o t her
as o l d ?
—
(4 a
1 2) x 3
I f To m h as
.
a nd
3n
+ 8 is t h e prese n t
m an w h o is t wi c e
o pe r a t i o n s
x
d o ll a r s
and
8 b)
+
( 6x + 1 20 )
4
Fr a nk
—
3x 20 ,
-
6
h o w m a ny h a s
F re d w h o h as h a lf a s m a ny a s b ot h t h e o t h ers ?
n
5
.
6
.
—
( 8 x 9y)
If
94
is
o ne n
b e rs m ay
a
,
t hird
m a ny pr o bl e m s , o n e o r bo t h m em
kn o w n a n d a n u nkn o wn n u m b e r T h us ,
s t at e m e nt o f
In t h e
.
(
u m b e r a n d 2 x 1 0 a n o t h e r w h a t is
w h i c h is t w i c e t h e s um o f t h e o t h e r t wo ?
x
um b er
( 2n + 1 5 ) X 4
—
—
3 5 b) I 5
—
5a
i
c o nt a n a
.
—
7x
4 = 8 + 5x
lvi n g it is ne c ess a ry t o h a v e a l l u n kn o wn n u m b e rs
in o ne m e m b e r a n d a l l kn o w n n u m b e rs in t h e o t h e r m e m b e r
B efo r e
so
,
.
I f by t h e
a
ddi t i o n
b ot h m e m b e r s
of
t he
a xio m ,
eq
15 , w e
u at i o n
w i t h o ut
add
+4
a nd
-
u n i t i n g Si m il a r
5x t o
te rm s ,
we h a v e
—
7x
Th e
5x = 8 + 4
l t m igh t h a v e b een o b t a i ne d by s ub t r a c t i ng
fr o m b o t h m e m b e rs of t h e e q u at io n
sa m e r es u
+ 51: an d
-
4
.
T h i s pr o c e ss of c h a n gi n g a t erm fro m o n e m e m b e r
o f a n e q u a t io n t o t h e o t h e r w i t h o u t d e st r o yin g t h e e q u a li t y
is c a ll e d tran s po s i tio n
95
-
.
.
id m e c h an i c a l w ork an d t o i m press upo n t h e mse l v es wh a t
a x i o m is i n v o l v ed in t h i s c h a n g e s t ud en t s Sh o u l d a l wa ys e xp l a i n t h e
wo rk b y t e lli n g w h a t t h e y add to o r s ubtrac t fr om b o t h m e m b ers
To
av o
,
.
Q UATIO N S
E
63
E xer c i s e 30
In l i k e m a n n e r
a pp
s
olv e
an d
l yi n g t h e a ddi t i o n a n d
1
3
5
7
9
11
13
15
17
.
.
.
—
5x
t he
ch e c k
fo ll o wi n g e q u at i o n s
s u b t r a c t io n a x i o m s a l t e r n a t e l y
—
=
3 2 3x 1 6
—
—
=
1 3 6s 2 5
2
9s
4
8 y + 14 = 4 y+ 74
.
—
9n 1 9 =
.
—
—
=
3 2 2 x 7 2 6x
6
44 + 2 n
.
6 b + 1 6 = 3 b+ 2 6
.
—
34 5 b =
—
49
.
—
13 =
93
4s + 2 7
.
—
23
8
10
12
8b
—
=
3x 7 l
7x
O r al
D o t his
if
.
an d
.
.
I f t h er e
4
.
as
5
.
fe et
,
use o f
6
.
Th e
sum o f
n
and n
sum o f
t he ir
.
.
16
.
.
9b+
—
=
1 2 6b + 4 0 b
—
15
—
=
3 x x + 7 5 9x
—
3s s
-
7a + 6
— 4a
88
—
=
1 0 + 9n 8 8 + 2 n 8
— 2x
6x
+2
—
—
—
=
4a
1 5 a 35 2 a
in 2 6 m i n u t e s
.
p a re n t h e s es t h e pr o du c t
n um b e rs , a s
re pr e s e n t
wh i c h
u a re
.
is t h e
the
a
°
8n
and
m
a nd n
a nd z
s um o f
of
t he
s um
.
u n i t s in a n u m b e r
fo u r
,
c o nse c
u t ive
,
od d
l a rg e s t ?
Ho w m a n y s q ua r e fee t
sq
.
.
h u n d re d s , y t e n s ,
r e pr ese n t t h e n u m b e r ?
of
.
,
P ra c ti c e
a re x
W h a t will
u m b e rs
i
ex e rc s e s
.
B h a s ya H o w m a ny would C h a v e
m a ny as A a n d B ?
I n dic at e b y
wh at will
t he
14
differe n c e o f a ny t wo
3
33
lis t o f
A h as x she e p
h e h a d t wi c e
2
n
i
e nt r e
—
—
=
14 4 n u + 3 2
14
18
E xe rc i s e 3 1
1
t he
a re
t h e r e in t h e wa ll s
of a
ro o m
fee t h ig h ?
of
4 m e n is l 0 x ye ars
the
a
g es
1 2 yea rs
g es
ag o ?
.
W h at w as
7
e
If
.
ven
9
pa rt
11
.
p a id
n
If
sm a
d o ll ars
x
h a r n es s
of b o th
an d
other
Th e
I f h e l o st
.
,
34
3
2
rt ?
If
o ne
W h a t is t h e
.
nu m
b er of
r o d s w id e
num
a c r es
in
f a rm
5n
the
a
rectan
,
and
t he
gl e of l an d
.
doll a rs
n
the
tw o
lo s s ?
b e rs is 2 5
l a r g er n um b er ?
two
,
a
follow i n g
u m b e r s is
l a r g e r n um b er
sum o f
Fi n d
n
d o ll ar s
an d
and
1 28 ,
a nd a s t
,
c o s t o f al l
p r obl e m s
o re
.
E quati on s
and e q
th e ir
u at i o n s
di ffe re n c e
is
.
7x
.
pa
r
a
t
?
p
E xpre ss in t w o ways t h e
.
an d c h ec k
Th e
.
o t h er
w h at w as h is
differ e n c e b et w ee n
A h ou se c o st
Solv e
.
a
.
E x er c i s e 3 2 — R ev i e w P r ob l e m s
1
doll ars fo r
4x
A b oy b o u gh t x o r a n g es at m c e nt s api e c e an d so ld t h em
d o ll a rs
4n
a
is 1 6, w h at is t h e
y is 4 5 , w h a t is t h e
.
.
fo r
c o st
of x
Re pres e n t t h e
—
ro d s l ong a nd x
5
14
.
.
ll er n u m b er is
13
an
,
i
.
r e pr es e n t
doll a rs a m a n p a id t wo d e b t s o n e of a doll ars
H ow m u c h di d he h a v e l e ft ?
o f b d o ll a r s
a t n c e n t s a p e ce
12
2n + 2
,
um b er ?
p art
o ne
i nt e g er d o e s
Sh o w wh y
an
R e pr esen t t h e
.
of
.
x
o t h er
A
.
h o r se
10
odd
Fro m
.
the
and
re pr ese n t s
n
or an
8
x
LE M E N TARY ALG E BRA
E
64
3
.
—
=
6s + 1 7 4 5 2 s + 8
Divid e t h e n u m b er 1 84 i nt o t wo parts s o t h at t h e g re a t e r
s h a ll e x c e e d t h e l e ss b y 4 8
4
.
.
.
5
.
9n
7
.
The
t i m es t h e
8
.
— n
+4
su m o f
t wo
sm a
.
ll e r
1 8 + 3x = 40
~
.
n u m b e rs
Fi n d t h e
-
e
x+ 7
is 2 7 0 ,
—
=
3 y+ 1 2 l 6 5 y + 4
and
n um b e rs
t h e ir
di ff ere nc e is 4
.
9
.
7b
— 2b
E Q UATIO NS
10
.
l ar ge
A
—
4n l
.
13
.
On e
6 m il e s
an
w as 1 20
14
.
16
.
t wi c e
.
B
s h are
a
11
17
a nd
f a rm
own a
B
as
65
A
w o rt h
H ow m u c h is B
.
—
=
5 + n 5 5n
’
h as 3 t i m es
as
s sh are ?
12
—
60 3 s =
.
68
u to m o b il e r a n 3 t im e s as f as t as a s ec o n d a n d
T h e s um o f t h e i r r a t e s
h o ur f a s t e r t h a n a t h i r d
Fi n d t h e r a t e o f t h e t h i r d
a
.
.
.
—
1 6 + 5x
15
x
—
=
8 a + 3 0 35 + 7 a 3
.
T h re e t i m es a n u m b e r di m i n i s h e d b y
Fi n d t h e
t h e n u m b e r i n c r e as e d b y 68
.
—
8y
y+ 6
18
5 7 , is
e qua
n um b e r
l
to
.
9n
.
-
19
m
.
A h o rse
th a n t he
o re
A
i ge
a n d c a rr a
i ge
ca rr a
.
c o st
$ 3 8 5 , t h e h o r se
Wh at did t h e h o r se
i g $ 95
c o st n
c os t ?
T h e y t r a v e l t o wa r d e a ch
o t h e r u nt il t h e y m ee t A t r a v e li n g t w i c e as m a n y m il e s as B
?
r
n
m
il
es
did
A
t
a
v
e
l
How m a y
20
.
a nd
B
5 7 m il e s
a re
a pa rt
.
.
,
21
.
t i m es
A h as t w i ce as m a n y a c re s o f l a n d as B
m a ny
as
h o w m a ny
a c r es as
a c r es
A
h a ve
1
.
A
.
c en t s
3
.
If
4
.
t he
a
c ost
.
O ral
ya r d s
.
of
a
h a lf- d o ll ars
Sil k fo r 34 5
a nd
w ill
s um
of a
and
di m i n i s h e d b y
r e pr ese n t
b,
.
Wh a t
w ill
b q u a rt e rs , h o w m a n y
di m i n i s h e d b y
c
.
Th e
y
the
.
s um
um b ers o f wh i c h 8 is t h e sm a ll es t ?
nu m b e r ?
n
P r ac ti c e
2 8 m i nu t e s
x
ac re s ,
pe r ya r d ?
t he
and x ,
Wh a t
ld
B h as t h r ee
H o w m a ny d o ll a r s ?
I n di c a t e
3x
so
m a n h as
h as h e ?
s um o f
li st in
m e r c h a nt
r e pr e se nt
2
i
e nt re
a nd
I f a l l o f t h e m h a v e 2400
a n d B t o g e t h er ?
C
E x e r c i s e 33
D o this
,
of
Of
t h re e Co n sec ut iv e
wh i c h
8
is t h e m iddl e
5
I f t h er e
.
r
r e p es e n t
6
po
a re x
n
60
at
pa per l ya r d s
A farm e r
d o ll a rs
a
a n
c e nts a
h ea d
l on g
b
at
,
w h a t w ill
and
po u n d , w ill pay fo r
c ents a
u m b e r of
fe e t w id e ?
n
w
sq
u are fe e t in
d o ll ar s fo r s h e e p
H o w m a n y did h e s e ll ?
r ec e
.
iv e d
n
po u n d ?
wh i c h h e
x
Fi n d t h e v a lu e o f a b u s h e l s o f a ppl e s
a n d b bu s h e l s o f pe a r s a t n c e n t s a p e e k
9
umb er
u m b er ?
W h a t w ill d e n o t e t h e
.
n
i
n
u
i
t
s
y
a nd
t en s
Ho w m u c h bu tt er ,
.
.
8
y
the
u n d s o f t ea
7
of
LE M E N TARY ALG E BRA
E
66
at
.
m
pi e c e
a
so
ld
at
pe c k
c e nts a
.
10
e
If
.
v e n n um b er ?
11
l ar g er
e
one
is
num b e rs o f
v en
The
.
su
m
b e t we e n t h em
14
Th e
.
t wo
w h a t is t h e sm a ll e r
w ill
wh i c h
Is
is t h e
sm a
n u m b er s
two
u m b e r s is
of
t h r ee
ll est ?
s
is 1 7 5 ,
.
t he
45
a nd
c o n se c
t he
u t iv e
l ar g est ?
the
and
differ e n c e
Fi n d t h e n u m b ers
ge s of 3 b o ys is 6x ye ars
s um o f t h e i r a g es in 8 yea r s ?
t he
re pr es e n t a n
um b er ?
5 t i m e s t h e s m a ll e r
s um o f
w h a t will b e t h e
3
n
n
s um
the
r e prese n t
of
do e s a + 1
a n o dd n u m b e r ?
Wh e n
x,
What
.
13
,
wh en
I f t h e di ffe r e n c e b e t w e e n
.
12
i n t e g er
a r e pr e s e n t s a n
a
.
.
I f t h e y liv e
,
C LEARI N G E QUA TI O NS OF FRA C TI O NS
of
96 Cl e aring
.
t io n s m u st b e
it
c an
be
so
Fr a c tio n s
c h an
lv e d
.
An
g e d so a s t o
O b se rv e t h a t
M ulti plyin g th i f c ti
s
.
ra
by 20 ,
on
a
u a t i o n c o n t a i n i n g fr a c
r e m o v e t h e f r ac t i o n s b e f o r e
eq
m u l tip l e
of
it s d eno m i n a t o r , t h e
M u lti p lyi n g a n y fr a ti o n by a m ul tipl e
pro d u t is a wh o l e n um b er
a n ce ls
o f it s d e n o m i n a to r gi v e s a wh o l e n um b e r , for t h e d e n o m i n a t o r
c
c
.
c
w it h
o ne
fa c t or
of
t h e m ulti p li e r
.
E
97
f
o
.
P rinc ipl e
.
If
any
Q UATIO N S
fr a ction
i ts denomina tor , the pr odu c t i s
98
.
Prob l e m
.
c ea r o f
l
To
x
a
67
i s mu l tipl ied by
w hol e
n u m ber
fr a c t i o n s
x
,
x
.
i
e qu at o n
x
2
M ultiply b t h
t he
mu l ti pl e
a
6
t h i s e qu a ti o n b y 1 2 t h e l e a s t
multipl e o f t h e d en o m i n a t ors b y m u lti pl y in g ea c h t e r m in it
c a n c e l l a ti on t o th e fr a c ti o n a l t er m s an d t h e r e sult is
o
m e m b e rs
of
,
,
,
co mm o n
a pp
l yi n g
,
6x
-
E v er y t e rm in t h i s
w o r k is
ca
ll e d c l e arin g
eq
an
60 + 2x
M
( u lt
.
Ax i o m )
u a t i o n is a w h o l e n um b e r
e qu ation of fr a c tio n s
.
(2)
This
.
In d es c ribi n g
t h i s t ra ns fo r m a ti o n o f a n e qu a ti on s t ud e n t s sh o u ld
t e ll b y w h a t t h ey m u lti p ly b o t h m em b e r s o f t h e e qua ti o n r a th er t h a n
“
th e y s h o ul d s ay : b y t h e
use t h e e xp r e ss i o n cl ea ri ng of fr ac tion s i e
”
u se o f t h e m ulti p li c a ti o n a x i o m
etc
,
,
,
,
,
So lvi n g
eq
.
.
,
.
ua t io n
x
=8
C h ec ki n g in
4
5
or ,
-
1
1
E x er c i s e 34
of
.
3
fr a c t i o n s
g
.
.
g
,
so
lv e
,
c he ck
t he
fo ll o w i n g
LE M E N TARY ALG E BRA
E
68
E x erc i s e 35
S o lv e
a n d c h ec k
A
the
P r ob l e m s
—
an
d E qua tions
fo ll o w i n g
w om an
boug h t sil k a t $ 2 a ya r d a n d h a d 3 1 4 l e ft
a ya r d w o u ld h a v e c os t 34
T wi c e a s m a n y y a r d s a t
m o re t h a n s h e h a d
Fi n d t h e c o st o f t h e s il k b ough t
1
.
.
.
.
3n
4
b er o f yar d s sh e bo u gh t ;
t h e n u m b e r o f d o ll a r s s h e h a d
t h e n u m b er o f d o ll a r s s h e h a d
3n
4
2n
n
1 8,
50
50
Le t
th e n
C h ec k
2
.
14
2n
and
n
the
n um
,
.
14
a nd
c os t
36
A h as tw i c e
x
m any sh e e p
as
B
and
9
3
.
B
and
9
11
the
y
3
9
5
and
x
.
Fi n d t h e
num b e r ,
13
The
d a ugh t e r
’
s
$ 3 50 ,
a
as
3s + 5
3
10
dim es
m o nt h ;
I f h e h as
.
—
6n 9
2n
3
3
A an d
C , 32 1 5 ;
4s + 5
.
i g
c o st o f ea c h
16
7
o n e c ost n
2s
1 5 t im e s
:
as
mu ch
a
n
,
,
.
s um
of
t he
a
g es
of
t
to f
.
m ot h er
an d
di ffe r e n c e b et w ee n t h e i r a g e s
Fi n d t h e m o t h er s a g e
ag e
the
’
.
as
.
u m b e r di m i n i sh e d b y 6 is e q u a l
i n c r ea s e d b y 2 Fi n d t h e n u m b e r
H a lf of
n
r
e
a
s
a
d
,
y
c o st
-
the
.
10
4
Tw o h o rses
.
a l l ea r n ?
x
5
.
$ 200
ea rn
How m u c h d o
.
x
.
.
t o ge t h er
B
C , 323 5
o t he r
12
of
y
y
A
.
l ess t h an C I f
+
“
3
.
35
2
h
m
n
a y 5 c e n t p i e c es
5 A b o y as i as
39 in a l l , h o w m an y c oi n s h a s h e ?
8
Silk )
A?
x
x
3
as
““
°
of
3 -1 8
h ave 63 5 , h o w m a n y h as
al l
2n
.
d augh t e r is 4 8
is f o u r t i m es t h e
LE M E N TARY ALG E BRA
E
G E N E R AL R E VI E W
O ra l
E xerc i s e 3 6
D o t h i s p a g e in 1 5 m i n ut es
1
.
t i m es
2
of
.
W h a t w ill
is t h e
x
.
.
.
ro o m
6
.
.
.
l arge
9
4
c
o n s e c u t iv e nu m b e r s
A m a n s c a pi t a l d o ub l e d fo r 3 s u c c e ssiv e ye ars
W h at is t h e
3x
fe et b y
sq
u are
A
and
r e p res e n t
,
.
b oy h ad
d o ll a rs
If
.
one n
a
A fa rm c o st
a
t i m es
t h e r e in t h e w a ll s
a re
s um o f
5
c o ns e c
.
He
of a
u t iv e n u m b ers
d b d o ll ar s
did h e h a v e l e ft ?
ea r n e
n a n d a n ot h er n
s um o f
w as
one ?
How m uch
is
ag o
f e e t h ig h ?
y
the
d o ll a r s
u m ber
w h a t is t h e
ya r d s
,
w h e n it
fi rst ?
m a n w h o y yea rs
a g e w a s x yea rs ?
fe et
2x
W h a t w ill
at
ag e o f a
b oy wh o se
Ho w m a n y
s pe n t c
8
i n c re a s e d b y 3
l a r ges t ?
wh i c h m is t h e m iddl e
7
b,
’
ag e o f a
5
of
sum o f
t he
r e pre se n t
How m u c h h a d h e
4
of a an d
.
wa s
t he
.
E xpr e s s s ix t i m e s t h e pro du c t
t h e s um o f x a n d y
whi ch
3
R e v ie w
the
3 t i m es
n
as
u m b e r is
t hen
and
4 t i mes
as
u m b e rs ?
m uch
as a
h o use
'
.
I f t h e fa r m
o st 36200 m o re t h a n t h e h o u se w h at did b o t h c o st ?
1 0 I f a fi e ld is x r od s s q u a r e h o w m a n y r o d s o f f e n c e will
b e re q ui r e d to e n c l o se it a n d divid e it i n t o 4 s q u a res ?
1 1 A gi r l h as x q u ar t e r s y di m e s a n d z n i c ke l s
G iv e a n
e x pres s i o n t o d e n o t e h o w m a n y d o ll a rs s h e h as
1 2 W h at will d e n o t e t h e n u m b er o f f ee t in t h e pe r i m e t e r
of a r e c t a n gl e 5x feet lo n g a n d 3 x fe et wid e ?
1 3 A m a n b o ug h t x S h ee p a t a d o ll a r s a h e a d a n d h a d b
d o ll ars l e ft Ho w m u c h m o n ey h a d h e at fi rst ?
1 4 A h o u se c o s t 3 t i m e s a s m u c h as t h e l o t , o n e c o s t i n g
W h a t did b o t h c o s t ?
$ 50 00 l e s s t h a n t h e o t h e r
c
,
.
,
.
,
.
,
.
.
.
.
.
.
G E N E RAL RE VI E W
E xe rc i s e 3 7
So lv e
1
A
.
’
al l
t h e pro b l em s
is t o B
s ag e
is 1 3 2 ye ar s
Fi n d
.
’
of
s as
Writt e n
R ev i ew
t h i s p a g e in 2 0 m i n ut e s
5 to 7,
.
t he ir
a
ge s
.
Le t 5 n = t h e n u m b e r o f ye a rs
and
7 n = th e n u m b e r o f e a r s
y
132
5 n + 7u
s um o f
the
an d
ag e of each
t he
71
in A
’
in B
’
s ag e ,
s ag e
.
Th e pu p il w ill un d e rs t a n d t h a t t h e n u m b e r s o ug h t is
o f n , b u t t h e n um b e rs r e p r e s e n t e d b y 5 n a n d 7 n
not
the
l
v a ue
.
2
B
.
t hei r
a
’
is t o A
s age
ge s is 2 7 ye a rs
’
s4
s a
to 7 ,
Fi n d
.
A
’
t he
and
s age
diff e re n c e b e t we e n
.
I f e ac h m an
Se v e n b o ys a n d 1 2 m e n ea r n $ 2 7 5 a w e ek
e a r n s 4 t i m es as m u c h as e a c h b o y , h o w m u c h d o t h e 7
b o ys ea rn per w ee k ?
3
.
4
.
6 to
.
A
B
h as 3 t i m es
m a ny
as
c ows a s
t h ey w o u ld t h e n h a v e t h e
,
A S h o uld
B ; b ut if
s am e n
u m ber
H o w m a ny
.
T h re e m e n e n g a g e in b u s i n e ss w i t h a c a pi t a l o f
B i n v e s t s h a lf a s m u c h as A a n d $ 2 0 0 m o re t h a n G
m u c h h a ve A a n d B i n v e s t e d ?
6
A
,
.
.
A B
,
,
C,
a nd
h a v e 2 90
D
C h as 1 5 m o re t h a n B ,
m a ny h a v e
A
a nd
an d
s h ee p
D
.
B h as
h a s 1 5 m o re t h a n G
.
.
c o st
.
h o rs e ,
i ge
c arr a
,
a nd
h a r n e ss
$ 95 m o re t h a n t h e h a rn ess ,
t h e h o r se
A
.
Fi n d
Ho w
3 t i m es
.
A
H ow
B?
,
8
.
1 5 m o re t h a n
T h ree m e n ra i s e d 1 684 b us h e l s o f o a t s A ra i s e d
a s m a n y b u s h e l s a s C a n d 1 8 5 b us h e l s m o r e t h a n B
m a ny b us h e l s did B a n d C ra i se ?
7
ll
h a ve bo t h m en ?
c o ws
5
se
th e
c ost
and
c o st
the
Of t h e h o rs e
$3 5 0
.
.
Ho w
T h e h o rs e
i g e $3 5 l e s s t h a n
c a rr a
.
g es a t 3 ¢ a pi ec e a n d h a d 20 ¢ l e ft
At 5 ¢ a pi ec e h e w o u ld h a v e n ee d e d 1 6¢ m o re t o pay for
How m any did h e bu y?
t h em
9
.
b o y b o ug h t
,
.
o ra n
.
1
2
giv e
Q u e s ti o n s
a l gebra i c
exp ress ion
s im i l a r
c a nn ot
w h at
va l ue o
f
mo no mia l ;
te rm ;
;
a n a l g ebra i c ex p r es s ion
m u st 9x + 6y
i
e x pr es s o n
and
9a
and
8 9
a re
Wh at
.
te r ms ;
P r ob l e m s
and
-
5z b e
8a be
a
dd e d in
th e
ar e
6 9
add e d
sam e
th e
of
f a c t o rs
in
ar
be
a
dd e d if
,
6
7
W h at
giv e
a n d show
it s
.
-
Di s t i n gui s h
a
,
f
equ a ti on o
di ff e r
2
( b
b
—
c
12
.
Is t h e
4,
c
—
5
2
fa c t o rs
it hm e t i c ?
In
d?
—
7a
5 b+ 4 c
c on di tion
.
G iv e
t es t
l
2d ,
)
and
— 4 b— c
(
)
.
u b t ra c t i o n pro v e d in
a pp li c a b l e in a lg e b ra ?
—
—
—
5z u a n d 3 y + 6 4 z 2 x
find
,
c o rr e c t n e ss o f s
sam e
ex am p es
.
—
Sub t r a c t 2 z + x 2u + y + 7
P erf o r m
ar
ub t r a c t e d fr o m
— c
How is t h e
m et i c ?
9 8 in
-
4
a
b
c
5
3
+
+
)
(
2
d
+
)
A dd
.
d by
.
—
+ b( b c ) , a n d a ( b + c )
11
.
fa c t or ?
s ub t r a c t e
s
Why
2 c?
id en tity ;
=8
e x pr esse
d f ro m
m u st b e
i
h ow t h e y
v a lu e if
10
s ub t ra c t e
Si m plify 9a
.
btra c t e d
m a nn e r ?
n um b e r ?
a
a re
a c om m on
e x pr es s o n
9a + 6b
D efin e
.
9
t h ey h a v e
whic h
w a y m ig h t it b e
ot h e r
.
8
n um b e r s
Ho w is 4 8
.
wh a t
to
H o w m ay
.
.
ithm eti c ?
b e t w ee n t h e pa r ts o f a n um b e r a n d t h e fa ctors o f it
5
su
p ol y
— 4x — 3
y+ 5 z ?
H ow
.
4
;
F ro m
.
3
E x e rc i s e 3 8
D e fi ne
.
n om i a l
to
LE M E N TARY ALG E BRA
E
72
fro m
th e
s um
of
ar
ith
—
4x 2 y+
.
di ffere n t o pe ra t i o n s o n a n e q u a t i o n s o t h a t
o n e t e r m s h a ll b e t r a n s pose d f ro m e a c h m e m b e r
13
.
two
.
14
.
v a l ue
D e s c rib e fou r o pe r a t i o n s w h i c h c h a n g e t h e form a n d
of t h e m e m b e rs o f a n e q u at i o n b ut n o t t h e i r equ a l ity
,
.
G E N E RAL R E VI E W
15
Wh at
.
9x + 4 y
16
18
an d
.
c
19
20
.
.
.
+ e + 2d
60 ?
5x + 3 x
,
—
s ub t r a c t
r o ot o f a n
a num ber
is
—
2a
Su b t r a c t
.
(
—
a
2)
if
the
be
a
dd e d
to
giv e
i
e qu a t o n
Ho w d o yo u d e t e r m i n e
.
a r oo t o f a n e q u a t o n ?
i
4b + 5
fr o m
and a dd
0,
t he
di ff e r e n c e
to
diffe re n t s t e ps in t h e s o l ut i o n o f a pro b l e m b y
e qu a t i o n
I ll us t r a t e
N a me t he
.
W h a t m us t b e
Ho w
Fr o m
tra c t t he
.
ano t h er ,
—
5a
3 c a n d un ity.
are
3 t erm s t h at
27
um be r
0?
.
t ru e
of
two
e q ua
.
.
from
t e rm
o ne
te rm s t h at
a re
a
r
t
p
s u m o f xx
A dd
—
a a
x
(
— a a — x
(
)
,
—
i
fo rm
to
or
der
i
or
m o r e t erm s
of a
.
a re
p a rt l y
s
i m il ar a dd e d ?
a nd a d d
2 x y + 3 xz
n
d
z
a
y
in
a n e quat o n ?
l i g t wo
I ll ust r at e
l y s i m il a r
sum o f
t he
l
n u m b er e x pr es s o n s
enc os n
.
26
n
l y s im il a r ?
S t a t et h e pr i n c i pl e fo r
po ly n o m i a l in a pa re n t h es i s
25
w h at
,
.
t h a t w e m a y pl a c e t h e m
24
giv
4
z
t
e
o
+
y
4b
gi v e
To
u se o f a n
.
r
t
a
p
a re
s um o f
.
3
Define
w h et h e r
23
—4 x
—
T o w h a t e x p r es s i o n m u st 8 a 4 b+ 9c
—
2b
5a +
t he
to
—
—
—
2 c + 5 d 4 e s ub t r a c t t h e s um o f 3 d 5 e 4 c
—
b
H o w d o yo u
t w o t e rm s
22
id e nt i t y
the
Fro m
.
+ 2b
t he
dd e d
x re pr ese n t ?
17
21
a
—
6x 5
7z ?
In
.
d oe s
—
m ust b e
i
e x pr ess o n
73
-
yz
5 z + 3 xy
-
t he m
a nd
Wr i t e
.
—
3z
2 xz + x y
sub
xx
.
)
( + 3)
and a a
H o w d o yo u pro v e w h et h e r t h e n u m b e r s fo u n d in
s o l vi n g a pr o b l e m s a t i s fy t h e c o n di t i o n s o f t h e pr o b l e m ?
28
.
C H A P T E R VI I
D ATA
S OLVI N G S I M ULTANE O US
E Q UATI O N S G RAP H I CALLY
G RAP HIN G
.
O R AP
99
G r aph in g ,
.
w as
as
i g b y pi c t u re s
r e prese n t n
1 00
Th e
.
di a gr a m s
H IN G
D ATA
ill u s t r a t e d in C h a pt e r V m ea n s
a n d di a g r a m s
,
.
an d
bel ow
i
e x e r c ses
show
how t o
nn
h
c
e
c
w
f
t
se
a
t
o
t
o
t
s
o
re
n
i
u
l
s
l
d
r
a
t
a
t
e
c
w
t
e
um b ers s uc h
p
a s p r i c e s a n d d a t es t e m pe r a t u re s a n d t i m e s e t c
when t he
l a w s c a n n o t b e expre s s e d a s e q u a t i o n s a s w e ll as w h e n t h e y
c a n b e s o ex pr e ss e d
,
'
,
.
,
,
,
.
E x e r c i s e 39
L
a
o
E
1
m
H
'
“
)
u
n
m
p
of
n e w s p a pe r
J a n u ary
,
1 9 1 6, t h e
fr o m Ja n 1 0 t o 1 5 o n a
B o a r d o f T r a d e w e r e giv e n a s in t h e fig ur e
T h e n um b e r s a l o n g t h e h o r i z o n t a l a r e t h e
“m “
d a t es a n d t h o se a l o n g t h e v ert i c a l t h e
W h a t w a s t h e pri c e o f w h e at o n Ja n
p er b u s h e l
w h e at
.
.
0
e
c
a
of
r
p c es
a
e
In
i
S
U
.
S
m
p
I
?5
.o
n
,
,
pr i c es
On Ja n 1 1 ?
10?
2
.
.
.
1 2?
13?
1 5?
14?
On w h a t
.
t w ee n w h a t
d a t e w as t h e pr i c e h ig h es t ? L o w est ?
d a t es did t h e pr i c e c h a n g e m o st ?
Be
Sh a r e fo r
d at e s Ja n 8 1 5 1 9 1 6 o f 20 l e a di n g
s t o c ks o f t h e N e w Y o r k S t o c k E x c h a n g e
Ho w m u c h
w as a s s h o w n In t h e figu re
did t h e pri c e f a ll fr o m Ja n 8 t o Ja n 1 0 ?
B e t w ee n w h a t o t h e r d a t e s did t h e pr i c e
fa ll ? R i se ? W h a t d ay w a s t h e r i se
g re a t e s t ? T h e fa ll g re at e s t ?
3
.
Th e
a
v erage
i
r
ce
p
p er
,
-
,
.
,
,
.
.
9
‘5
74
.
GRA P H I N G
4
w a s th e
Wh a t
.
On Ja n
11?
.
a
v er a g e
O n Ja n
14 ?
.
of
pr i c e
15?
5
J A N JS
D ATA
th ese
h o url y
Th e
.
fr o m
6
a
m
.
.
in
1 8 , 1 9 1 6,
shown
st o c ks
in
Ja n
on
t e m pe r a t u r e s
to 6 p m
of
.
Ja n
C h i c a g o , w er e
t he
.
figur e
.
as
O b se r v e
t h e d e g r e e n um b e r s a l o n g t h e
v e rt i c a l
and the
h o ur n u m b e r s
a l o n g t h e h o ri z o nt a l
a n d giv e
.
-
-
,
t h e t e m pera t ur e
m ; at 1 2 m ;
at 6 p m
9
a
.
.
.
6
At
.
6
at
a
at
.
m ; at
2 p m ;
i
.
.
.
.
w h at
h o ur
At
Wh a t
h o u r s h ig h est
t e m pe r a t u r e
Wh e n doe s
th e
w as
l o w es t o n Ja n
t h e g r a ph S h o w
t e zn p e r a t u r e
?
.
.
18 ?
the
s t a t o n a ry ?
i
g r a p h i n g t e m per a t u r e s t h e lin es c o n n e c ti n g t h e po i n ts t h a t r e pr e
s e n t h o ur ly r e a d i n g s d o
n o t r e pr e se n t t h e t e m p e r a t u r e s
fo r t h e
i n t e rm ed i a t e po i n t s T h e t e m pe r a t ur e w as pr o b a b ly n o t s t a ti o n ary
B ut fro m t he h o ur ly r e a d i n gs it w as a ppa r e n tly
at
a n y ti m e
s t a ti o n ar y
N e v e r t h e l e ss t h e g r a p hs gi v e a g oo d n o ti o n o f t h e g e n e ra l
tr en d o f t h e t e m pe r a t ur e fo r t h e d ay
In
,
.
.
.
,
.
7
m
p
.
The
.
.
h o ur l y t h e rm o m e t er
6
a
.
.
o u rs
R e ad i n g
6
7
+2
2
8
,
O
.
M
M
.
9,
10,
1 1,
1 2,
li n e is
giv e n in t h e fig ure W h e n w a s
it c o ld es t ? Wa rm e s t ? W h e n g r o w
in g c o l d e r ?
Warm er ? When
s t a t i o n a ry b y t h e g ra ph ?
as
.
t
no e a
ft er
P
.
.
M
.
1,
,
Sh o w t h a t t h e t e m pe r a t u r e
See
di n g s fr o m
Ja n 1 7 , 1 9 1 6 , in C h i c a g o , w e r e :
on
A
H
rea
pr o bl e m 6
.
JAN n
m
.
to 6
E
76
L E M E N TARY ALG E BRA
8
T h e h o u rl y t em pe r a t u re
.
fr o m
Ja n
th e
.
c ur v e
6 p 111 Ja n 1 7 t o 6 a In
1 8 , 1 9 1 6, w as a s s h ow n in
.
.
fig ur e
m o m e ter
.
.
W h at
.
di n g
.
th e r
w as t h e
7 p m ? At
8 , 9, 1 0 , a n d 1 1 p m ? A t m id
n ig h t ?
At 3 a m o f t h e 1 8 t h ?
rea
at
.
.
.
.
At
5
a
m
.
At
?
6
a
.
.
.
m ?
.
Fr o m Ja n 1 7 6 p m t o Ja n 1 8 6 a m w h e n w a s
it g rowi n g w a r m er ? C o ld e r ? W h e n st a t i o n a ry b y t h e g r a ph ?
1 0 A c l a s s s t u di e d t h e m o v e
m e n t o f a s n a il by h a vi n g it
c ra wl
a l on g
a
fo o t r ul e
The
in m i n u t e s w a s
o b se r vi n g t i m e
w r i t t en al o n g t h e h o r i z o n t a l
an d
t h e di s t a n c e s c r a w l e d
in
i n c h e s a l o n g a v e rt i c a l givi n g
o m
b zdr g
a p i c t u r e of t h e s n a il s r a t e o f
c r a w li n g a s in t h e fig u re
Ho w
fa r h a d t h e s n a il c r a w l e d t h e fi r s t m i n u t e ? Th e fi rs t 2
9
.
.
.
,
.
.
,
.
.
.
-
.
,
,
,
,
2
g
4
3
,
z
’
,
’
s
l
.
m in ?
In
.
m in ?
4
In
.
6
m in ?
.
In
12
m in ?
Wh at
.
did it c r a w l m o st r a pidl y ? M o s t s l o w l y ?
T h e d a il y g r o w t h s o f a t u li p in i n c h e s w e re :
m i nu te
11
D ay
.
.
.
H e ig h t
14
.
3
35
8
6
84
9
1 04 1 1 5
,
.
,
.
2
3
4
5
6
DA YS
7
8
9 10
12
M a rk o ff t h e d a ys a l o n g a
h o r i z o n t a l a n d t h e g r o wt h s a l o n g
v e rt i c a l s t h ro u g h 1 2 3 e t c
u s i n g a sc a l e of 1 s h o rt s id e t o
1 i n c h a n d d r a w a b r o ke n li n e
c o n n e c t i n g t h e po i n t s
W h a t w as t h e l e as t g r o wt h o n
?
n
Th e g rea t es t g r o wt h ?
a y d ay
,
1
:
-
,
.
,
E
78
17
a
ge s
.
Th e
and
LE M E N TARY ALG E BRA
r
n
e
c
e
t
p
ra t e
being
.
,
30 0 4 0 0
T h e r e is
s
an
a
60 0
B
S
z
ti
v
lg eb r a i c
we t ake
now
ub st it u t e
c a l cu l a te
l
a so
50 0
an a
lg
9 0 0 10 0 0
l
t l
l
$5
i
e x pr ess o n
eb r a c
i
iv e v a l u e s o f x
c o r r es p o n di n g v a l ues
s u c c es s
the
8 0 0
70 0
as e c a e
h or i o n a sp ace
e r c a sp ace
1
g
perc e n t
,
1
If
f o ll o w i n g
$ 1 00 , $ 2 00 , $ 300 , $ 400 $ 5 00 , $ 600 , $ 7 00 , $ 800 , $ 900 , $ 1 000
$ 50
$ 5 $ 1 0 , $ 1 5 , $ 2 0 , $ 2 5 , $ 30 , $ 35 , $ 40 , $ 4 5 ,
5 10 0 2 0 0
.
t he
b as e s :
B as e
P er c e n t a g e
18
g ra ph
5,
,
x = 1,
2,
3,
4,
5,
0,
— 1
,
=2
1
y
3)
4:
5;
6:
1;
0,
of
t h is l aw , t h us ,
=
k
e
li y x + 1 , w e m ay
r ig h t a n d l e f t f r o m 0 , a n d
l aw
of
h
s
t
u
y,
—
5,
etc
.
M a r k o ff t h e x v a l ue s a l o n g t h e
h o r i z o n t a l t o t h e r ig h t if po s i t iv e
M e a s ure
a n d t o t h e l e ft if n e g a t iv e
t h e c o rr e
t o a c o n v e n i e nt s c a l e
s po n d in g y v a l ues o n t h e v e rt i c a l s
u pw a r d if po s i t iv e a n d d o w nw a r d
if n e g a t iv e
C o n n e c t t h e po i n t s
T h i s li n e is t h e g ra ph
w i t h a li n e
-
,
,
.
,
,
-
,
.
.
G r a ph
of
y
=x+ 1
of
y
= x+ 1
.
GRA P H I N G
19
the
G raph
.
a
l aw
lg e b r a i c
iv e v a l u es fo r a:
co rr e s po n di n g v a l u es o f y
D ATA
and
s u c ces s
y
l
9
=
x
l
2
,
by
i g
s ub s t
in
c a c u at n
y$
it ut i n g
x
t he
2
,
.
x=
y
0,
1,
2,
0,
1,
4,
3,
4,
5,
— 3
,
9,
1 6,
25 ,
9,
G raph o f y
S l
1 h ori o t l p
1 v rt i l p
— 5 et c
,
2 5 , et c
1 6,
.
.
= x2
ca e
n a
z
e
M ark t he
20
by
an d
G r a ph t h e
.
l
l
i
a
y
-
O
$
v a u e s o ff o n
3
5
$
4
G r a ph
$
of
o r at
*
s
ig
Th e
n
po s iti v e
i
m e an s
or ne
t h at t h e
ga ti v e
.
'
?
4
5
3
0
1
n u m b er
—2
ab o v e :
—4
— 3
$
4
$
—5
3
3
—
V 25
x
‘
y
2
+y
.
— 2
—
l y
25
— x 2 m e an s th e
e x pr es s o n
a n d v e r t ic a l s
25 , or y
o f y a n d pl o tt i n g po i n t s a s
2
1
h o ri z o n t a l
l
lg e b ra i c l a w cc
l
1
10
ace
a ce
s
2
c a c u a t n g v a ue s
a:
y
x
ca
s
s
qu ar e
c a l c ula t e d
fo r
‘
roo
t
V 25 —
2
—
23
o f 25
:c
2
.
m ay b e
,
etc
.
0,
e tc
.
E
80
Fr o m
LE M E N TARY ALG E BRA
v e pr o b l e m s it is s e e n t h a t a g ro u p o f
fa c t s e x pr e s se d b y t wo diff e r e nt s e t s o f c o n n e c t e d n u m b e rs
li ke d a t es a n d prl c e s t i m e s a n d t e m pe r a t ur es a g es a n d
h e ig h t s x v a l u es a n d y v a l u e s in a n e q u at i o n m a y b e
T h i s is g en e r all y d o n e b y m e as ur i n g
pi c t u r e d o r g r a ph e d
o ff t h e n u m b e r s o f o n e se t h o r i z o n t a ll y a n d o f t h e o t h e r s e t
v e rt i c a ll y l o c a t i n g po i n t s a n d t h e n c o n n e c t i n g t h e po i n t s
101
.
the
abo
,
,
,
-
-
,
,
.
,
.
,
,
1 02
Pro b l e m s
.
.
and
1 8 , 1 9,
20 h av e
show n
t he
fo ll o w i n g
i m po rt a nt f ac t s :
i gl e e q u a t i o n in t w o u n kn o wn s is s at i s fi e d b y ma ny
s
f
f
h
n
k
n
n
a
i
r
v
a
l
u
es
t
e
u
o
w
s
o
o
p
2 B y m e a su r i n g o ff x—v a l u e s h o r i z o nt a ll y a n d y v a l u es
v e rt i c a ll y t o s u it a b l e s c a l es l o c at i n g po i n t s a n d c o n n e c t i n g
t h e m e qu at i o n s m a y giv e e i t h er s t ra ig h t o r c u r v e d li n e
g r a ph s o r pi c t u res
3 E v e r y p a l r o f v a l ues o f x a n d y t h a t s a t i s fi es a giv e n
e q u at i o n giv es a po i n t p i c t ur e t h at li es o n o n e a n d t h e s am e
li n e o r c ur v e
4 It is e a sy t o s ee t h at t h e x a n d y di st a n c e o f a ny
r
m
h
h
n
r
r
n
n
u
i
h
v
f
f
li
ld
n
n
e
c
u
r
e
o
t
e
c
o
s
e
e
e
e
c
e
e
s
w
o
o
t
o
t
p
if sub st i t u t e d sa tisfy t h e e q u a t i o n t h at g a v e t h e g raph
1
.
A
s n
.
-
.
,
,
.
,
.
-
.
'
-
.
,
,
.
,
In p r ob l e m 1 8 t h e g r a ph o f y = r + 1 w as f o u n d t o b e
T h i s c o uld b e s h ow n by s t re t c h i n g a s t ri n g
s tr a ig ht l i n e
1 03
a
.
.
l
a on
g
whi ch
the
ro w o f
A n y equ a tion in two u n kn own s in
*
—
ha s the expon en t 1 ( a s 3 r
2 y 1 ) giies a
Kn o w i n g t h i s , it is e a sy t o d r a w g r a ph s
i
n
o
ts
p
ea ch u n kn own
—
s tra i ght l i n e g r a p h
.
.
i
'
v a l u es o f x
c a l c ul at i n g t h e c o rr es p o n di n g two v a l ue s fo r y lo c a t i n g t h e
t wo p oi n t s a n d d r a w i n g a s t r a ig h t li n e t h r o ug h t h e t w o
n
h
r
r
i
i
l
s
w
t
a
u
e
o
t
p
su c h
of
eq
b y m er e l y
u at i o n s
c l oo s in g
two
,
,
,
.
W it h
li k e x 23 y y t h e s m a ll n um b er wr itten ( o r
un d er s t oo d ) a t t h e r igh t a n d a b o v e t h e l e tt e r is c a ll e d a n e xpon e n t
W he n no n um b er is wr itt e n as w it h x o r n o r y 1 is und ers to o d t o
‘
‘
1
b e t h e e xpo n e n t j u s t a s t h o u gh t h e wr itt e n f o r m s we r e 23 o r n o r y
*
n um b er s
2
,
2
,
,
,
.
,
,
,
,
,
,
,
,
.
GRA P H I N G
A
D ATA
l ul a t e d
t h i r d po i nt m a y w e ll b e
c he ck on the wor k
ca c
81
a nd
l o c at e d
as
a
.
It is b e s t
it is
diffi c u l t
i
n
ts
o
p
v a l u e s o f a: t o o n ear t o g e t h e r a s
a c c u r at e l y t h r o u g h t w o v e r y n ea r
t o t a ke t h e
n ot
dr aw
to
a
li n e
,
.
Lin e ar E q ua tio n s Si n c e e q u at i o n s in t w o u n kn o w n s
b o t h w i t h e x p o n e n t 1 h a v e s t r a ig h t li n e g ra ph s t h e y a r e
c o m m o n l y c a ll e d l in e ar e qua tio n s
1 04
.
.
-
,
,
.
1
.
li n ea r
G r a ph t h e
i
e quat o n
3x
x=0
T a ke
te
c o m pu
+ 3,
+ 4,
,
y
n u m b e r- p a r s
i
The
-
2,
and
—3
fo r t h e po i n t s
ar e
Q
wri t t e n t h u s :
<3
<0 ,
fi rst
be i ng t h e
th e
2’
-
,
n um b e r
l
x- v a u e
t he
in
r
n
h
1
a
e
t
es
s
p
A
frgrfgrgss
ph
G raph
3x _ 2y = 1
of
.
fi rs t tw o po i nt s ( 0
and (3
as a t A
and B
d r aw a li n e t h ro ugh t h e m w i t h a r ul e r a n d t es t
—
li e s o n t h e li n e as a t C
2
w h et h er t h e po i n t
G r a ph t h e
,
,
,
,
,
2
.
In
1
4
.
.
7
.
1 05
.
a s
i m il a r
way
=x— 2
y
y
=4—
g ra ph e a c h o f t h e fo ll o wi n g
5
x + 2y = 6
We
,
2
x
.
8
.
.
.
y
=
—
x
4
= 2x — 1
y
—
2x
h av e
y
j u s t s ee n t h a t
u n k no w n s is s a t i s fi e d b y ma n y
B ut two li n ea r e q u a t i o n s in t w o
=4
3
.
6
.
9
.
y
eq
u at i o ns :
= 2x
= 2r
y
—
3x 4
+3
~
y
=4
li n ea r e q u a t i o n in t w o
p a i rs o f v a l ues o f a: a n d y
u n kn o w n s s u c h as
o ne
.
,
2x+ y = 7
—
2y x = 4
c an
bot h b e
v a l ue s o f x
i fied
s at s
and
y
.
at
the
sam e
t i m e by
on l y one
r
f
a
i
o
p
E
82
e xam p e ,
l
Fo r
LE M E N TARY ALG E BRA
g ra ph
2x + y
i g
givi n g
x=
us n
an d
g ra ph
y
=
7,
—
1,
+ 3 , a nd
+ 1,
+ 5, + 1,
and
+9
—
2 y sc = 4 ,
u i g
givi n g
s n
y
+ 4,
0,
a:
=
—
and
3,
+ 2 , + 4, and
N ow , we
giv
s a t i s fy
to
e x
Th e
.
ask, c an a
po i n t l ie
both e q u at i o n s ?
a n s w e r is yes
Th e
.
is fi e s
t he
i
P,
po n t ,
P,
,
re q
x
=
so a s
—di s t a n c e s t h a t w ill
y
and
i nt e rs e c t i o n
these
( Se e figu re )
of
t he
ui r e m e n t
+2
and
po i nt
g r a ph s
.
y
=
of
sat
Fo r
the
+ 3,
a nd
i fy b o t h e q u a t i o n s
H e n c e t h e x a n d y di st a n c es of th e
p o i n t o f i n t e r s ec t io n o f t h e g r a ph s
a r e th e g r a ph i c a l s olu tio n of t h e t w o
giv e n li n e a r e q u a t i o n s Si n c e t h e
on e p o i n t t h e r e is on l y on e so lu t i o n
v a l ues
s at s
.
-
,
Si m ult a n eo us Equa ti o n s
I n t er s e c ti n g G ra ph s
.
g r a ph s
of
t he
ly
pa i r o f e q u a t i o n s
c r o ss
at
on
.
10 6 H e n c e , two l in ea r
.
fied by
sa ti s
on l y o ne
a
i
r
p
o
equ a ti n s
f
o
S OLVIN G S I M ULTANE O US
i n two
va l u es
E
f
o
u n kn own s ca n
the u n kn own s
Q UATIO N S
G RA P
be
.
H I CALLY
Sim ul tan e ous E qu atio n s
E q u at i o n s t h at c a n b e
sa t i s fi e d by t h e s a m e v a l u es o f t h e u n kn o w n s a r e c a ll e d
1 07
.
.
s imu
l tan e ou s
10 8
.
It is
u atio n s
eq
n ow
.
w o r t h w h il e t o
s ee
t hat
n ot a l l
i
p a rs
of
in t w o u n kn o w n s c a n b e sa t i s fi e d b y e v e n
i
o n e pa i r o f v a lu e s o f t h e u n k n o w n s
Tw o o r m o r e e q u a t i o n s c o n s id e r e d t o g e t h e r a re s a id t o
f o rm a s ys t e m
li ne a r
e q u at o n s
.
.
S OLVI N G
1
GRA P H I C ALLY
Q UATIO N S
1
s ys t e m
C o n s1 d er t h e
.
E
2
2y
.
6y
.
=4
-
x
—
3x = 6
g r a ph s o f t h e e q u at i o n s a r e s h o w n in t h e fig u r e
—
Dividi n g 2 t h ro ugh b y 3 g i v es 2 y x = 2 a n d t h e g r a p h o n
—
w h i c h t h i s is w r i t t e n is t h e g r a p h o f 6y 3 x = 6
Th e g r a ph s
T h ey
a re a p a i r o f p a ra l l el l i n es
The
.
,
,
.
.
do
n ot
m ee t ,
po i n t t h a t
li e s
on
is
t h er e
an d
no
b o t h g r a ph s
is n o pa i r o f
.
T h i s m ea n s t h e r e
v a l ue s o f x a n d y t h a t
b ot h e qua t i o n s
w ill
i fy
sa t s
X
1
.
1 09
.
I n c on s i s te n t E qu a ti on s
E q ua t i o n s w h i c h
c a n n ot
fi e d b y a ny p a i r
s im ul tan e ou s , o r
of
be
N o n Sim U It a n eo u s E
'
.
P ar a ll e l
i
sat s
v a l ues o f
in c on s i s te nt
the
G r a ph s
u n kn o w n s a r e c a
u atio n s
eq
qua ti o n s
ll e d
n on
.
T h a t t h e e q u at i o n s o f 1 0 8 a r e i n c o n s i s t e n t c a n b e s e e n
w i t h o ut g ra ph i n g b y dividi n g t h e s e c o n d t h r o u g h b y 3
T h i s d o es n o t c h a n g e t h e r e l a t io n b e t w e e n a: a n d y T h e n
—
—
=
4 a n d t he o t he r t ha t 2 y x
o n e e q u a t i o n s ays t h a t 2 y
x
is a t the s a me tim e e q u a l t o 2
T h i s is o b vi o us l y a b su r d
—
h
n
m
T e
u
b e r 2y x c a nno t a t the s a me time b e bo t h 4
.
,
.
,
i
.
,
and
2
,
.
1 10
to be
.
F or
a sys tem o
f
ca pa bl e o
f
111
.
two l i n ea r
s ol u tio n ,
the
equ a tion s
e qu a tio n s
in two
mu s t be
u n kn own s
mul ta n e o u s
si
.
D e pe nd e n t E qua tio n s It is h o w e v er n o t s ufitcien t
t h a t t h e e q u a t i o n s b e s i m ul t a n eo us
We s h a ll n ow se e t h a t
t w o li n e a r e q u a t i o n s in t w o u n k n o w n s c a n f a il t o giv e a
defin ite so lu t i o n b e c a u s e t h e y h a v e too m a n y s o l u t i o n s
.
.
,
,
.
.
1
.
C o n s id er t h e syst e m ,
E
84
B o t h g r a ph s
i
co nc
id e
E
.
L EM E N TARY ALG E BRA
a r e s h ow n
v er y
in t h e
figu r e
po i n t t h at is
Hence
on
y that
is
one
any
,
i gl e li n e
T hey
o t h er a l so
as a s n
pair
i fi es
t he
on
of
.
o f a: a n d
v a l ues
o ne of
.
i
Dividi n g
s a t i s fi es t h e o t h er a l s o
t h e s e c o n d e q u at i o n t h r o u g h b y 3
—
giv e s 2 y x = 5 w h i c h is i den ti ca l
w i t h eq u at i o n 1
O n e e q u a t i o n de
s at s
the
e qu at o n s ,
.
,
,
X
.
d
u
e
n
s
o
p
qua ti o n s
C o i n c i d en t G r a ph s
D e p e n d en t E
o ne
by
ca n
s
t he
d 1 v 1 sio n b y
.
c a l n u m b er
Su c h
1 12
i
e q uat o n s a r e c a
Fi n a ll y for
.
,
u n kn own s
to be
a
ll e d
s ys te
ca pa bl e o
f
s imul ta n e o us a n d
f
o
the
or
.
equ a ti on s
equ a ti on s
i n two
mu s t be both
.
So lv e t h e
no
u a tio n s
ind epe nd e n t
E x e r c i s e 4 0 — G r aph i c a l
is
a n a rit h m e t i
eq
two l i n ea r
s o l u ti on ,
o t h er
.
d e pe n d e nt
m
t h at
s e n se
d e r iv e d f r o m t h e
be
i m pl e
in t h e
o t h er
fo ll o wi n g
d e fi n i t e so lu t i o n
,
s ys t e m s
S ol uti ons
g r a ph i c a ll y
t e ll w h e t h e r t h e
,
or
sys t e m
in
c as e
t h er e
is i n con si s ten t
dep en den t :
—
x
—
3r
x
+y
3x + 3y = 6
2y = 9
y
—
=
—
3x
3y = 1
—
2x
=2
x
=
+y 5
—
x
5y
=2
y
= 2x — 3
y
5y = 1 5
x + 2y = 14
2x
x
+
3
5y = 1 1
—
=
5x 3 y
3
=
2x + y 1 0
g r a ph i c a l w ay o f s o lvi n g e q u at i o n s m akes t h e m e an
in g o f s o lu t i o n s c l e a r ; b u t t h e a lg eb r ai c w a y of t h e n e x t
c h a pt e r is s h o rt er a n d a s i t c a n b e a pp li e d t o e q u a t i o n s in
3 4 5 a n d e v e n n u nk n o w n s it is a l so m u c h m o re g e ne ra ll y
u se fu l t h a n t h e g r a ph i c a l w ay
Th e
,
,
,
,
,
.
E
86
LE M E N TARY ALG E BRA
In d e pe n d e nt e qua tio n s
1 16
.
d er iv e d o n e f ro m t h e
c a t io n o r divi s i o n by
n um b e r
Th e
are
qua tio n s gi v e n
fro m
t he
4x + 3y $ 2 8
1 17
A
.
i n v o lvi n g
By
by
po s i t iv e
,
.
e
d er i v ed
o t h er
a
u at io n s w h i c h c a n n o t b e
a ddi t i o n o f
o r m u l t i pli
o r n e g at iv e a r i t h m e t i c a l
are e q
n um be rs
th er
t wo
in
or
,
.
2r + 3 y = l 4
an d
s ys te m
s et
a
o
b ov e ar e i n d e pe n d en t for o n e c an n o t b e
b y s i m pl e m ulti pl i c a ti on a nd d i v i s i o n
So a l s o
a
of
m o re
.
u atio n s is t wo o r m o re
u n kn o w n n u m b e r s a s
eq
,
a s ys t e m
u a t io n s
,
roo ts is m e a nt t h e v a lu es
of
eq
of
t he
u nkn o w n
.
b e e n n o t e d ea c h e q u at i o n o f a sys t em whe n t a ke n
It w as n o t e d a l s o t h at o nl y o n e
b y i t s e lf is i n d e t erm i n at e
se t o f r oo t s w ill sa t i s fy t w o i n d e pe n d e n t e q u a t io n s
In t h e
t w o s ys t e m s ab o v e x = 22 a n d y 5 in t h e fi rs t a n d x = 6 a n d
= 8 in t h e se c o n d w ere t h e s e t s o f r o o t s
y
As h as
,
,
.
,
,
,
.
$
,
.
,
.
Si m ul t a n eo u s
in
C h a pt e r VI I
To
so
lv e
u n kn o w n
d e nt
as
1 18
.
.
two
i m pl e
Th ey
s
n u m b ers ,
s in g l e equ a ti on
Th is
s
i
w ill
i m ul t a n e o us
it is
eq
n e c es s a ry
i i g b ut
c o nt a n n
be
n ow
lv e d g ra ph i c a ll y
s o lv e d a lg e b r a i c a ll y
w er e
e q u at o n s
so
.
u at i o n s
to
i i g
c ont a n n
o b ta n
i
d o n e o nl y in c a s e t h e e q u at i o n s
1 12
w e ll a s s im ul t an e o u s ; s e e
c an
t hem
f ro m
n um b e r
on e u n kn o wn
be
a re
t wo
a
.
in d e pe n
.
E l imi n atio n is t h e pro ce s s
of c om b n n
i i g
t wo
or
m ore
i m u l t a n eo u s e q u a t i o n s c o n t a i n i n g t w o o r m o re u n kn o w n
n u m b ers in s u c h a w a y as t o o b t a i n a s i n gl e e q u a t i o n in w h i c h
o n e o f t h e u n kn o w n n u m b e r s d o es n o t a ppe a r
s
.
E
LI M I N ATIO N BY A D D ITIO N OR S UBTRA C TIO N
E LI M I NATIO N B Y AD D ITIO N O R
1 19
.
i
n a t on
87
S UB TRA CTIO N
f o ll o wi n g exa m pl es i n di c a t e
b y a dd i tio n an d b y s ub tr a c t io n
th e m et ho d
Th e
mi
of el i
.
s ys t e m s :
So lv e t h e
x+
y
8
(1)
3x + 3 y = 9
x
= 6
y
2
( )
3 33 + y = 5
-
=
2y = 4
14
2x
7
a:
y
2
m e m ber t o
We s ub tra c t ( 2 ) fro m ( 1 )
We a d d ( 2 ) t o
a n d t h e n fi n d t h e
e li m i n a ti n g x
m em b e r e li m i n a tin g y a n d t h e n
fin d t h e v a l u e o f 23
v a l ue o f y
W e t h en sub s tit u t e t h es e v a l u es in o n e o f t h e e q ua ti o n s o f t h e sys
t e m t h a t g a v e it a n d fin d t h e v a l u e o f t h e o t h e r u n k n o wn n um b e r
,
,
,
.
.
.
,
= 1
y
Fr o m
{
x—
In
(2)
3
.
( 1)
( 2)
=
7
y
exam p e
x
{
Ch e d ‘m g
333 +
3,
1
( )
3x + 2y = 1 7
( 2)
3r + 2y = 2 l
2x + 3y = 1 9
9x + 4 y = 4 3
( 1)
6w+ 4 y = 4 2
6r + 4 y = 3 4
(3 )
6r + 9y = 5 7
exam p e
4
.
1 20
.
a:
by
Rul e
n u mber s
n u m ber
si gn s
e qua tion s ,
f
o
fi rs t
co n v en i en t
B y the mu l tipl ica tion
If the
a nd
i g ( 3 ) fro m
i t is m or e
tha t u n kn own
( 2 ) by 3
an d
( 1 ) by 2
s ub t ra c t n
D eter mi n e
.
of
5y = 1 5
4 , w e m ul t i pl y
l
li m i n at e
tion
°
9x + 4 y = 4 3
In
the
=3 1+ 2
y
( 1)
= 5 ( 2)
,
9
e
=1
giv e n b e l o w w e m ul t i pl y b o t h m e m b e r s
a n d e li m i n a t e y b y s u b t r a c t i n g ( 3 ) f r o m
l
by 2
From
the
to
a xio
m,
whi c h
el imi n a te
,
u n kn own
.
s a me
the terms
the two
§ 1 5 , ma ke the
n
l
c
i
e
ts
fi
coe
i n both equ a tion s
to be e l imi n a ted a re
me m ber to mem ber ; if
from the other
f
o
f
o
.
u n l i ke ,
a dd
a l ike , su btra c t one equ a
member fr om member
.
E
88
LE M E N TARY ALG E BRA
E x er c i s e 4 1
So l v e t h e
fo ll ow i n g
9x + 8 y =
4y
—
6x
eq
u at i ons
,
c h ec
ki n g
s om e of
t h em
12
—1
P RO BLE MS
1 21
.
S ol v ing
wh i c h two
by t h e
or
In a lg e b ra m a ny pro bl e m s in
n u m b e r s a r e t o b e f o u n d c a n b e s o lv e d
Pro b l e m s
m o re
u se of a s n
.
i i g b ut o n e u n kn o w n
num ber
b ut in m a ny pr o b l e m s it is m o re c o n v e n i e n t t o
i nt ro du c e as m a ny u nkn o w n n u m b e rs a s t h e re a re n u m b e rs t o
b e fo u n d
Su c h s ol ut io n s i n volv e a s ys tem of si mu l ta n eous
equ a tions a n d t o m a ke a s o lu t io n po ss i b l e t h er e m u s t b e a s
ma n y i n depen den t equa tion s as t h ere a r e u n kn o w n n u m b e rs
u se d
i gl e
i
e q u a t o n c o nt a n n
,
.
,
.
,
E
90
9
$4
se
A
.
hea d
a
The
.
ar
e
s
y
11
t he
.
In
the
as
12
eq
.
l
did at e
did e a c h
u al
r es t a t
to
fo r $ 390 ,
$6 a h e a d
t h ea g es
se
s om e o f
lli n g
i
ag e
w o uld
is 92 yea r s
’
e x cee d A s
I f B wer e
.
.
5 1 63 m e n
,
of
the
n um
tw o
b e rs is
ll e r
s m a ll e r ?
sm a
.
2 55 ,
g
of
t he
d oe s
the
and
B y h ow m u c h
.
.
,
a
m i xt u re
of
a nd at
a
1 00 b u s h e l s w o r t h 7 2 ¢
H ow m a n y b u sh e l s o f e a c h d o es h e
liv e
ag e
it
r
a
y
E
ig ht
e
a
rs
ag o
y
.
A
m erc h a n t
at
.
A
so
a s o l d as
as ol d as
A
.
a
b ush e l
.
.
A
,
b ut
if
b o th
W h at w a s t h e
4 8 y a r d s o f s il k fo r $ 8 9 , s e lli n g pa rt
How m any
a n d t h e re s t a t $ 2 a ya r d
.
did
he
h a s 1 60 s h ee p in t w o
se
ll ?
fi e ld s
.
I f he
fi rs t fi e ld t o t h e se c o n d h e h a s t h e sa m e
fi e ld H o w m a n y are t h ere in e a c h fi e ld ?
,
.
d ay
ld
ya r d
b e t t er s il k
a
ds of t h e
17
t he
u se ?
B w as 3 t i m e s
8 ye a rs , B w ill b e o n l y t w i c e
?
h
a
r
8 ye s ag o
o f e ac
16
of
.
l a r g er
l a rge r
A m ill er m i xes c o rn w ort h 80 ¢ a b us h e l w i t h oat s w ort h
m a ki n g
15
by 1 6
ag e
.
f
h
t
e
o
5
}
d
A an d B
,
14
at
H ow m a n y s h ee p did h e
.
T w e lv e m e n a n d 6 b o ys e a rn $ 24 a d a y
s am e d a il y w a g es 7 m e n a n d 8 b o y s w o u ld e ar n
How m u c h d o es e a c h m a n e arn per d a y ?
13
t h em
v o t e d fo r t wo c a n did a t es a n d
e l e c t e d h a d a m a j o r i t y o f 567
Ho w m a n y
c a n did at e r e c e iv e ?
s um
n u m b er e x c ee
of
a g e of e ac h
a n e ec t o n
The
sh ee p
h e is , his
th e
c an
v o t es
80
s um o f
Fi n d
.
ld
pr i c e ?
a s ol d
twic e
so
and
a t ea c h
ll
10
is
m an
LE M E N TARY ALG E BRA
t a ke s 1 5
n
fr o m
u m b er in
t he
eac h
C H A P T E R IX
M ULTIPLI CATI O N
1 22
M ul ti pl i c a tio n is t h e
‘
.
as a n a
r
p o c ess
of
dd e n d a c e rt a i n n u m b e r o f t i m es
t aking
on e
n u m b er
.
15
1 23
.
1 24
.
T h e mul ti pl i c an d is t h e
T h e m ul ti pli e r is t h e
T h e pro d u c t is t h e
.
THE
1 26
T a ki n g + 5
.
w hi c h
the
a re
T a ki ng
—1 5
;
4
wh i c h
twi ce
—
‘
r es u
of
lt
A
5)
,
— 20
;
the
.
,
+ 48
.
1 0 ; three t i m es
dd e n d w e h a v e
—
v
m
2
h
e
t
i
es
5
T
u
s
fi
,
.
,
—
7
35,
,
— 45
,
9
sa m e a s
20 ,
.
Fro m
three
,
,
,
“
g a tiv e m ul tipli e r m e a n s
o ppo s i t e q u a li t y f r o m w h a t it
T h e r e f ore
w e r e p os i t iv e
4)
.
dd e n d w e h av e + 1 0 ;
fi ve t i m e s + 2 5 T h u s
ne
—
m ul t i pli c a t i o n
as an a
as a n a
20 ,
—
d en ot e s h o w m a ny
+ 20 ,
5 twice
t he
a re
.
s a m e as
fou r t i m es
-
dd e n d
.
+ 1 5,
-
as an a
S I G N OF T HE P RO D UC T
four t i m es
t im es , + 1 5 ;
t a ke n
wh i c h
n um b e r
t i m es t h e m ul t i p li c a n d is t a k e n
1 25
n um b e r
35 ,
—
5)
— 45
t h a t t h e p r o d u c t is o f t h e
w o u ld b e if t h e m ul t i p li e r
,
—
20
f o re g o i n g e x a m pl es
—
6)
+ 42
—
5)
,
-
—
35
48
+ 40
E
92
L E M E N TARY ALG E BRA
F r o m t h ese r esu
l ts w e m a y d e riv e a l aw of
m ul t i plyi n g pos itive a n d nega tiv e n u m b e rs
s
ig ns fo r
.
1 27
Law
S i gn
.
n u m ber s
i
v
g e
M ul tipli c ation
of
os
i
i
v
e
t
p
a
Like
.
ro
d
u
c
t
a
n
d
u
n
li
k
i
e
s
,
p
gn s
s ig n s
g i ve
f
o
two
a neg a tive
r
o
d
u
c
t
p
.
1 28
.
fa ctors
The p rod u ct
a l l the fa c tor s
Th u s
f
of
o
two
f the
=
2 a >< 3 b 2 3
,
G iv e t h e pr o d u c t s o f t h e
.
3 a:
—
2
ab
2y
3c
3
.
b = 6a b
ci s e 43
fo ll o w i n g
.
as
-
a
~
mus t con ta in
n u m ber s
n u mbers
ea ch o
E x er
1
m ore
or
i
o ra
ll y
4a
.
3n
i
t wi c e as a f a c t o r it is n ot wr i t t e n
2
a n d is r e a d : a s qu a re
a a b ut a
Wh en a t e rm c o nt a i n s a: 3 t i m e s as a f ac t o r it is n o t
3
w ri tt e n xxx b ut 23 a n d is r e a d : a: cube
Whe n
a
t er m
c o nt a ns a
,
.
,
,
,
,
.
,
is a s ym b o l o f n u m b e r w ri t t e n a t t h e
o
r ig h t a n d a li tt l e ab o v e a n o t h er s ym b o l o f n u m b e r t o s h o w
h o w m a ny t i m e s t h e l at t e r is ta ken a s a fa ctor
1 29
.
An
e xp n e nt
.
2a b c
2 3
Th i s is
its
s
igni fi c a ti o n
It m u st b e
t he
a
l
b at
l
=2
~
o nl
a
b b
-
-
-
c
-
C c
y w hen t h e
°
= 2 a bbccc
e x po n e n
1 is
.
l
O b s er v e t h a t
W hi l e
is
a
po s iti v e
i n t e g er
.
d t h a t w h e n n o e xpo n e n t is e x pre sse d
T h u s a br m ea n s
a l w a ys u n d e r s t o o d
re m e m b e r e
e x po n e n t
t
a
5
a
X a X a Xa Xa ,
LE M E N TARY ALG E BRA
E
94
E x e r c i s e 44
G iv e t h e
f o ll o w i n g
2
1 6a x
2
.
3a
.
pro d u c t s :
3
3
x
z
6xy
.
— 5 a 2x
2
3x y
2
ax
T h e r e are t h r ee i m po r t a n t f un d a m e n t a l l a w s o f m ul t i pli c a
c a t i o n w h i c h it w ill b e w e ll t o n o t i c e h er e
T h es e a r e : l a w of or der o r commu ta tive l a w; l a w of gr ou p
.
,
ing ,
1 34
sa me
.
i n wha tever
It is
fo r
e
each
In
O rd e r
of
La w
or d er
m em b er
of
the
f
o
.
s evera l nu mbers
.
e qu a
s am e n
li t y is t h e
um ber
b
-
G roupin g
-
a
.
in wha tev er m a nn er
b
The p rodu ct of s ev era l
they a re g rou ped
d e n o t e s t h at 8 is t o b e m u l t i pli e d b y
h
i
du
l
i
li
d
b
a
t
s
5
8
5
3
3
e
t
8
m
u
t
3
r
c
t
o
)
(
p
y
;
p
.
,
B y t h e l aw
of or
d er
,
,
g e n e ra l
n u m b e rs ,
a
b
-
c=
-
(
a
n u m bers
is
.
8 5 3
T h e re fo r e
.
-
-
In
is the
num b ers ,
of
Law
sa me
The pr odu ct
they a r e u s ed
th i s
a
.
.
dis tri bu tive l a w
vid e nt t h a t
g en er a l
1 35
and
l a w;
o r a s socia tiv e
(b
°
c
)
-
a
=
(a
~
c
) b
-
5
an d
the
M ULTI P LI C ATIO N
136
.
D is tri b utive
The pr odu ct of a pol yno mi a l a n d a
su m of the p r odu c ts obta in e d by
.
a l ge br a ic
m on om ia l i s the
m u l ti pl yi ng
Law
ter m
ea ch
f
o
95
,
the p ol yn omi a l by the m on omia l
.
6
-
In
n u m b e rs ,
ge n er a l
(b + c )
-
a
= ab + a c
T h i s is c a ll e d t h e dist r i b ut i v e l a w b e c au se t h e m u l t i pli e r
is di st r i b u t e d o v e r t h e t e rm s o f t h e m u l t i pli c a n d
,
.
1 37
.
A po we r
is t h e pr o d u c t o b t a i n e d b y t a ki n g
o f t i m e s a s a fa c tor
a n y n um b e r
1 38
.
t a ki n g
.
A s qu ar e
o r s e c on d
,
a n um b e r
twice
as a
po w e r , is t h e p r o d u c t
fa c t o r Th us ,
.
t a ki n g
A c ub e
a n
,
or
2
i
d
by
= 6a -6a = 3 6a z
th ir d po w e r , is t h e p r o d u c t
thr ee t i m es
u m be r
ob t a n e
.
( 6a )
1 39
a n um b e r
as a
fa ct o r
ob t a n e
i
d
by
.
5
-
d f a c t o r is t h e r oot o f t he pow e r a n d t h e
e x po n e n t i n di c a t i n g t h e p o w e r is t h e ex pon en t of the pow er
T h us
T h e p ro du c t is t h e pow er
The
r e pe a t e
,
.
.
,
ex
root
pon n t
e
t
2
3
8
e
o r
— p we
P O WE R S OF M O N O M IA L S
1 40
.
To fi nd
pr o d u c t
of
a
two
po w e r o f a n y n u m b e r
or
( 2a b )
?
4
m o re
2a b
z
e qua
2a b
2
l fa c tors
2a b
?
i m pl y
Thus
is
s
t o fi nd t h e
,
.
4 8
=
2a b
1 6a b
2
ig n s in m ul t i pli c a t i o n
1 2 7 a l l p owers of
r
a
e
r
d
r
n
u
m
b
e
s
a
n
e
w
r
t
n
u
m
b
e
s
i
n
a
t
i
v
e
v
e
n
o
e
s
o
e
i
v
e
os
g
p
f
p
w
rs
r
v
e
o
d
d
o
e
o
n
a
i
v
n
u
b
rs
a
e
e
a
i
v
e
e
m
e
n
o
s
i
t
i
t
e
t
;
p
g
f g
p
re
1 41 R ul e
1
a
i
r
c
e
c
i
e
n
t
t
o
t
h
e
i
a
i
R
se
t
h
e
n
u
m
e
c
l
o
( )
fi
r
r
2
c
tt
b
t
h
e
w
a
e
e
i
r
e
d
o
e
m
u
l
t
i
l
h
x
o
n
e
t
o
e
h
l
u
n
t
e
e
(
)
y
p
q
p y
p
f
exp on en t of the p ower a nd ( 3 ) give the r es ul t the pr oper s ig n
B y t he l aw
of s
,
,
.
.
.
,
,
.
E
96
LE M E N TARY ALG E BRA
E x er c i s e 4 5
G iv e t h e s e
i n di c at e d po w e r s :
MULTIP LYI N G A P OLY N O M I AL B Y A M O N O M IAL
1 42
.
O b s er v e
c a re
full y
3a b
4
2a b + 3 a b
3
2
3
2a b
2
2
2a b
2
4a b
6a b
5
1 43
.
Rul e
m u l tip l ier
as
3
s
M u l tipl y
.
6a b
5
ea ch
i n mu l ti p l i ca ti on
3
2
“
b
3
4a
o
the m u l tip l i ca n d by the
ter m
f
m o n o mi a l s
f
o
.
E x erc i s e 46
M ul t i pl y
1
3
5
7
2
.
3 ax + 4 a
2
.
5x y
3
.
.
3a c
6a
-
3
x
by 3 a
2
3
x
3 xy b y 4 x y
2
3
2
— 4 a 2 c b 5 a 3 c3
y
2
3 2
—
x
7 a x by 3 a x
3
2
4
6
8
.
3a b
.
—
5a n
a
.
6a b
ab
.
5b
z
3
3
c
i’
z
3
2
2
3
by 4 a b
3
-
—
n
2
3
+b
z
4a
3
n
s
by 5 a
2
n
2
+ 3 a b b y 6a b
—
c
3
4b
2
2
c
2
2
by 3 b
2
s
c
2
E
98
L E M E N TARY ALG E BRA
E xe rc i s e 48
M ul t i pl y
1
.
2
3
4
5
6
7
8
9
10
11
1 2x
3
.
—4
+ 2 x by 4 + 5 x
3
2
—
3a 2 + 4a
.
.
2a b
.
—
3a
.
—
4a o
3
3
-
2
—
by 4 a + 3a
2a
3
by 3x
2x + 3 + x
2
2
3b + a
2
2b + 3 c
2
x
—
3a b
b
.
—
3a
3
3
2
—
2b + 4c
by 4 a
2
—
c + 3a c
by 2 a
2
3a + 2c
2
2
—
3a + 2a
2
3
3
3
3
—
—
2 xy
x + 3x y
y
—
3a + 2a
.
3
.
4x
3a
2
-
4x
2
3—
—
4 by 5 3 a
3a
2
4
2
2
+ 2 x y+ 4 x y b y 2 y + y
x
3
I
2
2
3
— 2x - - 3 —
by 2a +
z
3
.
.
— 3x
—a 3
by y
3
+ 3 by
-
2
3 xy + x
2
—
3a a + 4
— 3x — 4 x3 + 1 b 3 x3 + x — 3 x3 — 5
y
P e r fo r m t h e f o ll ow i n g i n di c a t e d m ul t i pli c at i o n s
re s
ul t s i n t o
12
13
14
15
16
*
as
p o s s ibl e :
—x
3 (a
.
2 ( 3 a + x)
.
-
2
(
—
3a
—
—
( a 2 c) ( c 3 a )
.
)
— 2 a c — x2
(
—
5 (a
18
.
)
2
3 c) + 69c
2
2
3x) ( 3 a + 3x )
( 3 a + c) (
2
1 1x
—
2c a
)
2a o
— 1 9x
y
—
.
u n i te
*
.
17
e
as
.
Fi t d cid
rs
fe w t e rm s
a nd
e
h o w m an y
3 y>
—
—
( as syr
t e rm s th er e
ar e
in
ea
éw
w
— zz
c h o f t h e giv e n exer c i s e s
.
M ULTI P LI C ATIO N
E xe rc i s e 49
P e rf o r m
n
u
c
a
o
,
y
1
S pe c ia l
99
P r o d ucts
fo ll o w i n g i n di c a t e d o p e r a t i o n s
u s i n g p e n c il o n l y w h e n n ec essa r y
the
2
.
3
.
4
.
5
.
7
.
8
.
( 7 + x)
3
(a
.
6
a s rap
( 3x + 2 y)
(
(
—
a
b) ( a
3
y
)
3
z—
Z a b+ b )
2
9
.
—
13
(r + y )
15
.
( 2x + 1 )
17
.
19
.
4
21
x
— x
)
— 5 a— 5
)(
)
(a
23
27
— 3
)
— a 5
)(
(8
.
25
2
2
(
-
a
29
)
— 8
31
)
33
.
.
.
.
.
.
35
.
(
37
.
x
s
b)
2
4 2
2
(a
( a + b)
(a
(
3
— 1 a
)(
—
a
x
)
-
1)
2
2
—
( 7 x)
(
—
x
(
—
x
l 2) (
a
)(
( 4
9
04 2 9
( 2x
—
x
3)
—
x
3
—
(4 + y r
3
2
—
l x)
.
11
—
-
2
b)
idl y
as
CH A P T E R X
S I M P LE
E
Q UATI O NS
Th e d e g r e e of a t e rm is
e xp o ne n t s o f t h e li t e r a l f a c t o r s
1 47
.
i n di c at e d by t h e s um o f
th e
.
Th us
2
2
a x
,
is
a
t e rm
t h e fou r th d egr ee
of
.
d e g r ee o f a t e rm i n a n y pa r ti cu l a r l etter is i n di c at e d
b y t h e e x po n e nt o f t h a t l e t t e r in t h e t erm
The
.
Thus
1 48
3
a x
,
is
of t h e
Th e d e g r e e of an
of t h e highes t power
.
d e gre e
b
x
A
.
4x
s econd
o
n
i
n
i
t
of
the
x
.
u n kn o w n is
n um b er
qua ti o n of t h e fi rs t d egr ee
3 is an e q ua ti o n o f t h e s eco n d de gr ee
2
the
.
an e
.
or l in e ar
c l e a r e d a n d s i m plifi e d
o
,
e quati n ,
o
is
fra c tio n a l e qua ti o n is
b e d e t er m i n ed un til it is c l ear ed o f fr a c ti o ns
r ed uc ed t o it s s i m p l es t f or m
W h e t h er
on e
u n kno w n
s im pl e e quati n ,
w h i c h , w h en
degree in
e qua
—
2x a is
5x + 7
149
2
or n o t a
of
is
.
an e q
u atio n
t h e fi rs t d e g r ee
.
i m p l e e qua ti o n c a nn o t
a n d th e r e s u lti n g e q ua ti o n
a s
.
A l so
s
i m pl e
x
—
x
4$
or
li n e ar
eq
2
,
,
2
x
and
3
u a t io n s
2
2x
2
=
+x+5 x +x
( x + 2)
a re
.
T h ese ar e s im pl e e qua ti o ns b e c ause wh e n s i m il ar t erm s
t h e s q ua r e o f t h e u n kn o wn n u m b er d i sa ppe ar s
,
ar e un
it e d
,
.
o o t o f a n e q u at i o n is t h e
pr o c ess of pr o vi n g t h a t t h e r o o t sa t i s fi e s t h e e q u a t i o n
T h i s is d o ne b y s ub st i t u t i n g t h e ro o t fou n d in t h e e q u at io n
a n d a s ce rt a i n i n g W h e t h e r t h e r es ul t is a n i den ti ty
'
1 50
.
Ch e c kin g
or
v e rifying
a
r
.
.
8
5 03 3 3 )
9
—
5x
-
.
sh o u
ld
14
.
m u ch
15
r
e
p
11
5
5
+
1 1x + 9
.
_
x
1
_
or v e r
ifyi n g
the
l ti o n
of a
so u
pr o bl e m , t h e
2_
_
x
8
4x
x
_
l 1x + 5
:
b s tit uti o n
su
.
C
A
.
_
as
im et er
17
5x + 5
h a s t wi c e as m u c h m o n ey
Th e
.
)
A
u m b e r is
16
— x
b e m ad e in t h e pr obl em i tself
The
.
38
)
6)
1 1x
c h e ckin g
In
-
§
4
1
—
4 00 2
”
3 (x
x4-2
10
n
LE M E N TARY ALG E BRA
E
10 2
B
as
,
and
B h a s t wi c e
as
I f al l h a v e 35 95 , h o w m u c h h as C ?
.
sum
68
of
t he t hird ,
l e n gt h of
is 1 44 f e et
m an
the
Fi n d
.
fourt h
n u m b er
and
,
e
ig ht h
of
a
.
gl e is t w i c e it s
Fi n d t h e di m e n si o n s
a r e c t an
.
part s
w id t h ,
an d
t he
.
gave $ 1 2 5
t o h is 5
son s ,
ivi n g 35 m o re t h a n h is n ext
m u c h did t h e o ld e st so n r e c e iv e ?
re c e
of 4 of t h e m
b r o t h er
How
e ac h
n
ou
g
e
r
y
.
J am e s h as as m a ny m a r bl es as Fr a n k I f J am es buys
1 2 0 a n d F r a n k l os es 2 3 J a m es w ill t h e n h a v e 7 m o re t h a n
F r a n k Ho w m a ny h as e a ch ?
18
.
.
,
.
19
Th e
.
e x c ee
ds
s um
of
t wi c e t h e
Se v e n m e n
two
n
u m b er s is
l a r g er b y
20
.
85,
an d
3 t i m es t h e sm a ll e r
Fi n d t h e l a r g e r n u m b e r
.
g re e d t o sh a re e q u a ll y in buyi n g a bo at
b ut as 3 of t h e m w ere u n a b l e t o p ay ea c h of t h e o t h e r s h a d
Fi n d t h e c ost
t o pa y $ 3 0 m o r e t h a n h is o r igi n al sh a r e
o f t h e bo a t
20
.
a
,
,
,
.
.
T h re e m en i n v es t e d $ 94 00 in bu s i n e s s A put in $ 600
m o r e t h a n B a n d C i n v e s t e d $ 200 l es s t h a n A
Ho w m u c h
did A a n d C t og e t h e r i n v est in t h e b u s i n e ss ?
21
.
.
.
,
22
re c e
A farm er s old
.
iv e d
l am b s
.
30
l am b s
an d
60
sh e e p
for 3300
.
He
m u c h per h e a d fo r t h e s h e e p as for t h e
H o w m u c h did h e re c e ive fo r t h e 60 s h e e p ?
twi c e
as
S I M P LE
1 52
prece ded
n um
f
T h us
,
x
So lv e t h e
fo ll o wi n g
,
3 + 2x
—
8x
c h ec k n
i g
—
x
8
—
x
8
4
+ 23
2x
3
2
2
—
2x
—
2x
6
2
4
5
2x — 7
2)
4
2
5
9
2x
5 (x
3x + 8
—
x
1
72
3
3
7
_
4
2
x+
+
them
4x + 5
—
12
5x
+ 2”
5x + l 5
93
3)
0
—
+ 2 (x
—
3
3
4(
3
3
6 +
—
x
—
3 ( x 6)
2
3 (x + 2)
2)
2
3
I f each m an
N i n e b o ys a n d 1 6 m e n e a rn33 65 a w ee k
4 t i m es as m u c h a s e a c h b o y , h o w m u c h d o t h e 9
.
e arn
pe r w e e k ?
A bo y h as
4 tim es
he
v in c u l u m
3x
so m e o f
l
4
.
a
f the
o
P r ob l e m s in O n e Unk n own
an d
3
2x + 2
9
ter m
8
3
9
b o ys
ea ch
o
—
2 x
+l 2
2
E xe rc i s e 5 1 — E qu ati on s
ea rn s
f
s ig n
is
.
x
.
the
fra ction
a
,
-
—
1 2x
8
o
mus t be cha nged , for the fra ction l in e i s
n u mer a tor
the
si gn ,
1 03
r
f
f a c tio ns if
equa tion
an
by the min u s
er a tor
or
cl ea ri ng
In
.
E Q UATIO N S
a nd
as
m a ny 5
in
-
c e nt
w h a t is t h e
di m es
pi e c e s
as
5 - ce n t pi ec e s , a n d h e h a s
di m es Ho w m a ny c o i n s h a s
a nd
.
v a lu e o f e ac h ki n d ?
1 04
E
“
A
LE M E N TARY ALG E BRA
w a l ke d 9 5 m il es in 3
d ays g o i n g 4 mil es m o re t h e
s e c o n d d a y t h a n t h e fi r st a n d 3 m il e s m o r e t h e t h i r d d a
y than
Ho w far did h e g o t h e t h i r d d ay ?
t he se c on d
10
.
,
.
.
11
A is 3 t i m es as ol d a s B
.
as o l d as
B
Fi n d A
.
Le t
a nd
x
= the
w as 5 t i m es
.
n um
’
.
’
.
’
10
,
’
10
3x
s a g e n ow
A
ag o
b er o f year s in B s a g e n o w
t h e n um b er o f y e a r s in A s ag e n o w
t h e n u m b e r o f ye a r s in B s a g e 1 0 ye a r s a g o
t h e n u m b er o f ye a r s in A s a g e 1 0 ye ars a g o
3x
x
’
T e n ye a r s
.
.
—
—
=
10
3x
5 (x l 0 )
12
A is 4 t i m e s as ol d
.
ti m e s
A farm er s old c o rn
.
r ec eiv e d $ 800
an d
whe at ,
,
and
F o r h is
.
fo r h is wh ea t
a nd
o ats
c
ag o
he wa s 7
.
.
37 2 0 ,
F o u r t ee n
oa ts
ea c h
F o r h is
.
a nd oats
o rn
3584 0
ye a r s
he
c o rn a n d
re c e
iv e d
H o w m u c h did h e
.
iv e fo r a l l his gr a i n ?
15
s pe n
A
.
m an
it fo r
of
a
s pe n t
w at c h ,
3
h is m o n ey fo r
of
and
had 31 1 5
l ef t
a
s
ui t
How
.
o f c o t h es ,
l
m u c h did
he
d?
16
Th e
.
s um o f
u m b e rs is 8 2
t h e q u o t i e n t is 5
two
divid e d b y t h e l e ss
Fi n d t h e t w o n u m b ers
,
17
D is
.
B is 3
t h e ir
18
50 ¢
s age
5 ye ars
.
wh ea t h e
i
Fi n d t h e f at h er
.
’
so n , a n d
A
.
14
re c e
his
m a n is 24 y ears o ld er t h a n his s o n
h e w as 3 t i m es as o l d
Fi n d t h e a g e o f
13
ag o
as ol d
as
ye ar s o
a
.
a
6 yea rs
ld er
g es w ill b e
A g ro ce r
po u n
o
n
if t h e g re a t e r is
a n d t h e r e m a i n d er 4
and
.
.
t h a n C ; C is 4 yea r s old er t h a n B ;
A I f t h ey liv e 5 ye a rs , t h e sum o f
ld er
th an
.
1 35 yea rs
.
Fi n d D
m i xe d t ea wo rt h 7 0 ¢
d in s u c h
,
pr o p
1 00 po u n d s w as w o rt h 35 8
w e r e in t h e m i x t u r e ?
o rt i o n s
.
’
s ag e
a
po
.
und
w i t h t e a w ort h
t h a t t h e m i xt u re w e ig h i n g
H o w m a ny p o u n d s
of ea c h
ki nd
E
1 06
LE M E N TARY ALG E BRA
E xe r c i s e 5 2
f o ll o wi n g
So lv e t h e
eq
P r ob l e m s in
S im ultan e ou s
pr o b l e m s
in
s
E qu a ti on s
i m ul t a n e o u s
s
i m pl e
u at i o n s :
1
.
e x ce e
s um
Th e
of
ds
of
t he
two
sm a
Le t
an d
u m b e r s is 85 a n d t h e i r di ffe re n c e
by 8
Fi n d t h e n u m b ers
n
ll e r
,
.
x
= the
y
= the
l a r g er n um b er
s m a ll e r n um b e r
,
x
—
x
an d
2
s e on d
— 8—
y
y
5
e
I f 5 is
v a l ue is
v a lu e is
Cl ear t h i s
.
of
fr ac ti o ns
.
,
.
.
85
+y
qua ti o n c o n t a i n s a fr a c ti o n
t h e n w it h t h e o t h e r e q ua ti o n e li m i n a t e
Th e
c
.
dd e d t o t h e n um era t o r o f a c ert a i n fr a c ti o n it s
a n d if 1 is s ub t r a c t e d f r o m t h e d e n o m i n at o r it s
Fi n d t h e fra c t i o n
a
,
,
3
.
.
t
n u m e r a or ,
Le t
n
th e
an d
d
th e d en o m in a t o r
.
T h r e e t i m es t h e l a rg e r o f t w o n u m b e r s ex c ee d s 3 o f t h e
a n d 3 t i m e s t h e s m a ll er e x c e e d s 3 o f t h e
s m a ll er b y 66
l a rg er by 4 6 Fi n d t h e n um b e r s
3
.
,
.
4
v a l ue
is
3
.
dd e d t o b ot h t e rm s o f a c er t a i n fr ac t i o n it s
is 3; a n d if 4 is s u b t r ac t e d f r o m b o t h t e r m s it s v a lu e
Fi n d t h e f r a c t i o n
I f 3 is
.
.
a
,
,
.
A
m ill e r b ou g h t 5 0
b u s h e l s o f c o r n a n d 4 0 bu s h e l s o f
A t a n o t h e r t i m e h e b o u g h t a t t h e sa m e pri c es
o a t s fo r 364
Ho w
3 8 b u s h e l s o f o a t s a n d 7 0 b us h e l s o f c o r n fo r
m u c h did h e p ay fo r a l l o f t h e c o r n ?
5
.
.
6
at
.
A d ea l er b o ugh t
of
pa yi n g $ 1 2 for
3 fo r
b ut he
o ra n
so
ld
the
re
m ai n d er
H o w m a ny
o ra n
ges
al l
.
at
g es
,
so m e
2 fo r 5 535 a n d s o m e
d o z e n w ere u n s a l a b l e ,
at
Thr ee
30 525 a d o z e n m a ki n g
did h e buy ?
,
a
pro fi t
C H A P T E R XI
D IVI S I O N
Div i s io n is t h e pro cess o f fi n di n g one o f t w o n u m b e rs
t h e i r pr odu ct a nd t h e othe r n um b e r a r e kn o w n
1 53
.
wh e n
.
1 54
sent s
.
of
t h e pro du c t
1 55
.
fa cto r
n um be r
of t h e
T h e quo tie n t is t h e
re pr ese n t s t h e other fa ctor
.
to be
n um b e r s
t h e t wo
Th e d i v i s or is t h e
re pres e n t s one
1 56
n um b e r
T h e d i v id e n d is t h e
divid e d
a n d r e pr e
.
by w h i c h we
divid e
and
d by divi si o n
an d
divid e n d
n um b e r o b t a n e
i
of
t he
divid e n d
Si n c e d i v is i o n is t h e rev ers e o f m ulti p li c a ti o n
d er i v ed fro m t h e pro c e ss o f m u lti p li c a ti o n
,
,
.
the
l
ru e
fo r d i v i s i o n is
.
T h ree t h i ng s m u s t b e d e t erm i n e d : Th e s ign of the qu otien t
t h e coefiicien t t h e exponen t of ea ch l etter
,
.
,
DIVID I N G A M O N O M I AL B Y A M O N O M IAL
1 57
.
Th e S i gn
of
th e
Q uo ti e nt
.
t h e r e f o re
—
3 5 , t h ere fo re
— 35
)
—
t h e r e fo r e
— 35 t h e re fo re
,
1 58
.
S i g n La w
divisor give
a
of
Divi s ion
posi ti v e qu otien t
:
.
—
Like
35 )
s ign s
-
f
o
7
7
divid en d
u n l ike si gn s , a ne ga tiv e
and
i
n
t
u
o
t
e
q
.
LE M E N TARY ALG E BRA
E
1 08
G iv e t h e
fo ll o w i n g q u o t i e n t s
—
—
—
64 >
8>
<96>
<68 )
—9
>
7>
—
8>
—
4>
-
5>
Si n c e 5 a >< 3 x = l 5 a x , t h e re fo re 1 5 a x + 3 x = 5 a
The
ffi
ci en t o
f the quo tien t i s the
divi ded by the coefil cien t of the divis or
coe
fiicien t of the dividend
coe
.
Th e E xpon e n t in th e Q uo tie nt
n
f
h
h
i
h
r
d
u
c
t
e
f
a
c
t
r
i
s
e
o
o
o
o
w
c
t
,
p
1 59
is
a
of
th e
i
To
.
divid e n d
u
n
o
t
e
t
q
of
th e divi s or
Law
.
is t h e
s um o f
the di vi sor i s
the divi den d
e x po n e n t o f
fr o m
of
t h at
of
t h e q uo t i e n t ,
t h e d ividend
E xpo n e n ts for
s ubtr a cted
fr om
th e
e xp
o nent
.
Divi s io n
the
s ubtra ct
E
.
a ch
f the
exp one n t o
exp onen t
s a me
in
l e tter i n
.
t h e r e fo re
S in c e
In
the
,
.
fi nd th e
1 60
Si n c e t h e
divid en d
divi sor th e e xpo n e n t
exp o n e n t s o f divi so r a n d
.
g en e r a l
n um
b er s
a
5
-
a
z=
a
3
,
O b s e rv e th e f o ll o wi n g
m
2a b
4a b
4a b
2
B y t h e l aw
n
u m b er
,
‘
2
o f e xp
ex c e pt
0,
2
= ao
o n e nt s for divi s io n a + a
divid e d by i t s e lf a l so equa l s
3
,
a
0
1
3
1
.
.
M e an in g of E xpo n e nt 0 Si n c e a m a y r e prese n t a ny
n u m b e r it f o ll o w s t h a t a ny n u m b er w i t h a zer o expo nen t is
e qual to 1
Th u s
1 61
.
.
-
,
.
,
t his e qua ti on it is ev i d e n t t h a t any l e tt er w ith a z ero e xpo
n e n t m a y b e o m itte d fr o m a t e r m b e c au s e it s p r ese n c e o n l y m ulti p lie s
t h e r e s t o f th e t e r m b y 1
Fr o m
-
,
.
E
1 10
L E M E N TARY ALG E BRA
D IVID I N G A P O LY N O M IAL
1 63
.
Th e
r
ul e
fo r
dividi n g
d e du c e d fro m t h e
St ud y t h i s
pro c es s
e x a m p e c ar e
l
B Y A P O LY N O M IAL
a
po l y n o m i a l b y
of
m u l ti pli c a t i o n
3
2
2a b
3
.
15b
3
4
a
a
3
+ 4a b
3b
2
— 2a b
+5b
2
2
3 a b + 2 6a b
3 3
3
8 a b + 6a b
3
3
2a b
3
2
3
5 a b + 20a b
— 1 5 b4
5a b + 20a b
—
2
2
2
Arr a n ge t h e d i v i d e n d
r
f
a
o
w
e
s
o
,
p
ol
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at
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r
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igh t
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t h e d i v i d e nd
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t h e a lg e b r a i c s u m o f t h e pr o d u c t s o bt a i n ed b y m u lti p l yi n g t h e d i v i so r
b y t h e s ev e r a l t er m s o f t h e q u o ti e n t
H e n c e w h e n d i v i d en d d i v is or an d q uo ti en t a r e arr a n
ge d w i th
r e f e r e n c e t o t h e d es c e n d i n g p o w ers o f s o m e l e tt e r t h e fi r s t t e r m o f t h e
d i v i d e n d is t h e pr o d u c t o f t h e fir s t t e r m s o f t h e d i v i so r a n d q u o ti e n t
wh e n c e t h e fi r s t t e r m o f th e q uo ti e n t is t h e q uo ti e n t o f t h e fi r s t t er m o f
t h e d i v i d e n d d i v i d e d b y t h e fir s t t er m o f t h e d i v i s or
D i v i d i n g t h e fir s t t e r m o f th e d i v i d en d b y th e fir s t t e r m o f t h e
2
d i v i so r w e h a v e a for t h e fi rs t t e rm o f t h e quo ti e n t
Si n c e t h e d iv i d en d is t h e a lg eb r a i c s u m o f t h e pro d uc ts o bta i n ed
b y m u lti p l y i n g t h e d i v i s o r b y t h e s e v er a l te r m s o f t h e q u o ti e n t if t h e
pr o d u c t o f t h e d i v i so r a n d fi r s t t e r m o f t h e q uo ti e n t is s ubt r a c t e d fro m
t h e d i v i d e n d t h e re m a in d e r w h i c h is a n e w d i v i d en d is t h e pr o d u c t
o f t h e d i v i so r a n d t h e o t h e r t er m s o f t h e qu o ti e n t a n d t h e n e x t t e r m
o f t h e q u o ti e n t is t h e q u o ti e n t o f t h e fir s t t e r m o f t h e r e m a i n d e r d i v i d e d
b y t h e fi r s t t e r m o f t h e d i v i sor
D i v i d in g t h e firs t t er m o f th e r em a i n d er b y t h e fir s t t er m o f t h e
d i v i s or w e h a v e — 2a b for t h e s e c o n d t e r m o f t h e qu o ti e n t
R e pea ti n g th i s pro c e ss un til t h er e is n o r em a in d er we o bt a i n t h e
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19
a
+ 20 x + 7 5 by
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a
.
18
3
x
3
16
17
LE M E N TARY ALG E BRA
3
27
29
31
33
35
37
39
by 2 a + 4
—
36x
3
.
2 7 x + 64 y by 3 33
3
.
2
.
2 5x
3 —
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2
4
2
3
4
2
~
E
1 14
c an
d o t h a t ; b ut
of
d it io n s
F ro m
and
m a t t er
in
e x pr e ss
t h e pr o b l e m ,
t h ese
the
LE M E N TARY ALG E BRA
i
no
al l
b ol s
o
u
c
a
n
f
o
y
the
m a t t er h o w
e x p r ess o n s
f o rm a t i o n
s ym
yo u
of t h e
con
u se l ess t h i s m ay s ee m
w ill se e w h a t n u m b e r s a r e e q u a l
e q u a t i o n w ill b e c o m e a s i m pl e
.
,
.
VI I It is m u c h
m a ll
u m b e rs t h a n
I f t h e n um b ers in a pr o b l e m a re l a r g e
a b ou t l a r g e o n es
o r c o m pli c a t e d o r a r e g e n e r a l n u m b e r s s i m plify t h e pr o b l e m
b y r e pl a c i n g t h e m w i t h s i m p l e a r i t h m e t i c a l n u m b e r s ; t h e n
r er ea d t h e p r o b l e m u s i n g t h e s i m pl e n u m b e r s a n d t ry a g a i n
T o fo r m t h e h abi t of doi n g t h i s will
t o s e n s e t h e m ea n i n g
h e l p yo u g re at l y
VI II Sc h oo l w o r k t h at r e q ui r e s lit t l e o r n o effo rt o n yo ur
n
n
r
r
r
h
r
h
n
a
ill
o
t
i
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se
o
u
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t
i
g s Yo u
a
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t
w
o
p
p
y
s h o uld w e l c o m e so m e t a s ks t h a t t est yo u t o t h e li m i t ; a n d if
r
n
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w
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y
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y p on you rself
y
It is r ui n o u s t o yo ur pr o g r e s s t o re l y o n o t h e r s t o ass i st you
in s o lvi n g yo u r pro b l e m s
.
i
to
e a s er
r e a so n a b o ut s
n
.
,
,
,
,
.
.
.
.
.
,
.
IX
A ppe a l
t o y ou r t e a c h e r fo r ass i st a n c e o n l y a ft e r yo u
h a v e re a ll y d o n e yo u r b est , a n d t h e n as k o nl y for o ne or t wo
h i nt s t o st a rt yo u r ig h t
.
.
P r ob l e m s R e qu irin g
E x e r c i s e 56
So lv e t he
x+ 5
3
7n + 4
5
x
fo ll o wi n g
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6
+
—
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eq
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and
3
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m
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r
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4x
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A P P LI C ATIO N S O F S I M P L E
8
38
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each
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i n c re as e d 5 i nc h es t h e a rea w ould r e m a i n t h e s a m e W h at
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1 18
34
L E M E N TARY ALG E BRA
A t wh a t t i m es be t we e n 5
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a c o c k at r
l
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Th e h an d s
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In t h e fi r s t c a s e , t h e m i n u t e h an d m u s t p as s o v e i 2 5 s pa c es , p l us t
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ag o
he
I f he ha d
.
w o u ld h a v e
eac h
B
two
n u m b e rs
$ 10
cost
A wo m a n bo u g h t
the
Fi n d
t he
1 2 ya r d s
of s
.
is
l e ss
8 ya r d s m o re fo r t h e s a m e m o n e y ,
H o w m uc h did it c o s t ?
a ya r d l e ss
A
so
a nd
ld h a lf
h is
B had t he
fi rs t ?
ea c h a t
is 4 t i m e s t h e
and
,
E ach
.
w h e r e u po n
,
H o w m a ny h a d
.
B
s h e e p as
30 t o
f ro m t h e g re a t e r
re m a i n de rs ar e e qu al
46
m o n e y,
m a ny
t ra c t e d
the
5 ye a r s
,
H ow m a n y di d h e b uy?
.
.
as
.
m a n b o ug h t
bo ug h t 2
m o re
19
’
.
43
Q UATIO N S
.
A
.
E
Fi n d t h e n um b e r w h o s e d o ub l e di mi n i sh e d by 2 3 is
g re a te r t h a n 5 3 as 68 is g re a t e r t h a n t h e n um b e r
is 2 8 yea rs
3 times a s ol d
42
S I M P LE
OF
o t he r
I f 24 is
.
s ub t r a c te
n um b e r s
sub
d f ro m
66,
.
il k b ut if s h e h a d b o ug h t
it wo uld h a v e c o s t 60 ¢
,
.
47
re ce
.
A f a t h e r a n d t wo so n s ea rn $ 222 a m o nth
t he
ivi ng
sa
me w age s
t h e y w o uld t o g e t h e r
I f the
.
r ec e
iv e
Ho w m u c h d o es t h e fa t h e r
48
l e ft
.
m a n b o ug h t
At $ 1 0 5
.
a
p y
A
fo r it
49
.
A fr ui t d e a l e r
5 ea n d t w i c e
t h em
al l
as
3 6¢
at
.
m il e s
an
a nd
a c re
ne e
s o m e o ra n
a nd
do ze n
.
h a d 3 1 0 00
d e d 320 0
m o r e to
h e b uy ?
did
o t h e rs a t
than
,
t he
ges
ra t e o f
m a de
a
r
p o
t he
at
ra t e o f
He
2 fo r
so
ld
How
of
fit
3 fo r
di d h e b uy ?
m a n y o ra n ge s
50
a
a cr es
dou b l e d
t h e i r fa t h e r
m o nt h ?
390
at
b o ug h t
m a ny
pe r
s o ns
w a ges w ere
l y 36 l es s
h e w o u ld h a v e
Ho w m a n y
.
e a rn
l and
a n a cr e ,
on
so n s
’
,
t he two
A pe d e s t ri a n w a l ke d a c e rt a i n di s t a n c e a t t h e ra t e o f 1 %
an
h o ur
a n d re t u rn e
d
9 h o urs , h o w
.
at
He
t he
m any
r es t e
d
2 h o u rs
rat e o f
m il es
did
1
2 1;
at
m il es
?
he wa k
l
t he
an
e nd o f
h o ur
.
his
I f he
j ourn ey
w as
o ut
E
1 20
L E M E N TARY ALG E BRA
E LI M INATIO N B Y
1 65
e
.
The
l imin atio n
S UB S TITUTIO N
follo w i n g ex a m pl e illu st r at es
b y s u b s titutio n :
the
m et h o d
of
T ra n s po s i n g 2 y in
Dividi n g ( 3 ) b y 3
,
S u b s t i t u t i n g in
lu e in ( 1 )
1 66;
.
F rom either
kn own
f
n u mber
,
el imi na te
to
equa tion ,
nd
other
in the
fi
the
f
o
the two un kn own
f tha t unknown n u mber
va l ue for the s a me u n
v al ue o
other equa ti on
.
E xe rc i s e 57
E li m i n a t e by
s
ub s t i t u t io n
4x
6y
6
2x
3y
9
an d s
nu m
.
S u bs titu te this
.
t his
.
con v en ien t
the
( 5)
30
v a lu e of y a n d s ub st i t u t i n g
fi n d t h e v a lu e o f x
D etermine fi r s t whi ch
~
ber s it is m or e
o
we
or
R ul e
i n te r ms
3
we h av e t h e
S o lvi n g
v a
3y
olv e
ELE M E N TARY ALG E BRA
1 22
In 4 yea r s a s u m o f m o n e y a t
$ 7 68 , a nd in 5 years a t t h e s a m e
9
to
s
.
Fi n d t he
A
s um
and
i nv e s t e d
the
r at e
i m pl e i n t e r es t
it
ra t e
.
po
,
.
t o $ 8 00
.
.
u n d o f t e a a n d 5 po u n d s o f
n
r
f
n
h
c
e
s
20
h
er
u
s
ea
i
ig
d
3
o
o
t
a
d
p
%
p
Fi n d t h e pr i c e o f e a c h
w o uld c o s t 36
10
a m o un ts
a m o un t s
f
c of ee
c o st
1 1 pou n d s
32
At
.
f
o f c of ee
.
dd e d t o t h e s um o f t h e t wo digi t s o f a c e rt a i n
n um b e r t h e r e s ul t is 5 t i m es t h e t e n s
digi t a n d if 4 5 is
a dd e d t o t h e n u m b e r i t se lf
t h e digi t s a re i nt e r c h a n g e d
Fi n d t he nu m b e r
11
.
I f 7 is
a
’
,
,
,
.
.
um b e r s is divid e d b y 5 t h e q u o t i e n t
is 2 1 a n d t h e r e m a i n d e r 4 ; a n d if t h e di ffe r e n ce o f t h e n u m b e r s
is divid e d b y 1 0 t h e q u o t i e nt is 6 a n d t h e re m a i n d e r 3
Fi n d t h e n u m b er s
12
.
s um o f
I f the
t wo
n
,
,
.
.
13
.
A m a n pa id 3 1 4 for o ra n ge s
fo r 2 5 ¢
a
and
14
.
r est a t
m ade
a nd
d o ze n
o f ea c h
the
a
,
’
1 4 fo r 2 5 gr
.
of
pr o fi t
b uy i n g
som e
of t h e m
at
12
He s o ld t h em a l l a t 3 0 ¢
H ow m a ny did h e b uy
ki n d ?
I f t h e l a rg e r o f t wo
t h e q uo t i e n t is 6
a nd
u m b e rs is divid e d by t h e s m a ll e r
r e m a i n d er 8 ; b ut if 7 t i m es t h e
l a rge r t h e q u o t i e nt is 1 a n d t h e
n
t he
,
m a ll e r is divid e d b y t h e
,
Fi n d t he n um b e r s
re m a i n d e r 9
s
.
.
u m e r at o r o f a c e rt a i n fra c t i o n is dou b l e d a n d 3
a dd e d t o t h e d e n o m i n a t o r it s v a lu e is 3; if t h e d e n o m i n a t o r
is d o ub l e d a n d 2 a dd e d t o t h e n um e ra t o r it s v a l ue is 3
Fi n d t h e fra c t i o n
15
.
I f the
n
,
,
.
.
of
l a n d we re 2 0 fe et lo ng e r a n d
1 0 fe e t w id e r t h e a re a w o u ld be i n c rease d 3 000 s q u a re fee t ;
b ut if t h e l e n gt h w e re 1 0 f ee t m o re a n d t h e w id t h 3 0 fe e t
l es s t h e a re a wo uld b e di m i n i s h e d 24 00 s qu a re fee t Ho w
m a ny s q ua re fe e t a re t h e re in t h e pl o t ?
16
.
If
a re c t a n
gul a r
plo t
,
,
.
C H A PT E
R X II I
G E NE RAL NUM BE R S
F OR M ULAS
.
.
E
TYP F ORM S
G E N E RAL N UM B E R S
B y c o mm o n us a g e t h e
Re pr e s e ntin g N um b e r s
A ra b i c n um e ra l s o f a r i t h m e t i c a n d t h e l e t t e rs u s e d in a lg e b ra
It m u s t b e r e m e m b e r e d h o w e v e r t h a t
a r e c a ll e d n u mbe rs
a l l n um b e r s ym b o l s a re u se d s i m pl y t o r ep r esen t n u m b ers
1 67
.
.
,
,
.
,
.
Si n c e
l e tt e r s
l ette rs
t h ese
1 68
t h a t m ay
To be
a nd
f
o
to
a ny
a bl e
n um b e
re ad
Si n c e
a
of a ny
.
or
3 xy
l
.
me n ts i n
in
E
con ci s e
n gl i s h
a l gebra i c s ymbol s
is
,
b ers
c
t t h r ee ti m es
re pr e se n s
2 ( a — b)
nu m
b m ay
a and
o r o t h e r n um b e r s ym b o
.
A lso
t wo
s ta te
or
2 ( x — y)
t h e pr od u t o f
m a y r e pr es e n t t w i e t h e
c
.
r e pr es e n t a n y
two
u ne q u a l
n u m b e rs ,
t he
li t y
—
(a
b
+
)
(
a
e x pr esses
The
the
su m o
f
fo ll o wi n g
any
by twi ce the s ma l l er
If
l e tt e r
a l g ebra ic ex pr es si on s
3a b, 30 127,
n um b e r s
d iff er e n c e
r is
ma thema tica l
i mpor taf i ce
l
t wo
e q ua
to
e xa m p e ,
,
a ny n
.
r e pr e s e n t a n y n u m b e r
ex pres s
r
a
t
e
g
Fo r
a re
A g e n e ral
.
u m b e rs
d in a lg e b r a t o re pre se n t
c a ll e d gen er a l n u m ber s
a r e u se
a
a nd
b
i
n
r
c
p
a re a ny
t wo
2b
i pl e
two u ne qua l
n u mber
b)
n u m ber s exceed s
their differen ce
.
n um b e r s o f
w h i c h b is t h e
sm a
w h a t pri nc i pl e
d o es t h i s e q u a li ty expre s s
—
2 (a
2 ( a + b)
b) = 4 b?
i
n
n
t
e
s
h
r
i
n
i
l
e
f
ll
o
w
i
g
id
e
i
t
c
s
d
o
t
e
o
p
p
3
—b = 2 a
1
a
+
)
)
(
-
4
.
W h at
e x pr ess
ll e r
,
E
1 24
LE M E N TARY ALG E BRA
F O R M ULA S
1 69
A form ul a
.
ru l e
in
e q ua
li t y
The
g e n e ra l
n um
i
of a genera l principl e
s ym b ol s a n d in t h e f o r m of
a n e x pr ess o n
b er
,
or
an
.
io n of aform ul a in w o r d s is
o f it a s a di r e c t i o n is a r u l e
e x pr e s s
i
e xpr es s o n
Th e
is
a
n
i
r
i
n
c
l
e
d
p
p ,
a
the
.
ilit y t o e x pr e ss ge n e r a l pr i n c i pl e s a s fo r m ul as a n d
t o r e a d fo rm u l a s a c c u ra t e l y as pr i n c i pl e s a nd r ul e s is of t h e
grea t e st v a lu e t o s t u d e n t s o f a lg eb ra ph ys i c s e t c
T h e t r u t h o f t h e f o ll o w i n g a lg e b ra i c st at e m e n t c a ll e d a
fo rm ul a m ay b e v er ifi e d by pe r fo r m i n g t h e i n di c a t e d o per
ab
,
,
.
,
,
,
a t io n s
(
S u ppo s i n g t h a t
i
pr n c
)
x+ v
and
x
3
—
(
v)
2
= 4x
v
two
a ny
a re
y
i pl e d o es t h e fo r m ul a
—
x
n
u m b e rs
,
wh a t
e x press ?
E x e rc i s e 5 9
1
.
Ve r ify t h e
( + x)
a
2
.
of
t rut h
t his
fo rm ul a
2
t h e t r uth
H a vi n g v e rifi e d
—x
)
of
th is
2x ( a + x)
a
g e n e ra l pr i n c i pl e it e x pr e s s e s
Si n c e a f o rm ul a e xp resse s a gen era l
t o a l l pa r ticu l a r e x a m pl es o f t h a t t yp e
t e ll w h a t
s t a t e m e nt ,
lg e b ra i c
.
i
i pl e it
d
—
?
1
2
87
5
6
n
r
c
p
a pp
,
li es
.
3
.
B y h o w m uc h
h ow m u c h
4
of
.
.
3?
.
exc e e
By
d
e xc e e
sq
d the
s q ua re
.
i g t h e bi n o m i al giv e t h e diff e renc e b e
s q uar n
,
and
B y h o w m uc h
m uch
ex c e e
u a re o f
G iv e r es ul t w i t h o u t s q u a ri n g
Wit h o u t
t w ee n
6
does
H o w m uc h d o e s t h e
—
50
5
d o es
d o es
d o es
ex c ee
e xc ee
d
d
5 69
35 0 ?
B y ho w
2
Wh e n
.
ds
ya r
L E M E N TARY ALG E BRA
E
1 26
3
wh at is it s
,
A
.
gl e 1 8 fe e t
l en gt h ?
re c t a n
a
re c t a n
w id t h in
Fi n d it s
of
gl e
ro
ds
l an d
64
r
wid e
o
i
1 50
c n t a ns
od s lo n g
c
o n t a i ns
sq
18
u a re
a cre s
.
.
1 60 X 1 8
w
:
64
4
pe r
E xpres s in
.
i m et e r of
5
Us i n g
.
fi n di n g
6
t he
sq
the
rul e
.
.
If
ds
the
x is
wh i c h thi s
fo r
fi n di n g
t he
w r i t e t h ree
,
fo rm ul as fo r
fo rm ul as o f pro b l e m 5 a n d giv e
d er iv e d fo rm ul a s expres se s
t h e t h ree
eac h of
t he
.
a
l t it ud e is
24
fee t
i
52
c o nt a n s
lo n g is it s b a s e ?
h ow
,
l
ru es
.
t ri a n gl e w h o s e
a
yar
g e n e ra l n u m b e rs
o f a n y t r i a n gl e
o ne of
whi c h
If
u a re
8
a rea
tw o
.
a ny
So lv e
.
7
a ny
g e n e r a l n u m b e rs
r e c t an gl e
ag e
i
e q u at o n
of a
b oy
no w ,
m a ke t he probl e m
of
s t at e m e n t
is t h e
—
=
3 (x 7)
x+3
in g
Us i n g
g e n e r a l n u m b e rs w r i t e t h e fo rm ul a fo r fi nd
t h e volu m e of a n y r e c t a n gul ar pr i sm
9
.
10
.
.
So lv e t h e
e a c h of
.
A
pro bl e m
13
.
.
t h e t hre e
,
e
m
r
b
o
p
9 a n d giv e t he
l
d er iv e d fo rm ul a s express es
of
i
n
r
c
p
i pl e
.
fo rm ul a t h e r e l a t i o n of divid e n d divi s o r
a n d r e m a i n d e r in divi s i o n
a
,
h as
x
a c res
of
l and
and
of w h i c h t h e s t at e m e n t is 3 x
W i t h o ut
G iv e
i g
s q uar n
,
.
,
B 3x
a c r es
.
M a ke
t he
20
t he b i n o m ial ,
giv e
the
di ff e re n c e
and
b e t w ee n
14
f o rm u l a
E xpr e ss in
n
u
i
t
t
e
o
q
12
,
.
wh i c h
11
a ny
a
fo rm ul a fo r fi n di n g
w he n t h e pe ri m e t e r
a nd
the
di m e n s i o n o f a r e c t a n gl e
o t h e r di m e n s i o n a re giv e n
o ne
.
FOR M ULA S
15
If
.
a re c t a n
gl e
h as
fee t l o n g
64
1 27
a
pe r i m e t e r
2 26
fe e t
d s in a n y
giv e n in feet
box
of
,
w h at is t h e w id t h ?
R e pr e s e n t t h e n u m b e r o f c u b i c
s h a p e d e x c a v a t i o n w h e n t h e di m e n s i o n s
16
r
a
y
.
for mu l a
a re
a s a co mpa c t s hor tha n d
.
o f n um b e r
l aws
T h e fo ll o w i n g
is p e rh a ps t h e m o st pr a c t i ca l p a rt o f a lg e b r a
li s t o f pro b l e m s w ill gi v e pra c t i c e in fo r m u l a t i n g a ri t h m et i c a l
n
n
r
c
e
t
c
a
w
s
i
l
i
ifi
l
a
c
t
a
a
d
S
c
p
1 72
The
.
.
,
.
,
E x e r c i s e 61
1
D e no ti n g
.
s , an d
by m ,
of
A dd
.
,
s
l t i n g fo rm ul a m ea n s
l i
r e at o n
a
5
3
,
=M
.
St a t e
.
by
a
.
as
w ho l e
Sho w b y
t
b
f o r m ul a :
n u m b er , a ,
fra c t io n g by
a
a
fra c t io n
m by m ,
-
st at e
t he
Th e pro du c t
of
of
a
t he w hol e
d e no m i na t o r
f ra c t i o n
,
ca
lli n g
,
num b e r
.
t h e p r i n c i pl e fo r m u l t i pl yi n g
c
d
a nd
.
by t he
fo rm ul a
t h e pr o d u c t , p ,
.
is t h e pr o d u c t
di v id e d
of
t he pro d u c t p
a
.
St é t e b y a f o r m ul a th e re l a t i o n o f t h e pe r c e nt a g e , p ,
r a t e , r , a n d t h e b as e , b, a n d t r a n s l a t e t h e f o rm ul a i n t o
7
t he
a
n um e r a t o r ,
by t h e
6
m
,
Divid e b o t h Sid e s o f p
m e a n i n g o f t he re s ul t i ng f o rm ul a
4
a
a nd
,
.
,
,
by
d
—
= d a n d st at e w h a t t h e
m s=
fo rm ul a t h e
m ul t i pli c a n d M a n d m ul t i pli e r
Sho w b y
.
di ff e r e n c e
fo rm u l a t h e re l a t i o n
s u b t ra h e n
s how
of
id e s
re s u
3
l tin g Law s
orm u a
.
to both
s
iv e l y
F
and
m i n ue n d ,
r e s pe c t
n u m b e rs
t he se
2
d,
t he
Sta tin g
.
wo r d s
.
Divid e b o t h s id e s
o f t h e res ul t i n g f o r m ul a
8
.
.
of
= br b
,
p
y
r , a nd
t e ll t h e m e a n i n g
E
28
giv e m e a n i n g o f t h e fo rm ul a for t h e i nt e re st
in t e rm s o f t h e pr i n c i pl e p r a t e r a n d t i m e t ( in yea rs )
9
i,
LE M E N TARY ALG E BRA
St a t e
.
and
,
,
,
,
,
,
.
Divid e b o t h s id es of
r e s ul t i n g form u l a m ea n s
i = pr t b y
r t,
and
t e ll w h a t t h e
Divid e bo t h s id e s
r es ul t i n g fo r m ul a m ea n s
i = pr t b y p t,
and
t e ll
10
.
.
11
of
.
h t th
w
a
e
.
12
.
St a t e
13
.
St a t e
f ra c t io n s
14
of s
id e
is
16
,
s
s
17
a
fo r m ul a
t he l aw
a
f o rm u l a
t he
A,
of
of a
c
o f a re a ,
t wo
squa re
a
.
v o
lu m e
a
u ni t s in
fo rm ul a
t ent h s
t he
,
’
v a lu e f
,
l
p ac e
A ns
’
,
V,
ub e
w h o se
d ec i m a l fr a ct io n
u n i t s in h u n d re d t h s
of a
’
h
and
h
.
.
fo r m ul t i pl yi n g
l aw
.
h a vi n g t
e
ac
l
p
t he
.
.
St at e b y
.
f o rm ul a
a
St at e b y
.
ed g e
as
Sh ow b y
.
15
fo rm ul a t h e l a w fo r s ub t r a ct i n g t wo fra c t io n s
as a
f
_
10
So lv e t h e fo rm ul a in t h e
+
’
1 00
a n s w e r o f pr
o b l e m 1 6 fo r t ; fo r h
.
fo rm ul a t h e c ost l a w in w h i c h 0 is t h e t o t al
Solv e
c o s t n t h e n u m b e r of a rt i c l es a n d p t h e pr i c e of e ac h
t h e fo rm ul a fo r n ; fo r p
18
.
St a t e
as a
-
,
.
,
,
.
19
.
C a lli n g d t h e t o t a l
a nd
t the t im e ,
as a
form ul a
di s t a n c e
t h e di st a n c e
s t at e
r
,
-
t he
l aw fo r
.
,
,
.
fo rm ul a o f pro b l e m
Solv e fo r t
m e an i ng of t h e r es ul t
20
of m ov e m e nt
u n ifo rm m o tio n
rat e
Solv e t h e
.
19
for
and
r,
t e ll t h e
.
v e lo c i t y v of a fr ee l y f a lli n g bod y is t h é pr odu c t
of t h e g ra vi t y c o n st ant g by t h e t i m e t o f fa ll Fo rm ul ate
t h i s l aw
Solv e it fo r g ; fo r t
21
.
The
,
,
-
,
.
fo r h
.
So lve t h e
.
,
.
.
22
,
,
fo rm ul a A
,
2 7rr ( h + r)
for
7r ;
fo r 7rr ; fo r h + r ;
E
1 30
FORM S
1 73
l e a rn
form s
L E M E N TARY ALG E BRA
TYP E F O R M S O F ALG E B RA I C
A ND
N UM BE R S
-
f
M e aning o Type Form s
A v e ry i m po rt a n t t h i ng t o
in a lg e b r a is t h e m e a n i n g a n d u se o f form s a n d type
.
.
lg e b r a i c n um b e rs B y t h e fo rm o f a n u m b e r is
m e a n t h o w fro m it s w r i t t e n a ppea r a n c e it l o o ks as t h o ug h it
w ere m a d e u p o u t o f s i m pl er n um b e rs
A b it o f v alu a bl e
a d v i c e o ft e n giv e n b u t se ld o m a ppr ec i a t e d b y t h e b e gi n n e r
of a
.
,
,
.
,
,
,
is a l wa ys to l ook ca refu l l y in to a probl em or exerc ise befor e pu tting
”
“
n
c
i
l
a
r
k
e
t
o
e
L
b
f
r
l
e
a
i
a
g
d
m
e
u
s
oo
o
e
o
o
o
ott o
p
p
y
p p
.
lg e b r a i s t M a k e it a h a b i t T h e h a b i t is
T h e am o u n t o f u se l es s
pa r t i c ul a r l y v a lu a b l e in f a c t o r i ng
l a b o r it will s a v e yo u Will c om pe n sa te m a ny fo ld fo r t h e
e ffo rt
T h e w a y t o s t a rt t h e pr a c t i c e is t o l e a rn w h a t
n u m b er fo rm s m e a n a n d h o w t o u se t h e m
T h i s is n o t a n
e n t ir e l y n e w t h i n g
fo r n um b er f o rm s are u se d e a r l y in
a r i t h m et i c
fo r t h e yo u n g
a
.
.
.
-
.
-
.
-
,
.
exa m p e ,
l
Fo r
w h e t h er 5 is
it
to
a
w h e n yo u
fa c t o r o f
,
t o t e ll , w i t h o u t
n um
b er by not i c i ng
u s i n g t h e for m o f t h e
a
w er e
d e d in 0 o r 5 yo u
lig h t e n yo u r w o rk
en
l ea r n e d
,
dividi n g
,
w h e t h er
n
umber
.
L i ke w i se yo u h a v e pr o b a b l y l e a rn e d t o us e
a n u m b er t o d e c id e w i t h ou t dividi n g
w h et he r
is divi s ibl e by 1 0 1 00 2 4 8 e t c
,
,
,
,
,
In
lg eb r a
u se ful t h a n
a
m o re
I f we
two
,
we re
di ffe re n t
,
,
,
with
i
a n a c q u a nt a n c e
in
as k e
d
ar
i thm eti c
to
n u m b e rs
the
fo rm
the
n
of
u m b er
.
n umbe r
for ms
-
is m u c h
.
i n di c a t e t h e s um o r t h e di ff e r e n c e o f
in s o m e s ugg e s t iv e fo rm w e m ig h t
,
w rit e :
a nd
v e s s ugg e s t i n g t h a t a n y n um b e rs wh a t s oe v e r
m igh t b e w r i t t e n w i t h i n t h e m
B ut w h il e t h es e f o r m s s h o w
s um a n d differ en ce t h e y do n o t s ugg e s t t h a t t h e t w o n u m b ers
t he
e m p ty c u r
.
,
F OR M S
TY P E F OR M S
AND
in q u es t i o n a r e t o b e di ff e r e n t n um b e r s
o b j e c t i o n w e m ig h t s u gg e s t t h es e f o rm s
l
13 1
-
—
a nd
o bv at e
To
l
l
t hi s
i
,
d erst a n di n g t h at t h e c ur v e d a n d t h e s q u a re
c o r n er e d s ym b o l s a r e t o s u gg es t t h a t di ff e r e n t n u m b e r s a r e
t o b e w r i tt e n i n s id e t h e di ffe r e n t l y s h a p e d sym b o l s
I f w e h a d b e e n i n g e ni o us e no ug h t o s e e w h a t it t o o k
m a t h e m a t i c i a n s h u n d re d s o f y ea rs t o di s c o v e r t h a t b y
s i m pl y c a lli n g o n e n u m b e r x a n d t h e o t h e r y a n d w r i t i n g
w i th t h e
un
-
.
,
,
x+
We
y
and
—
x
,
y,
v eryt h i n g s h o w n eas il y a n d fu ll y t h e n o ur pr o b l e m
w o uld h a v e b ee n so lv e d
We m e r el y r e m e m b er t h a t t h e
d iff e r e n t l et t e rs a r e in g e n e ra l t o d e n o t e d i ffe r e n t n um b e r s
hav e
e
,
.
:
.
1 74
E xampl e s
.
b, m ig h t
a nd
t ion
B ut
.
as
i g
We
wel l
x an d
as a n y o t h e r
us n
of
y
Type Form s
A n y o t h e r l e tt e rs , a s a
h a v e b ee n use d as x a n d y in t h e l as t se c
-
.
l e t t e rs
so
,
t he m m o re t h a n
sa y
t h e n t ha t
o t h e rs
x + y and
,
.
—
x
y
a r e res pe c t
iv e l y
the
forms
diffe re n c e o f a ny t w o di ffere n t n u m b e rs
Si n c e x + y m ay s t a n d fo r ( t ypify) t h e s u m of a n y two n u m
be rs it m a y b e c a ll e d a type form fo r t h e s u m
Si m il a rl y
—
i
x
y s c a ll e d t h e type for m fo r t h e diff e r e n c e o f t w o n um b e rs
Th e type for m fo r t h e s um o f t w o pro d u c t s is a x + by;
—
fo r t h e di ff e r e n c e o f t w o pr o d u c t s d x by
T h e type for m fo r t h e s u m o f t w o pr o d u c t s h a vi n g o n e
f a c t o r common t o b o t h pro d uc t s is a x + a y a n d fo r t h e diff e r
—
e nc e o f su c h pro d u c t s a x
ay
fo r t h e
s um a n d
il y w ri t t e n a n d se rv e j ust a s w e ll
a lg e b r a i s t s f a ll i n t o t h e h a b i t o f
a re e a s
t he
.
-
.
,
,
-
.
-
.
,
-
,
,
.
Th e type form fo r t h e
-
t he
di ff e re n c e of
s um o f
two
—
x
s q u a res
is
3
x
3
+y
,
and
fo r
u a re s
O b se rv e t h a t x + y
y
m ea n s t h a t a n um b e r is m a d e b y t a ki n g t w o di ff ere nt n um
2
3—
b ers s q u a r i n g b o t h a n d a ddi n g t h e s q u a res w h il e 93
y
di r e c t s us t o fo rm a n u m b e r b y c h o o si n g t w o diffe r e n t n um
,
t wo
,
sq
3
,
2
2
.
,
2
E
132
L E M E N TARY ALG E BRA
i g bot h
be rs ,
s qua r n
a nd
u b t ra c t i n g C l e a r l y t he n s u c h
3
2
r
an d x
ar
e
c
m
v
e
o
y
y
pa c t w a ys o f
,
rt fo rm s a s x + y
s a y i n g a g rea t d ea l
sho
2
2
s
.
,
-
.
Su c h
a n
u m b er
t h e pr odu c t
th i r d
n omi a l
a nd
n
as x
up
.
u m be r
n o m i a l is
ca
form
trin omi a l s
n um
a n
,
,
,
Si n c e
b
i
h
+ a x + b s t e type for m fo r
by c h oo s i n g
As
.
3
x
B ut is m a d e
.
Th e
fo m
r
3
u m b e r s q u a ri n g it a ddi n g
of it a n d so m e se c o n d n um b e r a n d t h e n a ddi n g
b e r s to b e b uil t
a
-
o ne o f
ll e d
a
2
+
x
,
+a x + b
of three
up
r
(
s q ua r e -
+ b, is t h en
tr i
a
u m b e rs x a
is s q u a re d t h e t ri
d ifi er en t
n um b e rs , x ,
t h e se
d
t
i
u
a
a
c
q
ax
h as three t e r m s , it is
n
,
,
,
,
li ke )
tri n omia l
.
t ype- fo rm fo r qua d ra tic
a
.
Si n c e x + y st a n d s fo r t h e
Type - Form s In te rpr e te d
s um o f a n y t w o n u m b e rs , if w e m ul t i pl y it b y i t se lf w e g e t
1 75
s qua r e o f
t he
by
.
.
x+y
su m o f a n y
t he
giv e s
.
H e n ce t he
t ype
-
2
+ 2 xy+ 9
fo rm fo r
u m b e rs i s x + 2xy+ y
te ll s us m u c h
As
2
2
n
M ul t i pl yi n g
.
x+ y
us
x
1
two n u mber s
.
2
u a re o f t he
t ype fo rm t h i s
t he
a
sq
-
,
s um
3
x
of
t wo
+ 2xy+ y
2
.
1
.
It te l l s
n umber s
2
a re
3
.
is
a
It tel l s
tha t the
us
tr in omia l
ma d e by s qua rin g the
.
It tel l s
us
the
f
o
su m
tha t the
f
o
the three ter m s
n u m ber s
to be
to give the
4
.
su m
to
It tel l s
f
o
form
r es u l ts
.
two
us
ori gin a l
tha t
n umber s
the ir pr od u ct
two dif
a
ma in in g term
s hor t
i s to
a nd
the tr in omia l
.
f
re
su m
f
o
a dded s epa r a te l y
o
the trino mia l i s
n umbers
ma de by dou bl in g the pr odu ct of the two
a d ded
o
f
.
tha t two
us
s qu a r e
f r nt
e e
tha t were
.
wa y
f
o
s qua r e
doubl e it,
getting
ea ch
a nd
f
o
a
s qu a re
the two
the n to
a dd
f
o
the
nu mbers ,
the three
C H A PT E
R X IV
FA CT ORI N G
1 76
of
Th e fac tor s
.
m a kers
n um b e r
is t h a t
p rod u c t
of
t he
num b er ,
n um b e r
a
of
Fa c t ors
.
nu m b e r s
the
a re
by mu l tipl ica tion
a
n
um b e r
w h os e
are
the
.
F ro m t h e l a w o f t h e a lge b r a i c n o ta t io n a n d t h e m e a n
ing o f i n t e g r a l ex po n e n t s t h e f a c t o rs o f a m o no m i a l are t h e
fa c t o rs Of t h e c o e ffi c i e n t a n d e a c h l e tt er a s m a ny t i m es a s
T hus
t h e r e a re u n i t s in it s ex po n e n t
1 77
.
,
'
.
6a
3
=
bc 3 2
3
°
,
bb
-
aaa
M O N O M I AL FA C T OR S
Type fo rm :
-
1 78
a re
t h e pro d uc t
defi n i ti on
By
a nd
Po l yn o m i a l s h a vi n g
.
—
2a
3 b a re t h e
comm on
po l y n o m i a l
of a
of
a
fa c t o rs
ay + a z
ax
and a
fa ctor
(
f a c t o rs o f
—
6a
2
e
m o n om i a l
—
si n c e 3 a 2a
,
in
3 b)
v ery
t e rm
.
6a
9a b, 3 a
3
9a b
.
T h e m o n o m i a l fa c t o r is t h e grea tes t c ommon f a c t o r o f t h e
coefiicien ts m ul t i pl i e d b y t h e l ow es t pow er o f a l l t h e co mm on
l etter s
.
T h us
and
,
1 8x
Wh e n t h e m o n o m i a l f a c t o r is
t he
c o r re s po n
di ng
f
o ne
t erm
t e r m in t h e po l yn o m i a l
1 5x + 1 0 x
3
t h e p o l yn o m i a l ,
fa c t o r is 1 T h u s ,
of
.
2
D iv ide the p ol ynomia l by the mo n omia l
a n d wri te the div iso r a nd the qu otien t for the fa ctor s
1 79
.
R ul e
.
fa ctor
.
F a c t o rs m a y a l w ays b e
t o g e t h e r a n d c o m pa ri n g t h e
fa c t o re d
.
checked
r
p
odu c t
b y m ul t i pl yi ng t h e m
w it h t he
n
u mb e r
to be
FA C T O R IN G
1 35
E xe rc i s e 62
a n d c h ec k
Fa c t o r t h e fo llowi n g
1
5a
.
4
7
1 0a
2
2
2
4
6x + 1 5 x
.
6o
.
9
3
3
x
8a
3
4b c
6a
11
2
2
9a bc d
c
+a
x
2
3
3 2
—
x y
x y
3
8
4 a bc
3
2
3
3
a x
3
.
3
2o x y
x
2
3 2
.
l as t four
3
8a bc
3 3
.
5
3
th e
3
10
2
3a o
12
.
.
.
6
3
3
3
3
3a
ax
a
3
x
y
4a b
.
bc
3
14 a b + 7 a b
3
3
.
2a
2
3a
3
4
4a
x
2
xy
“
3a b c
2
3 3
1 0a b
3
2
3
x
3
o x
a
2
3
2
y
4
y
bc
3 2
C O MM O N C O MPO UN D FA C T OR
Typ e- form
1 80
.
T he te rm s
g rou pe d
as
to
of
a
s ho w a c
Co n s id e r
ax
po
l yn o m i a l m a y
o mm o n
ax
+ ay + b x + b y
co mp ou n d
+ a y+ bx
so
m e t i m e s be
fa c t o r
so
.
by
fi rs t a n d sec o n d t erm s of t h i s po l y no m i a l c o n t a i n t he
c o m m o n f ac t o r a a n d t h e t h i r d a n d f o u rt h t e rm s c o n t a i n t h e
c o m m o n f ac to r b
G ro u pi ng t h e t erm s in t h i s m a nn e r a n d
fa c t ori n g ea c h g ro u p we h a v e
Th e
,
.
.
,
B y t he
us e
t o two t e rm s ,
pou n d
pare nt hes e s , t h epol yn o m i a l is t h us r e du c e d
wh i c h are s im il a r w i t h r e fe r e n c e t o t h e c o m
of
fa c t o r x + y
r ul e fo r a ddi t io n of
,
.
C o m b i n i ng t h e t e r m s
te r m s pa rt l y
s
di ng t o
we h a ve
ac cor
i m il a r § 7 2
,
,
t he
fi rst te rm is n o t a l wa ys g ro u pe d w i t h t h e s e c o n d It
m a y b e g ro uped w i t h t h e t h i r d t e rm o r t h e fo urt h
F ac t o r
g ro upi n g t he fi rst w i t h t h e t h i r d
t e rm an d t h e s e c o n d t erm w i t h t h e fo u rt h
Thus
The
.
.
,
,
.
,
E
1 36
LE M E N TARY ALG E BRA
E IEerc is e 63
Wri te t h e fa c t o rs of t he follo wi n g
.
1
3
5
7
.
.
.
.
—
—
ac
ad + c n
a nd c he c k
dn
2
a x + 2x + a y + 2 y
a
a
3
2
+
3
a n
3
+ an + n
.
4
3
6
— mn — an
+ am
8
—
ax
cy+
—
cx
a
—
—
a n + bn
dx
bx
.
—
x
3
.
a
.
5
—
3 y xy+ 3 x
+a
3
x
+a
3
x
In t he prec e di n g e xa m pl e s , a p osi tive m o n o m i a l
is t a ke n o ut o f e a c h g ro up
O b serv e t h e fo llo wi n g :
181
y
.
2
+x
3
fa c t o r
.
bx
ax + a y
—
ax
a
by
— bx
=
b
+ y
y
b) ( x + y)
(a
(a
—
b) ( x
C o nvi n ce yours e lf t h at t h e e q u at i o n s
—
—
—
n
n
l
i
g
a
b
b
x
a
d a b by x y
+
py
y
y
-
y)
r
are c o r ec t
b y m ul t i
.
A polyno m i a l c ann o t b e fa c t o re d in
t h i s m a nn e r
u n l es s t h e
ou n d fa c t o r is the s a me in e ac h g rou p
—
h
n
i
n
To g e t t e sa m e c o m pou d f ac t o r
e a c h g r ou p
b is
t a ke n o ut of t h e s ec o n d g r ou p in e a c h of t he t w o exam pl e s
a bov e
c om p
.
,
.
E xerc i s e 64
F a ct o r t h e fo ll ow i n g
1
3
5
7
.
.
.
.
po
l yno m i a l s
a nd c h ec
—
—
an
bn a x + bx
2
—
—
by+ a y bx
ax
n
a
2
3
-
nx + n y
—x
4
.
.
6
y
3 — 3
—
a x
+ m x um
2
8
.
.
k:
3—
—
bx by+ y
ab
dx
+ xy
3
—a
y
—
xy
bx
—b
—
a
x
b
x
+
y
y
—
a br
bc + cn
-
a ux
o m e c a s e s t h e c o m pou n d f a c t o r in o n e g rou p is
li ke t h e r em a i n i n g te rm s o f t h e pol yn o m i a l o r li ke t h o s e
term s wi th their s ig n s cha nged
In s u c h e xa m pl e s t h e
—
1 as
m o no m i a l f ac t o r t a ke n o ut o f o ne g r ou p is + 1 or
fo r e xam pl e
—
—
ax
a y+ x
y)
—
—
ax
ay
1 ) ( x y)
1 82
.
In
s
,
.
,
,
-
,
LE M E N TARY ALG E BRA
E
1 38
The s qua re of the s u m of two n u mber s is the s qua re of
the fi rs t n u mbe r , p l us tw ice the produ ct of the fi rs t a nd secon d ,
185
.
l
t
u
s
h
e
p
f
s qua re o
the
secon d
.
E x erc i s e 67
G iv e t h e
1
.
4
.
b
( +
.
8
.
b ers
.
12
.
old
)
of
the
2
fo ll o w i n g
2
3
.
(a + c )
W h at
w ill
whi c h
( b+ x )
A
it
a
2
of
9
c
ul t s
m a n l iv es 8 y e a rs , h e w ill b e
8 ye a r s a g o ?
If
w as he
5
res
6
3
7
sm a
2
10
m a n w as
th e
ll
s um o f
c
+ b)
( a + by)
.
o nse c u t iv e odd
num
?
es t
11
.
r
l
a
ea
s
o
d
y
x
3
ax
How o l d
.
.
r e pr ese n t
is t h e
e
a
r
s
l
o
d
y
(
.
2
e
a
rs
a
y
go
2
a
b
+
(
)
.
2
I f h e liv e s , h ow
.
w ill h e b e in b yea r s ?
13
( x + y)
.
2
14
15
.
(x + 3 y)
.
SQ UARE OF T H E DI FFERE N C E OF T W O NUM B ER S
Type f orm : a 2 ab + b
‘
3
-
186
.
Si n c e
a
b
a nd
a re a n y
2
-
t wo
u m b e rs ( a b) is t h e
n u m b ers
T h e s q u a re o f
n
,
u a re of t h e di ffe re nc e o f a ny t wo
2
—
b is fou n d by m ul t i pli c at i o n t o b e a
a
m i n u s t w i c e t h e pro du c t of a
s q u a r e of a
s q u a r e of b o r
sq
2
-
.
~
,
2 a b+ b
2
a nd
or
,
t he
b, plu s t h e
,
(
1 87
.
The
s qua re
—
a
b
f
o
)
2
= a2 — 2ab + b 2
the differen ce
f
o
two
f the fi rs t n u mber min us twice the
a n d se con d p l u s the s qua re of the s ew n d
s qua r e o
,
,
.
n u mbers
prod uc t
f
o
is the
the
fi rs t
F AC TORI N G
1 39
E x e r c is e 68
G iv e t h e
1
4
7
10
13
r es
(b
.
.
.
(
(
—
b
(b
.
c
)
2
c
)
2
x
)
2
y)
2
-
—
a
-
ul t s
of
t he
—
(n l
5
8
11
?
—
( ti x )
.
fo ll o w i n g
14
(
.
(
.
.
.
—
x
3
n
w i t h o u t m ul t i pl yi n g :
,
—
) (n 1 )
—
) (x 3 )
2) (
-
3
—
n
2)
6
.
9
.
—
—
( x 4 ) (x 4 )
12
.
—
(n
15
.
6) (
—
n
6)
(
.
—
ax
(a
—
b)
by)
( 353
2
—
( x 3 y)
(4 a
i t h m et i c a l n um b e r m a y b e s q ua re d m e n t a ll y
by c o n s id er i ng it t o b e t h e s um o r t h e di ff e r e n c e o f t w o
n u m b e rs
Thu s
1 88
.
An
ar
.
,
2
46
2
46
21 16
:
7
E xer c i s e 69
E xpre s s t h e
t he n
1
6
.
.
as
t he
38
2
58
2
1 89
.
of
t h es e
7
.
.
47
2
64
2
3
8
trin o mia l m a y b e
a
nu m b e rs ,
fi r st
as
s um ,
t he
di ff e r e n c e o f t wo n u m b e r s :
2
A
t o m a ke
s q ua res
b i no m i a l
of it
3
.
65
.
76
4
2
9
by
u are d
T hus
.
sq
.
.
54
2
85
2
5
10
g r o up i n g
.
.
73
2
95
2
t wo t e rm s
,
a nd
E x e rc i s e 7 0
G iv e t h e
1
4
.
.
(a + b
(
1 90
t ho se
fo ll o w i n g
—c
—
a
x
.
l
If
)
2
s q u a res
2
.
5
.
wi t h o u t
(a
-
a c t ua
b+ c )
ll y
2
(a + x
3
.
6
.
(
a
—b
.
+ c)
2
(a + x
u m b er is t h e produ c t of equa l fa ctors
is c a ll e d a r oo t o f t h e n u m b e r
a n
f a c t o rs
m u l t i pl yi n g
,
o ne
of
E
1 40
LE M E N TARY ALG E BRA
roo t o f a n um be r is o ne of t h e two e qu a l
f a c t o rs wh ose pr odu c t is t h e n u m b e r s
2
4
3
G iv e t he s q u a re r oo t o f 9 ; 25 ; 64 ; 1 44 ; a ; x ; 4 b ;
3 “
4 3 3
1 6x y ; 1 00 a b c ;
(x
Th e
r
s qua e
root of a n u mb e r is o ne of t he three e q u a l
fa c t o rs wh o se produ c t is t he n u m b e r
3
3
9
G iv e t h e c ub e r oo t of 8 ; 2 7 ; 64 ; 1 2 5 ; a ; b ; 8 x ;
T he
c ub e
.
2 1 6a b ;
2 7a b
3 3
3
3 9
c
;
T RIN O M IAL SQUARE S
Type fo rm :
-
1 91
.
A trino mi al
r is t h e s q u a re o f a b i n o m i a l
s qua e
(
The
y)
2
=
2
—
2 xv + 9
x
of e
of
the
s q ua re a re
s qu a r es o f
the
.
t e r m is twice t he pro du c t o f t h e s qua r e roots
s q ua r es a n d m a y b e e i t h e r pos i tive o r n ega tive
other
t h e t wo
,
3
v e ry t r i n o m i a l
t h e b i no m i a l
Two te r ms
two te r ms
—
x
T hu s
.
of
.
'
T h e fa c t o r s
of a
t r i nom i a l
u a re a re t h e r efo re two l ike
bi nomia l s a n d t h e t e rm s o f e a c h f ac t o r a re t h e s qua re roots of
t he t wo s q u a r es in t h e t r i n o mi a l
Th u s
'
sq
,
.
4 a + 9b
3
Th e t wo
and
1 2a b is
q ua r es a r e 4a a nd 9b t he i r s q ua re roo t s
t w i c e t h e p ro d uc t o f t h es e s q ua r e r o o ts
2
s
,
.
of
f a c t o ri n g
a
tr i n o m i a l
Find the squa re r oots
conn ec t thes e r oots wi th the
.
wri te the bi n omia l twic e
T h e t wo
h
2a
d
an
3 b,
.
Rul e
s qua r es ,
e ac
are
sq
u a re is
st ate
d
as a
thus :
1 92
and
3
,
T h e m et h o d
rul e
2
2
a
b
a
b
3
3
+
+
)
)
(
(
l 2a b
2
,
fa c t o rs
fa c to r is
the
as a
f
o
the two ter ms tha t
f
s ig n o
fa ctor
th e
,
.
r
te rm ,
.
t r i no m ia l s q ua re b e i ng e q ual i t
s q ua r e r oo t o f t h e t r i n o m ia l
of a
o th e
a re
is
id e n t t h a t
’
ev
L E M E N TARY ALG E BRA
E
19
20
1
21
.
2
3
24
—
.
64x + 3 2x + 4
P ROD U C T
OF
.
3
2 5a
.
fi 4
c
+ 1 0a
3 2
c
+1
6 (x
2
.
9a b + 3 0 a b + 2 5
4 2
.
4 a + 4 a b+ b
4
25
26
2 ( a + x)
.
22
23
( a + x)
.
3
27
T HE S U M
A ND
1 2 1 + 4a b
3
‘
.
3
— 4 4a b
DI FFERE NC E OF T W O NUM BER S
Type form :
-
-
b)
L e t t i n g a a n d b r e pr e s e n t a n y t w o n u m b e r s t h e n
—
h
n
h
r
f
n
m
r
r
t
s
t
e
odu
c
t
o
t
e
i
r
s
u
d diff e r
a
b
e
ese
a
b
a
+
)
p
p
(
)(
3—
3
T h e p rodu c t is fou n d by m ul t i pli c a t io n t o b e a
b
e nc e
t h e di ff e r e n c e of t h e i r s q u a r e s i e
1 94
.
,
.
,
.
,
(a + b ) (
—
a
,
2—
3
=
b
b) a
The produ ct of the s u m
is the difi eren ce of their s qua r es
1 95
.
a nd
.
diflerence
f
o
two
n u mber s
.
E xe r c i s e 7 2
G iv e t h e
1
4
7
10
follo wi n g
—c
.
—c
.
pro du c t s
)
2
.
)
5
.
8
.
—x
)
.
(b
.
11
.
(x
3
.
(
n
6
.
(x
9
.
12
.
4
(
1 96 T h e pr odu c t of t wo
fou n d by w rit i ng th em as t h e
.
t wo
n
u m b e rs
.
Thu s
,
( 2a
—1
)
—4
)
(3
i t h m e t i c a l n u m b e r s m ay b e
s um an d di ffe re nce of t h e s am e
ar
FA C TORI N G
1 43
E x e rc i s e 7 3
G i v e t h e s e p ro d uc t s as t h e pro d u c t s
di ff e re n c e o f t w o n um be r s
1
.
3 8 x 22
2
.
4
.
66x 54
5
.
of
.
m
t he
4 7 x 33
t he
a nd
su
3
.
54 x 4 6
6
.
83 x 77
T w o t r i n o m i a l s m ay s o m e t i m e s b e g rou pe d s o a s t o
T hu s ,
re pr es e n t t h e s um a n d di ff e re n c e o f t w o n u m b e rs
1 97
.
.
( a + b+ c ) (a + b c )
( a + b c ) ( a b+ c )
( a + b c ) ( a b+ c )
(a + b+ c) ( a + b
.
H e nc e
,
(a +
a
—
b
c
)(
—
a
b
b + 2 bc
2
3
)
c)
c
-
3
C
E xe rc i s e 7 4
G i v e t he
1
3
fo ll o w i ng
pr o d uc t s
-
.
.
5
(a +
—
x
y) (
y)
—
a
x+
—
( x +v 2 ) x
(
2
y)
4
y+ 2)
-
B y m ul t i pl yi n g t h e
differen ce , t h e differen ce
f
o
a
sum
2
0
)
—x
)
— b— 3
)
(a
.
SQUARE S
T wo
Type f orm
.
(a
.
6
D IFFERE NC E OF
1 98
— b—
—b
of
‘
two
their s qua r es is
n um b e rs
o b t a ne
i
d
,
by t h e i r
t hus :
3—
—
=
(x + 2) (x 2 ) x 4
Si n c e t h e t e rm s
of
t h e p ro du c t a r e s q u a re s o f t h e c o rr e s
fa c t ors , t h e t e rm s o f t h e f a c t o rs a re
di n g t e rm s o f t h e
t h e s q u a re r oo t s o f t h e t e r m s o f t h e
po n
3
x
1 99
f
o
the
.
R ul e
.
Wr ite for
f
s qu a r e r oots o
pr
od u c t
-
the
the ter ms
,
or
1)
fa cto rs the su m a nd the difi erence
f
o
the bin omia l
.
E LE
14 4
M E NT A R Y
ALG E B RA
E xerc ise 7 5
Fac t o r th e follo w i ng
1
4
a
.
a
.
200
2
-
r
.
6
-
b
6
W hen b ot h
.
s q ua r e s ,
Th us
3
‘
t h at
t e rm s
of
t he
di ffere nce fa c t or a re
b e r esolv e d i nt o t w o o t h e r f a c t o rs
f a c t or m ay
-
.
,
— 1
>
s
a
s
E xerc i s e 7 6
F ac t or t h e foll o wi n g
1
4
16
a
.
a
.
a
.
2
— x2
4—
s—
b
4
x
bo t h of t h e s q u a res in t hi s t ype of exam pl e m ay
b e t h e s q u are of a bi n o m i a l
Th u s
20 1
.
One
-
or
.
2
—
( a b)
a
a nd ,
(a
2
— b
(
(a
2
(3
c
)
2
,
=
b
+
(
a
(a
—b
—
b+
c) ( a
c
)
c
)
—
—
b x
y)
L E M E N TARY ALG E BRA
E
1 46
E xe r c i s e 7 9 — R e v i e w
G iv e t h e
1
.
" f
9a :r
—
6
2
y
4
5
6 —-
.
8 1a l
a
4—
1 8a
.
a
.
8
4—
b
4
c
d
2
3
.
10
64 x + 1 6x + fc
-
6
y
9m + 4n + 1 2 mn
2
.
9
7
,
.
2
6
1 65
3
1 00 r
4
.
5
7
8
fo ll o wi n g
the
fa c t o r s of
.
(a
-
b)
2
—
(c
.
6
12
2
.
4 9x
— 42 x 2 ——9 4
y l y
13
9a
6—
b
4
c
4
15
17
a
2
— mz
.
(
—
—
b c
)
'
—
a
(
4
.
3y
—
.
.
a
—
(b
( a:
-
y
y)
9(a
— 4x
—b
) l
31
.
a
2 —-
l
2-
.
2
4 9a + 1 4 a c + c
a
x
) + 16
-
—
1 2c a
.
a
2
—
b
c
—
2 bc
2 -J-
f
2
1 6a
(
.
(a
—
l n
-
)
2
b) + 4 a
2
4
32
6
.
933
f‘—
2
2
—
2:13y l 2a c cr
y +c
35
.
.
— 4 x2 - - 2x — 8
l
3
- -
+4
2
2 -¥ 2
—
2 xy+ 9 x
y
—a 2 0 2
2
y
—81
2
c
28
.
.
22
2
1 44 b
4
—
—
3y 2y 2
2
36
2
xy+ x
2
~
a
.
4—
2 cx + 2 c
25
33
z
b
—
16
y
)
29
2—
—
—
ax
a + bx
x
2—
—
9
6x x
2—
2 533
b
4
—
b
c
26
(
a
8
19
23
a
.
6—
+ 2a + l
20
a
2
.
a
z
+
—
m
2
2
c
+ 2a x
2—
2—
—
b
2
x
2
a
+
+b x
y
y
2
38
2 - —
2
—
2 5a
20 a b l 4 b
4
.
F A C TORI N G
OF TW O B INO M IALS WIT H
P R O D UCT
14 7
TER M
A CO M M O N
Type form : (x + a ) (x + b )
-
2 03
t h at
M u l t i pl yi n g m+ a b y
.
a re s
i m il ar w i t h r e fe r e n c e t o as
,
(x + a )
fi rs t t e r m
The
t h e prod uc t o f x
a an d
f
o
of
a nd
we fi nd :
2
=
(x + b ) x +
th e pr o d u
the
tw o t erm s
it in g t he
su m o
f
ct
is t h e
a an d
f
s qua r e o
t h i rd t er m
b, t h e
c
t e rm
is
is t h e pr od u c t
of
the
x,
s e on d
b
.
The pr odu ct of two bi n omia l s wi th a com
term i s the squ a re of the common term , p l u s the pr odu ct
204
mo n
x + b an d un
P rinc ipl e
.
common
the
p r odu ct
f the
o
‘
.
te rm
a nd
u n l i ke
su m o
f
the
terms
the un l ike term s , p l u s the
.
E x e rc i s e 80
G i v e t he
1
4
13
16
(x
(
4) (
—
x
3)
— 5
)
.
11
.
14
b
(
.
.
.
8
.
22
.
5
—
x
.
.
28
d uc t s
2
.
19
25
r
p o
.
7
10
f o ll o w i n g
—4
)
(a
(x
.
(
.
a
—
.
17
.
20
.
23
.
2)
26
.
— 5 a— 2
)
)(
29
.
w i t h o ut m u l t i pl yi ng
( 2 a + b)
( cc
—4
2
y)
2
b
3
+
(
)
2
a
—
(4 x 2 )
?
—
( ar t by)
3
.
6
—
n
6
—
) (n 5 )
(
— l
.
)
9
12
15
.
(
n
—4
)
.
4
1
3
(
18
.
(a d
21
.
( Sr
24
.
2
—
b
a
i
H
(
)
27
.
(4 a
30
.
(
—
n
8 ) (n
-
2)
(x
—4
)
(
a
i t h m e t i c a l n u m b ers m ay so m e
t i m es b e c o n v e n i e n t l y f o u n d b y ex p r ess i n g t h e m a s binomi a l s
Thus
w i t h a commo n term
20 5
.
of
Th e pr o du c t
.
46 x 3 6
57 x 4 2 =
two
ar
,
—4
)
-
24
—8 = 2 5 00 — 50 — 5 6
)
E
1 48
LE M E N TARY ALG E BRA
E xe rc i s e 8 1
G iv e t h e
1
4
follow i n g produc t s
3 8 x 23
.
2
5
.
.
—
( 80
.
2)
SPE C I AL QUAD RA TI C T RI N O M IAL S
Typ
e-
fo rm :
x
2
+ ax + b
T h e pr odu c t of a ny t wo bin omia l s w i t h
is r e pr e se nt e d b y t h e followi n g t ri no m i a l :
20 6
.
2
x
a commo n term
+ ax + b
vi d e nt t h at t h e fac t o rs o f s u c h a t ri n o m i a l a re t h e
two bi no mi a l s o f w h i c h it is t h e pr od u c t
2
T h e fi rs t t e rm o f e a c h f a c t o r .is t h e s qua r e r oot of $ i e x
t h e se c o n d t e rm s ar e t h e two fa ctor s of b whos e s um i s a
2
x + 9x + 1 8
S i m il arl y
( a: 6) ( x + 3 )
2
—
—
x
6) ( x 3 )
2
—
x + 3x
3)
2—
—
x
3x 1 8 = ( x
T h e f a c t or s o f b w h o se s um is a in t h e s e f o ur e xa m p l e s ar e in o r d e r :
It is
e
.
,
.
.
,
,
.
,
’
,
If
th i r d t er m
the
f
ac tor s
+6
an d
+3
6
an d
—3
of
a nd
—3
and
+3
t r i no m i a l is pos i tive t h e
th e t h i rd t e r m is n eg at ive
the
h a v e l i ke s ig n s ; if
.
t h e factor s h a v e
+6
—6
u n l i ke s
,
,
igns
c o nd t er m s o f t h e
t h e s e c o n d t erm s o f
'
se
.
E x e rc i s e 82
G iv e t he
1
4
7
10
13
.
a
2
2
fa c t o rs o f t h e foll o w i n g
— 7a + 1 2
— 7a — 1 8
.
a
.
—
a
2
2
5
l 15
8
+ 9a + 2 0
11
.
a
.
—
a
Q
—
8a
2
a
132
14
.
.
.
n
n
n
2—
2
2
—
n
12
—
+ u 30
+ 6n + 5
-
.
.
71
2
+ 2n
56
—8
2
— 1 7x + 3 0
2
+ 1 4x + 4 8
2
— 1 1x — 1 2
2
+ 1 3x + 1 2
x
x
x
x
—
1 1x + 30
x
2
E
1 50
LE M E N TARY ALG E BRA
E x erc i s e 83
G iv e t h e
f ac t o rs of
—2
2
1
.
3a + a
—
—
4x
x
6a
fo llo wi ng
2
5
2
—
+a 2
2
5x
— 1 7x + 1 4
7a
2
— 1 7a — 1 2
2
— 4 5x — 1 8
8x
'
2
—
x
z—
7x +
8a
6x +
2
11
.
13
.
2a
2
-
6
9a
—
a
9
—
4x
—
x
5
1
.
2
—
—
—
1 l 6x 7 2:c
O
O
I
‘
a
O
6x + 3 1 x + 35
D
C
13
15
17
19
21
23
25
27
4—
—
1 1x
42
4—
2-
14 a l 4 5
4—
2 1x l
x
.
a
.
x
.
3a
.
.
a
2
2
R ev i ew
of
-
.
9a
8
x
a
6
6b
2
2
9y
m
Fac tori ng
3x
2a
2—
2
2-
2
—
—
—
l 5 a ac 3ar
3
—
l 1 6 + 7 2at
-
4 9a + 98 a + 4 9
4
2-
—80
1 2 1 b + 88 b+ 1 6
-
— 8 a 91J— 2 0x2
+ 1 1x
3
2
—
3ab 45 b
1 8a l
— 8a c
4
2
3
2—
—
1 2a
1 2a
9a
— 60
+ 1 5a + 5 6
-
2
— l 3 a — 30
2
1 3x + 1 2
'
-
— 9ac l—8 0 2
2
5a b
-
8 90 4 6fcy
2
-
l
4 9:cy
— az —
2
—
—
—
l
8 :c l 1 2a:
a
.
2
2
+ 1 60
.
.
6a
-
+ 1 9a + 84
2
Q
11
2
2
2
- -
foll o w i ng
— 1 2a — 2 8
2
a
the
8 xy l 3 y
2
5a + 3
E xe rc i s e 84
F a cto r
—
+ 3 2a 1 6
2
—
2
4 9y
b + 2a b
2
—- 1 6 f -l- 3 2 2
i
y
y
2
1 2a + 3 1 a az+ 9ar
2
2
8
36x + 2 5x
—- 2 3 13
l
— 60 x5
2
1
2
+
y
y
2
4 9x + 7 0xy+ 2 5 y
2
F A C TORI N G
?
1 5x d i r y
a
—4
g
2
’
"
c
b
.
-
2b l
2—
2-—
4c l 4
8a c 4a
34
a
.
2
+4 c
—
—
—
2 (x
4 ( a :c)
37
a
z
—
b
ac
z—
(
a
(
9x
)
36
by +
.
2
2
y +z
—
a b + br
.
)
2
— 9x ?
bm l
-
.
19
.
— a2 -
c
z—
b + 7b
3
.
2
—
39
—
z
—
bz
—6
2
3 13 4
.
br + y
42
z—
2
—
l
x
2
:
c
+
y
y
.
—
by a c
—
'
2
45
y
—bn —
2a c
—
3b 21
—
—
16
.
2-
yz
—1 2c r
4zc l
—
48
3
p
.
—
3 pq
-
.
—
bp c rn
— n ? z2
2
b
2
z
—
by
—
a sc
60n x
—
—
+ 6x 1 4 a c
— 4: b2:r 2
46
—
m
2
l
43
(fe
—
z
—
—
c m
a bm
— b2
z—
a
—
ar
.
40
z
33
2
.
2
—1 — d 4
—2 0 d2
2-
—
3 6n + 2 5 x
2
30
4—
31
151
—
2
9c
2
p
—3
p q l
-
2
q
I NC O MPLE TE T RI N O M IAL SQU ARE S
Type - form :
208
So m e t ri no m i a l s
.
tri nomia l
s qu a r es
re solv e d i nt o
b y t he
x
4
2
2
4
+x y +y
b i n o m i a l s w h i c h ma y be ma de
a ddi t i o n o f a sq u a r e t o t h e m m a y b e
a nd
two t ri no m i a l
fac t o rs
Fo r
.
e xa m p e , c o ns
l
id e r :
9a + 2 a b + b
4
t ri no m i a l wo uld b e t h e
T hi s
c e nt o f
i
9a
4
t he
se c o n
2
4a b
9a + 6a
2
n ot
t e rm we r e 6
2
“
b +b
4
2
2
.
u a re of 4 a b if t h e
Pr o c ee d t h u s
2
,
c o e ffi
2
,
and
we
z
( 3a + b + 2 a b) ( 3 a + b
w h i c h e q ua l s z er o t o 9a + 2 a b
a
2 2
—
b
4a b ,
c h an g e d
sq
2
4a b
2
A dd in g 4a
is
d
4
2
“
b
2a b
2
2
4
2
2
b
2
2
2
2
4
,
t hen
2
2
2
2 a b)
+b
4
,
h a v e t h e d iffer en ce of two s qu ares
the
.
l
v a ue
E
152
209
When t he
.
L E M E N TARY ALG E BRA
se c
on d
t erm
two difi eren t s qua res m ay in
’
4a
4—
5a b
a
4a
T he
4
—
l b
4a
a
dd e d
2
2
b
4
f ac t o rs of
a
b
z 2
‘
4a
t h es e t w o
( 2a
r es u
l ts
2
2
t hus
9a b
2 2
2 z
—
l 4a b
b
4
-
z—
b
2
( 2a + b + 3a b) ( 2 a + b
2
,
9a b
2
z
a re
b + a b) ( 2 a
2
4
g a t iv e
4
9a b
2
2
,
ne
,
5a b + b
4
b
2
be
c as e s
2
4a b
4
om e
2-
2
2
s
t he t r i n o m i a l is
of
2
2
)
ab
3 a h)
2
T h i s would s ee m t o i n di c a te t h a t t h e e xpress i o n h as two sets
of pr i m e fac t o rs bu t this i s impossibl e
We find th a t e a c h o f th ese fa c to rs m ay be f a c tor ed b y t h e pr e c ed i n g
c as e 20 7 giv in g th e fo ll o w in g f a c tors :
.
,
,
,
—
2
a
<
-
b><a
T hes e fa c t ors th o ugh a rr a n g ed d iffe ren tly
b>
— b
>
li ke a n d w e c o n c l ud e
t ha t when two s qua res ca n be added to th e express io n it c a n b e r es o l v ed
i n to four bi nomia l fa ctors a nd it is i mm a t er i a l w h i c h o f t h es e t wo s q uar es
is a dd ed to t h e e xp re ss i o n
Wh e n a bi no mi al c an be f a c to red b y this m e th od it c an g e n era l l y
be r es o l v ed in to four bi no m i a l f a c t ors
,
,
ar e a
,
,
,
.
,
.
E xerc i s e 8 5
F a c t or
1
4
6
8
10
12
14
4
+4
2
.
x
.
—
4x
1 7x + 1 6
4—
3 4a + 2 5
4
.
9a
4
.
x
4
2
2
1 9a
.
a
.
“
—
8a
9a
4
.
8 1x
r
.
64a + 1
3
5
7
4
— 1 0 x2 2
y + 9y
-
4
9
2
4
—
b l 25 b
2
b + 1 6b
.
3 6a
4
4
33
.
24a
2
:r
.
2
2
2 5 33
4
2
z
4
“
+ 4 9y
4
2
2
4
2
2
4
64 x + 7 6:1: y + 4 9y
4
2
4
8 1 a + 2 6a b + 2 5 b
4
2-
2
+ 4y
7 2 :1: y + 4 9y
2
.
2
4
4
4 0x y + 4 y
l 6a
4
7 6a
x
2 5 17
E
1 54
L E M E N TARY ALG E BRA
S UM
SA M E O DD
THE
or
Type fo rm :
-
21 1
s um of
The
.
of a
t h e pr o du c t
T he
s am e o d d
th e
b i no m i a l
fo llow i ng
r
o
p
x
a nd a
3
P
O WER S
‘ 3
f y
- -
o
p
we r s
of
po l y no m i a l
d u c t s m a y b e v erifi e d
t wo
n
u m b e rs
is
.
b y m u l t i pli c a t i o n
2
— x
y+ v )
3
=
x +
x +v
x
The bin omia l
two ter ms
3
6
5
5
(x + y) (x
+v
fa ctor
i s the
the bin omia l
f
o
The term s
a n d n ega tive
f the
o
su
4
-
m of the
r
4
wy + y
)
sa me odd roots o
f
the
.
pol yn omia l
fa ctor
a r e a l ter na te l y
pos itiv e
.
E x pon en ts i n the pol yn omia l fa ctor decr ea s e
by th e e xpo n e nts in th e bino mial fac to r
a nd
in cr ea se
.
E x erc i s e 8 7
G iv e t h e
1
4
7
10
13
16
19
22
25
3
a
.
a
.
fa c t o rs of t h e f o llo wi n g
+8
2
3
—
Ib
9-
l l b
- -
.
6
3
+y
.
x
.
—
8
.
5
—
a
x
l b
5
x
.
7
+b
.
a
.
—
1
x
23
7
3
a
.
3
x
.
20
7
5
+ 27
+ 32
8r + 27
.
17
+1
9
+ 64
6
14
5
a
.
11
3
x
.
8
9
n
.
5
9
x
.
+ 12 5
+ 216
6—
—
a
3
.
26
5
36
.
34 3
5 12
4 24 3
-
3
.
6
.
9
12
15
18
21
24
27
5 1 2 l 64 a
- -
.
»
l
—
27
9
—
7 2 9 F643:
3
.
.
2 7 x + 64 y
8a
3
3
—
l l 25 b
3-
y
7 2 9x + s
3
.
.
l 000 a
3
—
l b
8 x + 34 3 y
3
64 a l 2 7 b
‘
5- -
.
s
3-
3
.
s
6
bi n o m i a l s espe c i all y t h o se t h at a re t h e di ffe r
e n ce of t h e s a m e po we rs h a v e m o re t h a n o n e b i n o m i a l f a c t o r
fi—
fi
h as 5 bino mia l divi s o rs
x
Sho w
T h e bi no m i a l a
“
G—
x
w h a t t h e y a re a nd wh y t h e y are divi s o rs o f a
212
.
So m e
,
.
,
,
,
.
.
FA C TORI N G
2 13
.
S umma ry of Fac torin g
I Fi r st t a ke
1 55
.
m o nom ia l
al l
f a c t o rs a n d re t a i n t h e i r
pr o d u c t a s on e f a c t o r o f t h e giv e n e xpr e ss i o n
A ft e r t h e m on o m i a l fa c t o rs are r em o v e d n ext n o t i c e t h e
n um b e r o f t e rm s in t h e r e m a i n i n g f ac t o r
o ut
.
,
.
,
.
I I B i no m i a l s a r e f a c t o re d
.
(a)
Th e
di ff e r e n ce
—
a
2
b
h
T
e
( )
( )
c
s um o f
Th e
a
a
=
b
+
sq
—ax —
—
l b
2
,
o r ax
re
du c e d
2
4
+X Y +Y
g r o u pi n g
(
a
2
,
as
t hu s ,
2
+ b x + c , by inspection a nd tria l
t o IIa , t h u s ,
=
( KY)
or
m o r e t e rm s
2
a re
fac t o re d
t e rm s , t h us ,
a+b
)
x+
(
a+h
)
y
=
(
a+h
B y IIa , t h us ,
+b +
2
.
t ri n o m i a l ,
.
( b)
2 z—
3
3- - 4
—
—
( a a h l a b ab l b ) , e t c
u a re
IV Pol yn o m i a l s o f fo u r
By
+b )
2
2
( 0 ) A fo rm
X
po w e r s ,
zab + b =
zi
4
.
4
f a c t o re d
a re
H
(a )
et c
sa m e o d d
(a + h )
t ri no m i a l
x
po w e rs ,
)
—b
)
the
A q u a d rat i c
b
( )
sam e o d d
— ab
5
.
a
the
H
I I I T r i no m i a l s
(a ) A
of
—b 3
5
t hus ,
—
b
a
b =
(
5—
5=
b
a
(a
a
s q u a res ,
—
=
b
(a + h ) (a b )
3
-
t wo
2
di ff ere n c e
3
of
as :
2 =
—
2a b
c
(a + h )
2
-
c =
2
(
a+h
— c
)
)(
)
x+y
.
E
L E M E N TAR Y ALG E BRA
RE V IE W
214
O b se rv e t h e
.
foll o wi n g rul es
and
fa c t or t he
r i
exe c se s
b elow
I If the
s hou ld
11
f
o
in
ev er y ca s e
f
o
fi rs t be
s hou ld
a n expressi on un til
you
E x e rc i s e 88
4
a
4
x
x
6
fa ctored
are su re
fa ctor
G en e ra l R e v i e w
+
a
.
4—
81
5
11
fa ctors
the
.
Factorin g
2
4
—
1 0x
f
a s one o
of
+ 15 a + 44
—x
the d ifi er en ce
as
it i s p rime
2
3
fa ctor
.
a compou n d
wri te
n ot
.
tha t
,
.
two squa res , if this is pos s ibl e
III D o
fa ctor
be removed fi rs t
A l l bin omia l s
.
m onomia l
expression co n ta ins a
.
7
.
8a
z
2
—
—
l 3 7a b 1 5 b
-
5-
.
.
x
x
50a
2
—
x
l
y
y
6
-
— 3 5 a h — 41 b2
fi
8
9
2
x
.
2
+ 2 xy + y
-
—
4x
4y
2
- —
—
1 8a b l 8 b
9a
2
.
11
a
2
— m2
+ an
-
.
2 5a
9 (3 b
-
-
ax
+ cy+
4x
— 6l
z
x
4-
y l
2
16
(
a
z-
l
-
2
x
—
y)
2 2
.
.
.
.
.
5x y
( + x)
—
—
2
8
.
20 x +
—
8x
—
1 8a
dx
2
.
y
a
2
a
3
5x y
2x ( a +
20
—
x
23
— b2 —
2
-
9
-
2
4x
c
40 l
-
-
.
a
—2 b c - - a — b-ll
l
26
3
x
.
1)
z-
.
5
— a4
c
1 + 1 9a
2
— 2 0a 4
2
2—
—
—
—
2a b l 4 c
1 l b
29
.
30
2
—
—
6xy + 3 y + 3x z 3 yz
3x
2
.
2
6
a
—
—
—
b)
3 (a
( b a)
30
2
z—
z—
4—
—
z
2z
2 xy
1
x +y
— 4d 2x2
27
32
4
2
—8 1
24
x
— cx —
y
y
13
19
21
.
—
4x
3
.
2 c)
17
f
—
a
x
mn
14
2
10
6—
3
—
0
2
—3c
-
l 1
-
S E C O N D H A L F- Y E A R
C H A P TE R XV
E
Q UATI O N S
E XE R CI S E S FOR
.
RE VIE W AND
P RA CTI CE
S OLUTIO N O F
21 5
.
kn o wn
E
Q UATIO N S
B Y FA CTO RI N G
M a ny e q u a t io n s c o n t a i n i n g t h e s q u are o f t h e un
n u m b e r m ay b e s olv e d b y m e a n s O i f a c t o r i n g
Th u s
.
,
2
—
=
7x
8 4x + 1 9
2
t e rm s t o t he
fi rst m e m b er u n it e t h e
2
t erm s c o nt a i n i n g x a n d divid e bo t h m e m b e r s of t h e e q u a t i o n
2
b y t h e c o e ffic i ent of x we h a v e t h e foll o w i n g :
I f we
t r a n spo se
al l
,
,
,
9
2
x
0
F a c t o ri n g t h e fi rst m em b e r t h e
pr odu c t o f t w o f a c t o r s e q u a l t o 0
,
r es
ul t s h o ws t h e i n di c a t e d
.
—
=
3) 0
if on e of i ts fa ctors is 0 Si n c e t h i s
—
n
f
i
i
i
r
s
a
t
l
a
s
t
o
e
f
ac
t
o
t
s
0
0
r
d
u
c
t
3
e
o
o
)
p
—
=
I f x + 3 0 t hen x
t hen x = 3
3 ; a n d if x
3
S i n c e b o t h n u m b ers s a t i s fy t h e e q u a t io n x = + 3 a n d
A pro du c t
is 0 ,
.
.
,
.
,
-
,
B oth
a r e ro o t s
.
It is i mp orta n t to
the
f
s qu a r e o
21 6
.
The
the
n otice
un known
st at e m en t
lit t l e g r a ph i ng
.
.
°
here tha t
n u mber
in i t a li c s
e qua tions
ha ve two
can
roots
con ta in i ng
.
be m ade
l
c e a re r
by
a
.
S OLUTIO N
I
We
.
,
OF E
b e gi n b y Sh o w i ng t h e
x
We fi rst
l ul a t e
ca c
a nd
BY FA C TORI N G
Q UATIO N S
2
g ra ph i c a l
4
—
4
x
s o ut o n o f
l
i
( 1)
0
m a ke t h e
159
g r a ph
Of
fi rs t m e m b e r
t he
2
l ul a t e
T h us
ca c
x
?
3
1
.
1
x
—3
ash
—4
2
x
2
—4
3
2
4 =0
x
2
4
+5
v a lu es h o ri z o n t a ll y a n d t h e c o r re spo ndi n g
2—
4 v a l u es v ert i c a ll y a n d c o n n e c t i n g t h e po i n t s w i t h a
x
w e o b t ai n t h e c u r v e o f
s m oo t h c u rv e
t h e figu re
W h a t v a l ue
E q u at i o n ( 1 ) re a ll y as ks
2
Si n ce x
o r v a l ues O f x m ak e x
2
4 v a l ue s a r e t h e v ert i c a l di st a n c es
t he x
f r o m t h e h o ri z o n t a l t o t h e c u r v e t h i s
“
n
h
a
re
t
n
m
n
s
as
k
a
i
u
i
g
W
es
t
a
o
t
t
o
o
u
q
=
fffll mn
“
m m“
t h e x v a l ue s w h e re t h e c u r v e c r o sses t h e
4
G ra p h O f x
”
T h e a n s wer is se e n f ro m t h e
x a xi s ?
A P a rab o m
2
figu re t o b e x = + 2 a n d x
2
—
sa t i s fy it
2 s ub s t i t u t e d fo r x in x
B ot h + 2 a nd
F o r t h is e q u at i o n t h e n t h e r e a r e t w o v a l ue s o f x b e c a u se
t h e c u r v e c ro ss e s t h e h o r i z o n t a l in t wo po i n t s
2—
O b serv e t h at w e g ra ph f( x) x 4 a n d O b t a i n t h e pa ra bol a
T h e s o l ut i o n s O f f ( x) = 0 a r e t h e x v al ue s o f t h e c r o s s i n g
xis
n
t
a
a
r
h
h
r
z
i
n
of
h
r
b
l
a
v
t
e
o
i
o
l
e
s
t
e
a
a
o
o
o
t
p
p
G r a ph i
n
g
the
x
-
-
,
,
.
,
'
-
-
,
,
-
l
1 h
s p ac .
2
1
P
r
2 “
-
-
.
.
,
.
.
-
.
I I Le t
.
l
u s n o w so v e
g ra ph i c a ll y t he
x
G r a ph t h e
=1
,
2,
=
4,
3
x
+
10 ,
x
x
2
fi rst
m e m ber
2
eq
u at i o n
=
x
0
3
+
x
2
+ 3 x fi r st
,
( 2)
l
l
i g
c a c u at n
E
1 60
LE M E N TARY ALG E BRA
G r a ph i n g
a nd c o n ne c t n
i g t h e poi n t s
,
g ra ph o f x + 3x as in t h e
figu re
C l e a r l y t h e x v a l ues of t h e
i e t h e poi nt s w h e r e
c r o ss i n g po i nt s
2
x + 3 x is e q u a l t o 0
a re x = 0 a n d
—
x
T h ese v a lu e s b o t h sat i sfy
3
2
and
x + 3x = 0
w e a g a i n h av e t w o
v a lu e s be c a u s e t h e re a re t w o c ro s s i n g
we g et t h e
2
-
.
-
x
.
,
.
,
,
h
1
I
v
ore r oc na a spspa
ce
ace
iz
t
ti
l
.
l
,
G r aph Of x + 3x
A P ar a b ol a
~
m
o
t
s
p
.
= x2 + 3x
g ra ph f ( x)
o b t a i n i ng a
c r o ss i n g po i nt s ov e r
t h e h o r i z o nt al giv e
t h a t a re t h e so lu t i o n s of f ( x ) = 0
H ere
we
p a ra
the
-
bol a w h ose
x di st a n c es
,
-
.
I I I Th e g ra ph i c a l
.
so
x
is
ob t a ne
i
1,
x
x
2
T he
2
6x + 8
d b y fi rs t g r a ph i n g
C al c ul at i n g t h e
t h e m o re
lu t i o n o f
—
x
2
g e n e ra l f o rm
O
6x + 8
.
v alu es
2,
4,
5
.
0,
0,
+3
,
is s h o w n in t h e
figure a n d
t h e re a r e a g a i n t w o c r o ss i n g p o i n t s
x = + 2 and x = + 4
a n d t h e s e s a t i s fy
c
u rv e
-
,
,
x
2
u at i o n c o n t ai n i n g t h e
s q u a r e of t h e u n k n o w n t h e g ra ph o f
t h e fi r s t m e m b e r w o ul d b e s u c h a
c u r v e as w e h a v e f o u n d a b o v e
Fo r
any
eq
.
H e n c e equa ti on s con ta i ni n g the squa re
,
of the u n kn own ha ve, in gen er a l , two
T h e p a ra b ol a is
its
o ss i ng poi n t s
o f f ( x) = 0
cr
-
r oots
W
Y
,
2
e
ye rt i c a l
l
spa ce
G ra ph of x —6x + 8
A p ara bo l a
2
.
x
l
t h e g r a p h o f f ( x)
6x + 8 a n d
ov er t h e h o ri z o nt a l giv e t he s o lu t i o n s
a so
2
-
,
L E M E N TARY ALG E BRA
E
1 62
E x e r c i s e 90
Q ue s tio n s
an d
O ral W ork
A nswe r
t h e q u e s t io n s in n u m b e r s y m b o l s
o p e r a t i o n s i n di c a t e d in t h e e v e n n u m b e re d
the
1
.
'
W h at m a y
w h i c h is t w i c e t h e w id
of
l e n gth
— 1
)
2
3
.
4
.
5
.
.
7
.
a
m an
yea rs
0
liv es
x
r
s
e
a
,
y
9
.
m a n t o b uild m yar d s
a
)
t i m es
4
(4 x + 5 y)
pr o d uc t o f
t he
( 2a
5
n e xt s m a
of
H o w ol d
.
—
-“
a
a sq
a n d x c ube ,
u a re
.
m
(
2
odd n u m b e r W h a t
re pres e n t s a n
5)
3y
,
—
x
will
5) (
—
x
5
)
r e pr es e n t
ll e r o dd n u m b e r ?
( x v + 5 x)
.
—
2
E x pres s 7 t i m e s t h e t h i r d po w er
3 t i m e s t he s um o f 2a a n d 5 b
11
Ol d
2
—
b y n t i m es t h e n u m b er 2 x
I f 2x
t he
— 1
)
2
h e w ill b e y ye ars
(a
8
gl e
ag o ?
—3
E xpres s
re c t a n
a ny
.
2
—
( 23a 4 b)
(
If
i ses
d ay ?
—
—
n
n
a
a
)
)
(
i n c r ea s e d
10
a
of
exer c
?
th
H ow m a ny d a ys w ill it t a k e
was h e
6
are a
( 2x + 3 9)
if h e b uild s n fe et
w a ll ,
the
t he
re pre s e n t
f
p er o r m
and
.
of x,
4)
di m i n i s h e d b y
.
12
13
—2
)
.
.
A
in
14
15
.
.
(
.
17
.
sma
(
(
a par k
—
n
a
If
z
5
-
—5 a
) ( +4 )
c e nt
pi e c es ,
.
x
If
(a
b o y h a s x Silv e r d o ll a rs , y di m e s , a n d
W h a t e x pr e s s i o n e q ua l s 5 8 0 ?
al l
t i m es m u st
16
( 3a
— ab z
)
a
is l
m an
s
on
f o rm e d
(
z
x
w id e , h o w m a n y
n
m il e s ?
(
—
n
9
)(
—
n
5
)
ddi n g 3 feet o n a l l s id es o f a
h o w m a ny s q u a re f ee t a r e a dd e d ?
a s q u a re
,
to
)
n
+
)
)(
ll er s q u a re
ds
t ra v e l
—
b
a
a
5
(
a
is
0
a n d -w r o
g
w a l k a r o u n d it
ro
d l
x i/+ 4 1
2
by
a
S OL UTIO N O F
E
BY F A C TORI N G
Q UATIO N S
1 63
E x e rc i s e 91
S i m plify t h e
th e
1
3
5
r e st
3x
.
fi r s t six O f t h e fo llo wi n g
fa c t o r i n g :
by
2
1 5a
-
2x + 1 7 a
2
.
2
2
= 2 x2 — 1 1 a 2
= 5 x2
2 =
2
—
4 3a
3x
7x
2
.
So m e
1 0a
~
t
2
2
4
1 6a
t h at
6
i both
eq
uat ions
2
.
3y +
a n d so
lv e
2
2—
=
4y
13 b
12b
2
2=
2
—
5y
4 0n
3y + 3 2n
2
.
2
2
=
6y + 24 b
8y + 22 b
2
.
2
t he
fi rs t a n d t h e sec o n d
po w e r s o f t h e u n k n o w n nu m b er m ay b e so l v e d b y fa c t o r i n g
eq
u at i o ns
c o nt a n
.
2
x
the
4 x + 1 9x
+x
—
=
4
x
7
2
a nd
The
.
sq
num b e r
8
9x
.
2
-
3
x
o f a c e rt a n n
u a re
is 1 3 0
i
Fi n d
.
—
12 =
6x
t he
9
.
t he
8x +
2
13
16
is 240
2
2
=
8 0 4x
sq
.
12
.
3y
—
2—
—
=
15
3 y 2 5 2y
i
a c e rt a n n
um ber
.
5x
.
T he
.
—
1 4n
2
.
5n +
2—
=
18
6 3n
,
.
s um o f
the
Fi n d
t he
15
s q u ar e s o f
“
n
u m b e rs
.
.
.
c o nse c
u t iv e e v e n
-
15
n um
.
18
.
2
—
=
7 y + 5y 4 5 2y
2
t wo
T h e q uo t i e n t of o n e n u m b e r
Fi n d t h e
a n d t h e i r p r o du c t is 2 5 6
19
2
u a re o f a n u m b e r t h e n u m b e r i t s e lf is a dd e d
Fi n d t h e nu m b e r
2
=
+ 2 1 x + 3 6 2x
b e rs is 5 80
17
—
4x
I f to t he
s um
14
.
5 ti m es
a nd
b y 3 t i m es
.
.
.
num ber
3x
.
11
u m b er di m i n i s h e d
2
T h e pro d uc t o f 3 t i m es
Fi n d t h e n u m b e r
is 7 3 5
10
= — 5 a nd i
2
.
8y
—
2 —
=
52
3 2y + 8 4 y
divid e d b y
n u m b e rs
a n o t h er Is
4,
l e ng t h o f o n e s q u are fi e ld is t w i c e t h a t o f a n o t h e r
a n d b o t h t og e t h e r c o nt a i n 1 2 8 0 s q u a re ro d s
W h at is t h e
l e n gt h o f e a c h s id e o f t h e s m a ll e r s q u are ?
20
.
T he
,
.
E
1 64
L E M E N TARY ALG E BRA
E XE R CI S E
F O R RE VIE
S
O ra l
E x e rc i s e 92
A nsw er
in
n
um b e r
AND P RACTI CE
W
P r ac ti c e
s ym b ol s a n d
pe r
form i n di c a te d o pe r
a t io n s :
1
.
Wh
at
w ill
h u n d re d s , y t e n s ,
the
as
2
.
3
.
.
5
.
.
7
.
(a + b )
I f 2x + l
.
r e pr ese nt s a n O d
wh i c h t h e r e
2 2
d
n
—
s
5
-
u m be r
,
w h at
(a
A h as n c ows B
,
.
11
.
b o ys ,
12
.
13
.
w a ll s
( a + 5) (
A
h a s 5 m o re t h a n
—4
)
(a
r e pr e s e n t
4-
,
14
.
15
.
w h i c h m is t h e m iddl e
(a
b)
(a + 8 ) (a + 2)
5 2
5
(i
xx
.
—x
H o w m a n y S qu a r e yar d s
4 x ft b y 3 x ft
.
W h a t m a y r e pr e s e n t
At
t o pl a s t e r
feet
W ha t w ill
.
(
2
—
b 9
are
an d
t he
u a r e yar d
c e ili n g l f e e t l o n g a nd
,
t h e r e in
a
)
we e k fo r
(m a )
t h e c e ili n g a n d
y ft h ig h ?
.
(a
2
r
a ea
of
a ny
fee t
rect a n
gl e
t he
wid t h ?
wh a t w ill it
w
)(
—
b 6
—3
?
i n c h e s g reat er t h a n it s
a c e nt s a s q
a
v)
b
a
3
5
+
(
)
)
w h i c h is 8
.
)
one ?
d o ll a rs a w e e k for m e n a n d b doll a rs
h ow m u c h will 6 o f e ac h ear n in 4 w e e ks ?
—7
m a ny
—8
At m
roo m
as
)
of fi v e c o nse c u t iv e e v e n
( 3a + 2 b)
(b
l en gt h of
16
C h as
an d
—
a
3
“
2
—
l b)
s um
t he
(x
of a
—
s
H ow m a n y h a v e a l l ?
t o g et h e r
W h a t w ill
— b3 2
)
3
.
10
2
) ( 4)
(
will r epr ese n t
Q
T h e pe r i m e t e r of a s q u a re is l 2 x
d e no t e t h e nu m b e r o f s q u a r e fee t in it s ar ea ?
9
a re
ll e r O d d n u m be r ?
(x
n u m b e rs o f
8
u m b er in
u ni t s ?
(a
A a nd B
6
a nd x
n
a
2
n e xt s m a
4
r e pr ese nt
w id e ?
c
o s t in d o ll a rs
L E M E N TARY ALG E BRA
E
1 66
O ral
E xe rc i s e 94
P r ac ti c e
F o r m ul at e t h e o d d n um b er e d exe r c i s e s
p r o d u c t s in t h e e v e n n u m b e r e d e x e r c i se s
and
giv e
the
.
1
.
If
gl e is 6 in lo ng er t h a n
if e ac h is i n c rease d 8 in
a re c t a n
di m e n si o n s
.
4
.
5
.
l
en ar
id e s o f a
(
ge d
6
.
7
.
(
x
W h a t m ay
(
—
x
7) (
W h at
n u m b e rs o f
8
.
9
.
.
11
.
13
.
(
the
—
x
15
5)
8) (
r e p res e n t
t he
s um o f
n
is t h e
—
a
of
the
8)
fi rs t
and
fo ur
c onsec
l argest ?
is p l a c e d o n t h e ex po n e nt s
l a w o f e x po n e n t s fo r m u l t i pli c at io n ?
i i
8) (
—
x
(
—
s
t he
7) (
s
—7
6)
id e o f a
—
s
2
sq
of
re pr ese n t ,
u s e d in
)
if
x
m the
i
e xpr e ss o n
u are ?
(b
7)
W rit e 5 t i m e s t h e
sq
(
u a r e of
—
t h e b i no m i al s , x
a
-
b,
—
7 a nd x 9
—
x
9
)(
W h at d o es (
i
.
17
.
—
4)
x
di m i n i s h ed by t he
.
1)
s o n r e pre s e nt s
16
the
u t iv e e v e n
.
.
)
(x
r e st r c t o n
W h a t d oes ( x + 2 )
produ c t
14
t h e pe r i m e t e r s
fi r st p r o b l e m ?
—
(a
whi ch
W h at
r e pr ese n t s
.
w ill
)(
—3
1)
a
f e et
3
(a
vi n g
10
—
x
—
2
a
r e p re s e n t
gl es in t h e
re c t a n
u a re f o rm e d b y a ddi n g
lo n g ?
u a re x f e e t
sq
-
are a o f a s q
W h a t is t h e
o n al l s
12
—
l 2)
— 1
)
3
t he
a re
.
,
2
p ro
w id e , w h a t
.
(8
t he
—
x
s
4) (
—
x
id e of
a sq
—9 s—9
)(
)
W h a t will
3 t i m es
r e prese nt if
,
x
in t h e
e xp re s
u a re ?
(b
t h e q uo t i e nt o f a n u m b e r
t h e s um o f t h e digi t s ?
r e pr ese nt
fig u re s divid e d by
3)
of
t h r ee
RE
VI E W
E x e rc i s e 95
So lv e t h e
P RA C TI C E
AN D
P r ob l e m s for R e v i e w
follo wi n g pr obl em s
a nd ex e c ses
r i
—
4
x
x+4
3 (x + 4)
4x + x
2
8
x
3
1 28
.
s um o f
T he
t wo
n
Fi n d t h e n u m b e rs
.
4
.
Th e Sum
b e rs is 3 7 1
5
.
t h e ir
t he
Fi n d t h e
.
s um o f
The
sq
of
u a res is
.
a nd
24 ,
t he i r pro du c t is
.
u a res o f t h re e c o n se c u t iv e O dd
n u m b e rs
sq
ve n n um b ers is
Fi n d t h e n um be rs
e
18,
a nd
t he
s um o f
.
Fi n d t w o n um b ers w h o se diff ere n ce is 8
m ul t i pli e d by t h e s m a ll er n u m b e r is 2 80
6
n um
.
t wo
1 64
u m b e rs is
.
and
w h o se
s um
.
7
sq
.
Fi n d
u ares
t wo
e xc e e
ds
c
10
o nse c u t iv e n u m b e rs t h e s um of
t i m e s t h e s m a ll er n u m b e r by 1 5 5
w h o se
.
Fi n d t h e s id e o f a s q u are wh o se a re a is d o ubl e d b y
i n c reasi n g it s l e ngt h 6 in a n d it s wid t h 4 in
8
.
.
9
.
e xc ee
Th e
sq
d s t he
A
.
u a re of t h e
s um o f
.
s um
of
t wo
u t iv e n u m b ers
Fi n d t h e n u m b e rs
c o ns e c
t h e ir s q u a re s by 1 12
.
.
m a ny d ays as h e re c e iv e d
doll a rs pe r d ay a nd e a rn e d $ 2 7 2 Ho w m a ny d ays did he
wo r k a nd h o w m u c h did h e re c e iv e pe r d ay ?
10
.
m a n w o r ke d 1 7 t i m e s
as
.
u a re fo o t it c o st $ 56 t o l ay a pa r q u e t
floo r in a ro o m w ho s e l e n gt h is 6 fe e t m o re t h a n it s wid t h
Fi n d t h e di m ens i o ns of t h e fl o o r
11
.
At 20 ¢
a
sq
,
.
.
r
e
p
A
m aso n wo rke d 3 2 d a ys m o re t h a n he re c e iv e d doll a rs
How m a ny d ays did
d a y fo r h is l ab o r a n d e a r n e d $ 1 0 5
12
.
.
he w or k
an d
h ow m u c h
did
he
re c e
iv e
pe r d ay ?
o pl a ne flew 50 m o re m il e s a n h o u r t h an t h e
n u m be r of h o u rs it fle w
It fle w 3 99 m il e s o n t h e t ri p in
?
H
n
i
i
n
m
a
k
n
h
r
i g t e t ip
o w l o g w as t
q u est io n
1
3
.
An
aer
.
.
L E M E N TARY ALG E BRA
E
1 68
O ral
E x e rc i s e 96
An swer
1
t h e q u est io n s
t he
i
s o n r e pr es e n t s
(
—
x
6)
i n di c a t e d o pe r a t io n s :
s
id e of
a
) re prese nt if
s q u ar e ?
—
(x 2 )
(
,
—
Zi a 5 b
x
in t h e
r
e xp e s
)
3 At x c e n t s a r o d h o w m a ny d o ll a rs w ill it c ost t o e n c l o s e
r ec t a n gul a r fi e ld I r od s b y w r od s ?
2
.
.
a
pe rf orm
and
—2
What do es
.
R ev i e w
,
—
4
5
str
ip
u are fo rm e d b y c u tt i n g off
r
r
l
l
f
r
a
d
s
w
id
e
f
o
m
a
s
id
es
o
a
s
u
a
x
r
d
lo
n
g
e
a
s
?
q
y
y
2
.
7
.
—7
)
W h at w ill
h u n d re d s , y t e n s ,
(a
(
t h e q u o t i e nt o f
u n i t s , divid e d b y 8 ?
r e pres e n t
an d
z
—
8
9
W h a t is
.
a nd s
5)
a re a o f a s q
Wh at is t h e
.
6
2
old
at a
4)
iv e d fo r x s h ee p boug ht
f
r
h
r
t
b
a
s
a
a
?
ofi
doll
d
o
e
p
re c e
—
10
a
—
n
8
at a
u m b er
n
(
) (n
—
n
3)
-
of x
—
9) (n
doll a rs
a
a
7)
h ea d
(
2)
n
'
11
.
A m a n w o r ke d 8 d a ys o f n h ou rs ea c h at x c e nt s a n h o u r
He s pe n t b doll a r s
12
.
13
.
a nd s o
r
14
.
15
.
od s
—
) (x l
.
17
.
H o w m u c h h a d h e l ef t ?
(a
)
W h a t is rec e iv e d fo r y h o rses b oug h t
ld
at a
l oss of q doll a rs
( ”cm
— 2
)
A
a
(a
r
h
doll
a
s
a
ea d
p
h ea d ?
—8 a
) ( + 1)
(
g ul a r fi e ld 5 x r o d s l o n g h as a
w ill d e n o t e t h e are a in a c res ?
—5
—
n
7
pe r i m e t e r
(
)
)(
—
n
8
,
,
—
n
of
)(
r
n
h
uo
i
n
i
s
re
r
e
t
e
d
b
e
ivi
s
o
t
t
e
t
es
d
p
y q
q
?
i
n
n
r
b
r
h
h
i
d
divid
d
m
w
a
s
t
e
re
a
e
t
e
y
I f th e
the
at
re c t a n
W h at
.
16
a nd
(
—
x
3
.
.
6)
1 8x
—
n
1)
by d,
E
1 70
16
.
su m
t he
R ea d
LE M E N TARY ALG E BRA
of
( a + b) ( x + y) a n d ( a
—
1 ) an d ( c
d ) (a
l)
R e a d t h e s um of ( a + c ) (n
of
a
17
Writ e
t he
s um o f a n y
.
that
i
a n ex pr ess o n
n um b e r s
two
.
19
.
t iv e
i nt e g e r s
is
a n O dd n u m b er
u are
of
rul e fo r m u l t i pl yi n g
t he
t he
sq
u are s o f t w o
c o n sec u
.
R e pre sen t 3 t i m es t h e s um o f t h e
n u m b e rs m u l t i p li e d b y t h e i r diff e r e n c e
20
sq
.
diff e re n c e
S h o w t h at t h e
5 t i m es t h e
re p r ese n t s
F r o m w h a t l aw d o w e Ob t a i n
pol yn o m i a l b y a m o n o m i a l ?
18
b)
.
sq
u are s of
any
t wo
.
Sh o w w h e n t h e pro du c t o f
po sit iv e a n d w h e n it is n e g at iv e
21
se
.
v era l
ne
g at iv e n u m b e r s is
.
22
.
—
3a b
23
.
Fro m
—
2 bd
bc ,
W h at
r e p re se n t ?
—
4a h
3a o
and
+ 2 be
—
bd
d o es
a
3
s ub t r a c t
3 bc +
,
—
2a o a b
.
+b
3
W h at
r e pr e s e n t ?
do es
x
2
-
2
y
— 1 r e re s e nt ?
) p
W h a t do es
D e fi n e coefil ci en t ; exponen t ;
t h e ir m e an i n g o r s ig n ifi c a t io n
24
s um o f
the
—
bd a o
.
a n d sh ow
the
di ff e re n c e in
.
W h at d o e s 2 ( a + b) r e pre s e n t ? W h a t do es
r e pr ese nt ? W h at d oes ( a + b) ( a b) rep rese nt ?
25
26
2
.
.
f ro m
27
z er
.
v a lue
28
l aw
.
—
f r o m 3 x 8 y+ 6 z ,
—
4 x 3y+ 2 z
—
S ub t r a c t 7 x 5 y+ 3 z
o
,
and a d d
Si m plify
wh en
to
—
=
=
a
7, b
3,
St at e t h e
law
s ign
of m ul t i pli c at i o n
.
su
b t ra c t r esul t
.
2
b
(
1 2a
c
4
c
+
)
—
c
of
4
b
a
5
3
+
(
)
fi nd
it s
.
m ul t i pli c at io n
P r ov e
an d
bo th l a ws
.
St at e t h e ind ex
.
R e prese nt 5 t i m e s t h e s um o f t h e sq u ar es of
n u m b e r s m ul t i pli e d b y t h e s q u a r e of t h e ir s um
29
2
—
b)
3 (a
.
a ny
t wo
.
30
of
.
HOW m u c h do e s t h e s q u a re O f
d t h e pr odu c t
G iv e r es ul t wi t h o u t s q u a ri n g
e xc e e
.
RE VI E W
.
31
—
4n
a + 4 b + 6m
fr o m
t wo
9n
171
and
5 b + 5n +
—
a
4m
.
R epre se nt t h e pr odu c t
o f w h i c h di ff e r b y 2
32
PRA C TI C E
—
—
5m a
s um o f
S ub t ra c t t h e
.
AND
.
O f a ny
t h ree
n
u m b er s
,
t he
l ast
.
i
h
h
n
n
n
e
i
H
s
h
divid
e
d
fou
d
w
e
t
d
vi sor , q u o t i e nt ,
33
t e
ow
,
a n d r e m a i n d e r a re k n o w n
.
.
34
.
te t he
St a
divi si o n
35
.
s ig n
W i t h o ut
i g
squar n
w he n
37
a
the
v a lu e
= 1 b= 3
,
,
Wr i t e t h e
.
t h e s um
St a t e t h e i ndex l a w
.
of
.
the
bi no m i a l giv e
of
(a
,
t he
di ffe re nc e
a nd
Fi n d
.
d iv isio n
of
Pro v e b o th l aw s
.
be t w ee n
36
law
and
—4
and c
pr o du c t
of
c
) + (a
2
b) + 2 c
2
.
51
a nd
4 9 by
diffe re n c e of t w o n u m be rs
i g t hem
e x press n
as
.
Ho w d o yo u d e t erm i n e w h e t h e r a t r i no m i a l of t h e f o r m
2
o f x + bx + c is t h e pr od u c t o f t w o b i n o m i a l s ?
38
.
R e pre se n t 4 t i m e s t he s um o f t h e c u b e s
n u m b e r s m ul t i pli e d by t h e s um o f t h e i r s q u a re s
39
.
o f a ny
t wo
.
40
t iv e
Sho w t h a t t h e
.
O dd n u m
41
A dd ( a + c )
.
c
+
)
(
a
42
F r o m t he
.
b¢
Fi n d
.
i
2a ( b +
) b(b
a nd 4 ( b
s um o f
—
o f 3a c
t he
a p e ce , a n d
c o n se c u
.
—
( b c)
t ra c t t h e sum
43
b e rs is
differe n c e o f t he s q u a re s o f t wo
t w i ce t h e s u m o f t h e n u m b e r s
c
c
,
)
—
—
2 a b ae + 2 bc a n d 2 a c
4 bc
a b and
2a b
2 a bc
o st o f x b oo ks a t a rt
—
x
3 b o o ks at n ¢ a pi e c e
i
2a o
a p ec e , x
c
.
—
bc 3 a b s ub
.
+ 5 b oo ks
at
C HA PT E R XV I
H I G HE S T C O M M O N FA CTOR
LO WE S T
.
C OM M O N M ULTIP L E
H I G H E S T C O M M O N FA CTO R
217
n
A c omm o n
.
u m b e rs is
T h us
a
,
a n exa ct
or c
divi s o r of
a co mmon
is
2
div is or ,
fa c t o r
of
o mm on fa c tor , o f
e ac h o f
t he m
3 a b,
and a
2a
3
4
,
Th e h i g h e s t c o mm o n fac tor
n u m b e rs is t h e p rodu c t of a l l t h e i r c o mm o n
218
x
is t h e h
.
f
o
of
3
x
4
,
x
y,
and
or
m o re
or
m o re
.
5
bc
.
of
.
3
t wo
t wo
f ac t o rs
2
3
2x y
z
Thus
.
,
.
t erm greates t commo n divisor is us ed in a r ith m e ti c b ut it is n o t
3
a ppl i c a bl e in a lg e b r a
For e xa m p l e x a bo v e m a y o r m ay n o t b e
3
Th us if
and x
is t h e re f or e l ess t h a n x
gr ea ter t h a n x
3
In a lge b r a t h e t er m hig hes t common fac tor is use d
Th a t is x is hi gher
‘
t h an x ( m ean i n g x ) in t h e sense t h a t its exponen t is higher th an t h a t of x
Th e
,
.
.
,
.
,
,
.
.
HI G HE S T C O MM O N FA C T OR
219
In o n o
.
Th e h ig h e st
c
m ia l s m a y b e
8a
Th e h
.
c f
.
of
.
3
c
the
‘
.
4 a bc
,
M O N O M IAL S
o m m o n fa
c to r
o f t w o o r m o re
d et erm i n e d b y I n spe c t i o n C o n sid e I
2
3
OF
1 6a b
4
4
,
c o e ffi c i e nt s
is 4
2
0
2
1 2a
,
4
3
c
c o mm o n
T h e h ig h es t
.
T h e h c f is 4a c
f a c t ors ar e a a n d c
O b ser v e t h a t t h e po w er O f e a c h l e tt er in t h e h c f
p o wer o f t h a t l e tt er f o un d in a n y o f t h e m o n o m i a ls
l iter a l
2 2
2
2
.
.
.
.
.
.
.
.
is t h e l o wes t
.
.
22 0
.
R ul e
To the h
.
highes t power
f
o
ea ch
.
l etter
c
.
f
.
f
o
commo n
the
'
to
fiic ien ts
coe
al l
.
,
a n n ex
the
LE M E N TARY ALG E BRA
E
1 74
6
7
8
9
2
.
.
x
— 6x
-
—
x + 2x
1 5 , an
2
+ 9,
—
2d x 2 a ,
—
6a x
2
.
.
— 2a
1 8a x + 6a , 1 8a x
2
3—
2 7a
4x
x
3
2
2
64 , 9a
— 2 0x
—
24x
4
— 16
,
+ 2 5 , 8x
—
x
+ 27,
2
—
2 a bx
and
6a x ,
a nd
,
and
3a
— 1 25
,
3
d
z
— 1 8x
,
8 1 x , 1 2x
— 28a
2
48x
2
4—
3
—
a + 3d b
18b , a
2 7a b , a nd
a
a
2
+a
2
4 a b+ 4 b
—
9x
—
x
8a bc ,
4
2
—
1 8x
15,
+ 2a c + c
+ ax
—
64 a 3 2 a x
5a + 5 a b
4
x
8a
a
3
—
+a
8b
z
c
a
xy
a
,
-
2
x
+ 5a
4
— 2 a 2 b2 - - a 4
l
4x + 4
—
1 6 56a
2
,
2
+ 4 9a
—
2x
8x
and
3
—
—
a nd x
3 x 54
2a b
3 bx
2a y
2
3
—
x
,
3r y
x y,
ax
+ a y + 2 a xy
and
2
2
x
+ 2x
-
24
3—
—
a y
a bx + a by, a n d a
a bx
,
(
—
x
y)
z—
—
+ a b ab
2
an d
2
4c
2
z
3
+y
2
an d
4
,
b
—
x
4
2
4
2x y + y
2
2
—
2a b + a
and b
2
3
,
?
3
—
—
—
ay
cx + cy, a n d a
ac
c , ax
2
2- —
3
—
—
2x , 2x
1 6x l 3 2x , a n d
1 6x + 4x
3
—a 4
2
5 (x
2
2
3
2
+ 4 ax
,
— x2 —
y
3
,
3
— ax
9d x
3
5a
,
a nd
,
2b
2
—
x
2
+ 4x
4
2 —
2—
2
3
—
a
a c
a c + c , a nd 4 a
2
3
3x y,
2
3
a nd
3
—
—
2x 4 8 ,
x
2
b
4
,
—
1 4x
8 a + 4a
2
“
b and
—
3 b)
6
—
8 a , and
2
1 2a x + 4 a
9x
3
fi—
1 08x
2
a nd
4,
—
4 9a
1 6, a n d
3 6a ,
2
ag
—
1 0x
2
+ l 2x + 3 6,
3
a
,
2
24 x +
a
2
,
3—
81
—
—
ab
a
a nd
3
1 0x + 1 6, 1 2 xy
—
63 a
2
8b
—
4
a nd
8
—
16 a ,
5
4a b + 3b
2
2
x
,
6x + 1 , 6x +
—
1 6a b c
2
—
a
3
2
3
4—
8 a b,
3
2
-
2
a
—
8a b
2,
-
4
—
(a
2
4
25
—
x
and
and
+ 32
—
4x
a nd
2
2
2a h
5 4a x + 2 a
9, 8a x + 24 a x,
2
—
x
27
3
2
1 6x
3
—x
LO WE S T C O M M O N M ULTI P L E
LO WE S T C O M M O N
223
A
.
mul tipl e
divi si bl e b y t h a t
n
o f a n um b e r
u m be r
and
4a b, 8a c ,
224
A c o mm on
.
numb e r
t h a t is
2 25
.
on l y
If two
tha t
.
m ul ti pl e s
are
of
tw o
by
m u lti pl e
m ore
c
.
m
or
eac h
of
of
2a
.
m o re
u m b ers is
Thus
o f th em
n
.
2 a , 3 b,
n u m ber
.
o f 3a , 9a
.
an d
2c
.
f
o
6a
m o re
Thus
,
3
n u mber
a
or
c on ta i n s
.
mu l tiple
nu m bers
,
a
,
.
and
2
mu l tipl e
v er y
fa ctors of a l l
the
or
E
.
co mmon
The l owes t
l
is t h e
3
exa ctl y
l
.
fa cto rs of
com mon
t h a t is
ex a m p e ,
l y divi s i b l e
com m on
a numbe r
.
P rin c ipl e s
22 6
ta ins
is
lo we s t c omm o n m ul tipl e ( l c m ) o f t wo
is t h e pr o du c t o f a l l t h e i r diffe re nt f ac t o r s
1 8a
the
a
M ULTIP LE
The
.
n um b e rs
al l
2a x
m ul tipl e
e xa c t
is
Fo r
.
175
the
f
o
two
n u mber s
ha ve
or
mor e
co n
.
no c o m mo n
m u l tipl e is their prod uc t
n u mber s
fa ctor
,
the ir l owes t
.
LO WE S T C O MM O N M ULTI PLE OF M O N O M I AL S
l o w e s t c o m m o n m ul t i pl e o f t wo
m i a l s is d et e rm i ne d by i n sp ec t i o n
C o n s id e r
22 7
Th e
.
or
m o re m o n o
.
6a b c ,
2
lo wes t c o m m o n m ulti pl e
2
1 2a b c is t h e l o w e s t c o m m o n m ulti pl e
o f t h e lit e r a l pa r t s is a
O b se r v e t h a t t h e e xpo n e n t o f e a ch l e tt e r is t h e hi gh es t e xpo n en t t h a t
l e tt e r h as in an y o ne o f t h e m o n o m i a ls
The 1
.
c
.
m
.
of
the
2
c o e ffi c i en ts
bc
He nce
3
is 1 2
.
The
3
.
.
.
22 8
.
Rul e
a nne x a l l
.
To the l owes t commo n mu l ti pl e
the l ette rs
f
o
mo n omia l , gi vin g
a n y mono mi a l
ea c h
h ig h e s t exponen t it ha s in
f
o
.
the
s
c
i
e
n
t
i
fi
,
coe
ea c h
l etter the
E
1 76
L E M E N TARY ALG E BRA
E x e rc i s e 1 00
G iv e t h e
1
3
5
7
9
2a
.
lo wes t
2
3x
.
.
.
6a
3
5n
4
4a
3
5a h
3a
,
6y , 9x y
,
2
,
5x
,
2n
5
50
2
,
2
2
2
,
o n m ul t i pl e of t h e foll o wi ng
2
3
,
3
.
c o mm
3a
4
4
6
x
8h u
2
,
8
7a h
5
,
10
4ax
2
.
2a
,
2
x,
9a h, 4 a b
3
.
2
2
4 xy
.
3a
2
,
c,
3 x y, 5 xy
,
8 a b, 5 b x , 4 a x
,
,
3
2
4
5 x y, 7 x y
.
5a y
3
4
.
,
3
3
3
5
,
2 x y,
LO WE S T C O MM O N M UL TI PL E OF P OLY N O M I AL S B Y FA C T ORI N G
l o w e st c o m m o n m ul t i pl e o f po l yn o m i a l s is fou n d
b y r eso lvi n g t h em i nt o t h e i r prIm e f a c t o rs a n d fi n di n g the
i
r
n
c
a
l
t
h
e
d
re
t
f
a
c
t
o
s
F o r e x a m pl e
l
e
r
o
d
u
t
o
f
f
f
p
229
The
.
,
.
(a + 3) (a + 4)
a
4
4
a
+
+
(
)
)(
2—
—
—
=
4
4a 3 2
a
a
a
8)
+
(
)(
—
m is ( a + 3 ) ( a 8 )
+ 7o + 1 2
2
a + 8a + 1 6
a
T he l
23 0
.
.
c
.
Rul e
2
.
Fi nd the pr odu ct of a l l the difi er en t pri me
the n u mbers , ta kin g ea ch fa ctor a s ma n y times a s i t i s
.
fa ctors of
fou nd in a n y of the g iv en
n um ber s
.
fa c t or s O f t h e l o wes t c o m m on m u ltipl e
w ith o u t wr iti n g t h e f a c to r s o f t h e e xp ress i o n s
m ay
Th e
2 xv
T h e d i ffer en t
a nd
th e
f a c t o rs
l ow es t c o m m o n
th es e
m ulti pl e
in
2
21
45
,
6—
i
d e t er m i n e d
11
3
y, 2x + y,
e xpr es s o n s a r e
is y ( 4x2 — y2)
e
C o n s id e r
.
2x + v,
O ft en b
and
2x — y,
.
E xer c i s e 1 0 1
Fi n d t h e l c In of e a c h of t h e foll o wi n g
m i ni ng it w i t h ou t wri t i ng t h e f a c t o rs a s far
.
.
.
,
1
2
2
— 3x — 4
.
x
.
—
6a 6 b
a nd
an d
4a
50
2
2
—1
-
4b
2
i
d et er
b
s
e
o
s
i
l
:
p
ex e r c s es ,
as
LE M E N TARY ALG E BRA
E
1 78
28
.
29
30
.
.
.
4b
an d
a nd
.
—
4a c
4 b c , 2 a + 2a b,
.
.
x
+ 8 a + 1 6,
3
2
x
x
+ 2x
2
+y ,
a
xy
1 +x +x
,
—
1
y,
—2
5a
3
— x2
x
.
—
—
a
a
.
—
x
2
2
3
a
2
x
6,
—
2x 3
a
,
3
(
2
a nd
2
+y ,
2
)
x
a
ft
-
8
1 +x+x
2
—
4x 4
and
3a
2
( a + x)
2
z—
3
—
x +x y
xy
y
and
+ 2x + l
i
,
—4
z
and
—
a
2
—
6a
16
2
—
and 9
6x + x
,
—
—
4x 45 ,
x
2
)
+ xy
x
and
,
— 1 1a
+ 24 ,
— 3 a2 — 4 a + 1 2
,
+ 7x + 1 0 ,
+ 4x
2
2
x
2
8a + 1 6
x
—
y
(1
-
— 2 x2
3
—
a
x
3
—3 b2
3a b
2
8x + 8 ,
,
1 2 a + 3a
—
a
and
2
x
+
y
y
.
.
2
4b
2
a nd
2
— x
+ x , a nd
1 2 x + 1 2 , 2 95
.
xy
—
bx
1 + 2x + x
and
and
z
2
.
— 16
,
—4 x — 8
4
2
.
+ 4x + 4 ,
2
5
2
4
—
a nd 4
4x + x
2
2
—
5a
2
2
4
—
1
2x + x ,
,
2
3 c ( x + y)
an d
z
1 + 2x + x
a
— 1 2a
,
,
z
c
+ xy + y
and
2
a
3
2
and
,
5 , 1 6a
4,
—
b
—
4a +
dx
— 1 1 — 2 x x2
+ ,
,
4
“
+2
+ 8ax + 1 5a
x
(
—
x
2
2
—
3a 3
2
2
.
,
2a + 2 ,
—
3a b
and
3
y
x
—
ac x
y) ,
2
46
—
3
and
.
.
3a
— 2 7 b2
2
a nd
—
20a
.
45
b, b+ a ,
x
.
44
52
—
1,
a
a
a
x y, x
4
43
,
-
— 5 a x — 24a 2
2
39
42
,
b
-
and
—
4a + 4 , an d a
a
2
x
2
9 b,
.
38
51
313
2
37
—
3a
2
2
36
50
.
—
a
.
35
49
a
z—
34
41
64 ,
—4
.
.
33
48
2
—
6x 1 2
—
4x
1 6, a n d
2
2 a + 6 b,
2
32
47
.
2
31
40
—
8x
3
and
a nd
—
—
a
a
2
—
x
2
6
—
7x 1 8
C H A PT E R
X VI I
FRA CTI O NS
23 1
.
An
al g e b aic
r
f r a c t i o n al f o rm o f
e x a m pl es o b se rv e :
fractio n is t h e
o ne
n
u m b er b y
i ndi cat e d divi s i o n
a n o t h e r ( se e
,
a
+b
—
a
23 2
.
a
b
x
num e at
d e n om in ator is t h e
l 2 xy + y
n um
b er
a
n u m b er a b
b el o w t h e
ov e
li ne
n
u m era t o r
t o g e t h e r a r e c a ll e d
d i v ision
.
a nd
n
R e c a ll t ha t
2 33
+b
li n e
Th e
.
t h e divi den d ,
a nd
.
T he
o ne o f
t he
— b3
.
u m erat o r o f a frac t i o n re pre se nt s
d e no m i n at o r re pre se n t s t h e d ivis or
The
t he
2
is t h e
r or
T he
2 ——
3
the
.
See
152
a
f ra c t i o n
A fr a c ti o n
,
or
as
t a ke n
.
th e d i v id i n g
A n in te g e r ,
w h i c h is
d e n o m i n a t o r o f a ny frac t i o n
terms of the fra ction
li n e
is
a s ym
bol
of a
ggr e ga ti o n
as
w e ll
as
.
inte g ral
n umb e
5, 1 1 , 16
r , is
a n
u m be r no
r
f
a
t
o
p
.
th i n g is d e fin ed in a r it h m e ti c a s o ne o r m o re O f th e
e q ua l p a r t s o f it ; b ut s i n c e t h e t e r m s o f a n a lg e b r a i c fr a c ti o n m a y b e
a n y n u m b ers
p o s iti v e o r n e g a tiv e i nt e g r a l o r fr a c ti o n a l it is q u it e
e v i d e n t t h a t t h e a r it h m e ti c a l d e fi n iti o n d o es n o t a cc ur a t e l y d es c r i b e
a n a lg e b r a i c fr a c ti o n
T h e v a lu e o f a n y a r it h m e ti c a l fr a c ti o n is t h e q uo ti e n t o f t h e n u m er a t o r
T h is is t ru e o f a ny a lgeb ra i c fra c ti o n and
d i v i d ed b y t h e d e n o m i na t o r
2 3 1 ab o v e
fo r t h i s r e as o n it is d e fi n ed as in
A fr a c ti o n wh o se n um er a t or is a + b and wh o s e d eno m i na t o r Is a — b
o f any
,
,
,
.
.
,
.
is
r e ad :
a
+ b ov er
a
-
b,
or a
—
b
i
d
v
id
e
d
b
a
b
+
y
.
1 79
,
E
1 80
234
‘
.
T he
li ne t h at
235
of s
sign of a
se pa r at e s
S i nce
.
ig n s
LE M E N TARY ALG E BRA
fr ac tio n is t h e
t h e t erm s
Sig n
w ri t te n
b efore
the
.
fra c t io n is a n i ndi c a t e d divi s io n b y t h e l a w
in divi s io n
1 5 8 t h e foll o wi n g is t r u e :
a
,
,
+
9
—
3
—3
9
—
3
+3
Cha n gin g the sign s of both n umer a tor
n ot cha n ge the si gn of the fr a cti on
9
and
9
den omina tor d oes
.
Cha nging
the
the si gn
cha n ges
f
s ign
f
the
o
o
n u mer a tor
ei the r
fra ction
or
denomin a tor
.
it h er t erm o f a f r ac t i o n is a pol yn o m i al it s Sig n is
c hang e d b y cha n gi n g the s ign of ever y ter m
If
e
,
.
a
—
x
f rac t io n
a re
b
—
b a
y
— x
+y
—x
y
b ser v e d whe n t h e t e rm s of a
expr esse d b y t h e i r f a c t o r s v iz
23 6 Tw o pr i n c i pl e s
.
b
— a
+
are _t o
be
o
.
,
Cha nging the si gn of o ne fa ctor in n umer a tor
n ator Cha nges the si gn of the fr a ction
Fo r
1
or
.
den omi
:
.
—
—
b
b
a
(
(
)
c
—
a
b) ( b
—
i
v
(
yfl
(
)
—
—
( x y) ( y z )
z
-
c
—
—
( a b) ( c b)
)
(
y)
-
—
iv
yfly
—Z
)
T h i s is evid e n t for c h angi n g t h e s ig n o f one fa c to r c h a n g e s
t he s ig n of t h a t term of the fr a ction
,
.
Cha ng ing the s ign of two fa cto r s i n n umera tor or den omi
Fo r :
n a tor does no t cha n ge the s i g n of the fr acti on
2
.
.
(
a
—
(x
b) ( b
y) ( y
c
-
)
z)
(
a
b) ( b
c
)
—
z
x
(y
)(
y)
(
—
b a
—
(x
)(o
y) ( y
b)
—z
)
T h i s is t ru e fo r c h a n gi n g t h e s ig n s of two fa ct or s does
c h a ng e t h e s ig n of t h at ter m of the fr a ction
,
.
no
t
E
4
C h an g e
.
is
n at o r
5
na t o r is
238
.
to
f
i ?
an eq
i
y f
to
an e q
uiv a l e nt fr ac t io n
wh o se d e no mi
uiv a l ent frac t io n
w h o se d e no m i
(x
Ch a n ge
.
LE M E N TARY ALG E BRA
x
2
—
z
2
y
.
A fra c t i o n I S
i n i ts
l owes t ter ms w h en t h e
n
um e ra t or
d e n o m i n a t o r h a v e n o c o m m o n f a c t o r e x c e pt 1
T O red u c e a fr a c ti o n t o it s l o w es t t erm s w e m us t r e m o v e
and
.
,
fo un d
in b o t h
t
d en o m i n a t o r
n um er a o r a n d
is d o n e b y c a n c e li n g t h e c o mm o n f a c t o rs , whi ch is
to d i vid i n g b o t h n um er a t o r a n d d en o m i n a t or b y t h e m , t hu s ,
l 5a
2 0a
2
23 9
x
3
x
R ul e
.
3a
—
x
4x
—
x
2
2
2
Resol ve
.
3x + 2
(M x
—
x
nu me r a tor a nd
B)
equi v al en t
x
-
—
x
l
3
den omina tor in to their
) a l l fa ctor s common to both
W h e n t h e n u m era t o r of a fr ac t i o n is a fa c t o r of t h e d e n om i
na t o r t h e n u m e r a t o r of t h e r es ul t is 1
Fo r exam pl e
i
r
p me
fa ctor s
—1
)
5x + 6
f a c tor s
.
Th i s
3
al l
a nd ca n ce l
i
d
( vi de
ou t
.
,
d
.
,
l
+x
2
—
a
x
2
a
— x
dvi sa bl e t o c h a n g e t h e s ig n of a fact o r in
o n e t erm t o m a ke it li k e a f a c t o r in t h e o t h er
Thus
It is O ft e n
a
.
—
—4
>
—x
4>
—
5 (x 4 )
)
We c h an ge
5
—
4
ig
n
f
h
f
c
t
or
x in t h e d e n o m i n a t o r
the s
o t e a
t h e S ig n o f t h e fr a c ti o n a n d t h e n c a n c e l t h e co mmon f ac tor
,
,
‘
a n d a l so
.
,
E x e r c i s e 1 03
R e du c e t h e fo llo w i n g fr act i o n s t o
givi n g res ul t s a t s ig h t as far a s po ssibl e
2a
3
4 nx
2
3x
O
8a
z
3
6x
8 xy
thei r
lo we st
t e rm s ,
F RA C TIO NS
E x er c i s e 1 04
R ed uc e
po ss i b l e
4
a x
a
+ 64 a x
2
4a
a
2
l o w e st
to
16
+a
3 b+ 3 a b
2
3
9x y
x
3x y
8x + 1 5
2
x
1
2
5 xv + 5 v
x
2
2 x y+ y
x
2a
—4 a
2
3a b
a
a
y
2
2
2
Z
6b
2
b
2
+ 2a h
2
2
3
3
b
2
4x y + 4
5x y
ar
a
2
z
a
6a b
a
2
6a b
3
2
d
3
2 ax + x
9a
2
5
s
4b
2
3
x
2
2a x + x
t e rm s ,
givi n g r es ul t s
s
ig ht
as
far
as
LE M E N TARY ALG E BRA
E
1 84
2x
(
2
3
4x y
—
(a
3
—
a hx
x
2 40
+ 2ax + x
A
.
i nt e g r a l
re
A
du c e d
-
b)
x
a
a
+1
2
+ 2 + 3a
z
+ 5 + 6a
wh i c h is
of
pa r t
2a + 3x +
y
fr a c tio n is
or a
m i xe d
fr ac t io n
a
+b
5
_
—
x
3
wh i c h
c a nn o t
be
n um b e r , a s
—
x
a bc
+y
a
xy z
y
A n im pro pe r
.
6
'
fra c tio n is a fra c t i o n
r e du c e d t o a w h o l e or a m i x e d n u m b e r a s
2 42
1
.
a
2 2
4
,
who l e
a
3
u mb e r is a n um b er o n e
o t h e r p a rt f r a c t i o n a l a s
ro
e
r
p p
to
b)
+b
—
x
.
2
a
n
t he
a
2 41
(a
2
m ix e d
and
a
2
'
a
)
—
a
x
(
3
—
3xy
6y
2
2
x
a
2
3
—4
whi c h
c an
be
,
a
a
4
b
2
+b
4
x
2
2
5x + 9
—
2
x
RED UC TI O N OF I MP RO PER FRA C TI O NS
A n i m pro per f r a c t i o n is re du ce d t o a w h ol e
n um b er
b y pe r f o rm i n g t h e i n di c a t e d divi s i o n
2 43
.
.
re
du c e
to
a
a
3
a
3
m ix e d
n
u m b er
,
c
e
ro
d
e
p
+x
+ 2a
2
— 2a 2 x
+
— 2a 2 — 4a
4a + x
4a + 8
—
x
T h er e for e
,
a
2
8
2a + 4
as
or a
m ix e d
Thus ,
follows
to
ELE M E N TARY ALG E BRA
86
—
n
3
+1
n
a
—1
3
— l
a
x
3
x
1
+1
+l
2
1 5x
—
+ 5x 1
5x
RE
D UC TI O N OF
M IxE D
EXP RE SS I O NS
M i x e d e x pr e ss i o n s a r e r e du c e d t o i m pr o per fr ac t i o n s
a s in a r i t h m e t i c
e x c e pt t h a t w h e n t h e f r a c t io n a l pa rt is
O b ser v e :
minu s t h e n u m e rat o r of it is su btr a cted
24 5
.
,
.
,
a
_
3+
a
2
a
(a
+9
a
A ddi n g
+2
a
2a
a
H enc e
2
a
d
Al so
a
2
x
+9
(
+x
—a—6
2
+9
2
—a
—
2a
a+3
a
+2
—
—
a
x) a
x)
(
Su b t ra c t i n g
a
a
2
2
— 2ax
+x
2
2
+x
—
H en c e ,
+3
2
+2
2
2
2ax
—
a
x
E x e r c i s e 1 06
R e du c e t o i m pr o pe r fra c t io ns
1
.
a
+1+
26
2j
5
— 2a
+3 x
F RA C TIO N S
3
5
7
9
.
—
x
3
.
—
a
.
.
11
.
13
.
15
.
a
7
—3 x — 2
y
4+
6
—
5a 3 b
.
—
2x 3
+5
—
x
2+
—
4
a
10
—
2 a 4x
.
—
3x 4y
+
4x + 3 y
x+6
a
+h
16
LO WE S T C O MM O N
246 Tw o
or
.
m o re
3a
.
-
2x
D E N O M I N AT OR
f ra c t i o n s
h av e
a
commo n
de no min a tor
d e n o m i n a t o r s a r e t h e s a m e n um b e r s
T h e lo we s t c o mm o n d e no m in ator
Of
m o re fra c t i o n s is t h e l c m of t h e i r d e n o m i n a t o rs
wh e n t h e i r
.
.
.
(
a
-
(
x
.
Rul e
.
a
x
)
(
a a
x
)
—
( a + x) ( a x)
+x
Find the l owes t
common
mu l tipl e
ina tors for the l owes t common denomina tor
mu l tip l y both terms
f
o
the giv en
f
o
the denom
.
D ivi de this denomin a tor by the den omin a tor
a nd
or
— x a
x
+
)
)(
a
d
two
.
.
o u
C o ns id e r
24 7
y
fraction
f
o
ea c h
fr a ction
by the quo tien t
.
LE M E N TARY ALG E BRA
E
1 88
E xerc i s e 107
R e du c e t h e f o ll o wi n g fra c t io n s t o e q uiv a l e n t fr a c t i o n s
h a vi n g t h e l o w e st c o m m o n d e n o m i n a t o r
3a
2
3
2ax
4 xy
2
6
a
b
5d x
2
3
2
2a c
4x
b
3
,
2 a h 6a
3a
—
a
a
4
2
’
3
4ax
x
a
+4
2
l
x
1
4
+1
a
—
,
a
+2
?
a
b
a
C
c
3a b
a
’
5a
4d x
)
?
2
3x
3
2
4 bx
a
5
Sa x
a
2
l
’
4 a b 2 xy
2
s
+1
x
i
b
—
x
,
a
a
,
a
c
6b
2
’
2
+4
—
a
4a h
2
4
a
’
a
—
2
+2
a
,
+2
—
a
2
ADD ITI O N A N D SUBT RA C TI O N OF FRA C TI O NS
fra c t i o n s a r e a dd e d o r s u b t ra c t e d b y pe r fo rm
in g t h o s e o pe r a t i o n s u po n t h e n u mer a tors a n d wr i t i n g t h e
r es ul t o v er t h e c o m m o n d e n o m i n at o r
We h a ve l ea rn e d in divi si o n t h at
248
S i mi l a r
.
.
a
+ c+e
n
x
b
I nt er c h a ngi n g
t he
24 9
R ul e
.
s ig n s o
f
al l
.
b
t h e m em b e r s
ru l e fo r a ddi t i o n
+ +
b
of
t his
a n d s u b t ra c t o n
i
Red u ce the
the terms
a
f
o
fra ctio ns
n u mer a tor s
c
n
x
_
b
b
b
i
o f fr ac t i o n s
to the
f
a
c
t
i
n
s
r
o
f
o
we
e qua t o n ,
o b se r
ve
.
cha n ge
tha t
a re
the
r
e
p
EL E M E N TARY ALG E BRA
1 90
5
2
x+ y
x
—
b
2a + b
—
2x
3
2x + 3
2a
—
x
x+3
3a + b
—
a
3
—
x
—
4a
1
6a + 2
2a + 2
3 + 3a
x
+y
x+
b
v(
b
( a + b)
+b
4d x
—
(a x
a
+
‘
f
—
a
x
a
b
'
°
s
( x + y)
3
4
2
—
2)
a (a
—
a
4
y)
2
2
3
—
b
a
—
a
+4
-
n
n
z
a
2
+ a b+ b
—
n
+31
—
—
2n
15
n
2
3
x
+
y
x+y
2
33
2
n+3
—
x
1
3x
+b
2
2
5
3
a a
a
b
z
(
2
2
3
n
—x
y
2
y+ y
y
—
x
z
—
z
x
—
x
y
—
x
°
b
2
4a b
°
)
— x2
v
(
—
x
b
2
—
( a x)
2
—
2
a + 4a
2
3
—
a
l
y
x+4
4x
+
2
—
x
16
( a + b)
z
4
2
x
-
—
x
4
y
2
2
9x
—
x (x
y)
2—
2
x
v
x+ y
y
3
2
a:
y
a
y
—
a
'
y
5
a
3a + b
2
y
y
.
b
‘
x
-
1
a
+b
ii
b
2
b
2
— a2
F RA C TIO N S
3n + 6
2
°
+1
n
+ 4a + 9
2
d
a
n
—
u+
n
2
2
3
a
a
3
— n3
—
n
2
+ 8y
x+
+1
+4
— 3a
+9
2
n
2
+ 2n + 4
—
x
2
2y
6
n+
1
3
x
+3
3
a
+
+2 n + 2 8
4 xy
+ 3n + 5
u
2
+ 27
8
3
1
+
n
2
1 91
x
2
-
y
2
2 xy + 4 y
MULTIP LI C ATI O N OF FRA C TI O NS
nu
Th e pr odu ct o f t w o fr a c t i o n s is t h e p rodu ct
er a tor s o v e r t h e pr od u ct of the de no m i n a tor s
m
the
.
Fro m
Fro m
a
f
o
m
first
the
= bm b
y
a nd
§
= n w e have
,
i
equa t o ns , a
fi d
mn
x
= bm
a nd
‘
c
Wh y ?
.
= dn
M u l t i p l yi n g
.
dn , m e m b e r b y m e m b e r , w e h a v e
a c = bd m n
C:
.
Dividi ng bo th m e m b e rs o f a c
z
g
—
a
b
x
ma ,
0
ac
s
bd
bdmn b y bd , w e h a v e
a nd
c o m par s o n a x o m ,
by t he
i
i
T h i s m e t h od is a ppli c a b l e a l s o wh e n e i th e r fa c t o r is
i n t e g ra l fo r i nt e g e r s m ay be e x pres s e d in f r a c t i o n a l fo rm
Si n ce t h e prod uc t o f t h e n um er a t o rs is d i v i d ed b y t h e pr o d uc t o f t h e
d e n o m i n a t ors c a n c e ll a ti o n m a y b e e m p l o yed
.
,
.
,
E xe r c i s e 1 09
Si m plify t h e
3a
4x
8b
"
lid
b
2
X
5a
3a
x
fo ll o wi ng
5a c
6a b
3 cy
4a b
a
a
2
— 25
—
a
2
— 3a
2
a
2
a
— l
2 a + 8a
+ 5a
2
2
9
4a + a
—
a
2
LE M E N TARY ALG E BRA
E
1 92
—
x
1
—
I
$
°
y
—
2
X
2
1
—
4
x
2
—
x
2
x
2
a
X
+xz
2
—
1 +y
113
—
x
l
—
x
xy
x+
b
2
3x z + x
x
—
x
3
><
2
x
(
y
—
a
a
(
2
b
a
—x 2
6a
(
-
a
(a
a
a
n
°
2
)
2
il
+ 8y
z
2
10b
3 ax
— x
5 by
a
—
a
2
2
—
2a
—
4
( a + x)
—
ah
b
ab
z
4
—
ab
b
2
( a + b)
2
—
a
2
4x + x
z
d
2
2
_
y
’2
-
b
2
abx
2
a
4a + 4
2
+ 27
2
a
+x
+ 2a
b
2
3
x
y
2
+a
x
2
_
8
><
( a + b)
b
2
a
a
—
2
+8
2
—
+ 3 4a
6a
+ 2n + 1
2 —
—
n
n
3
ax
+a
—
4a
.
4
3
2
2n
‘
23
2
2
6x
a
2
2
+x
2a
3
—
-
ax
3a
2
22
.
24
.
26
.
x
3a h
2 xy
1+
1+
2+
4a
2
x
3
—9x2
+a
2
2y
i
2
2—
z
4 by
+ab
b)
x+
—
25
x
2
—
c
x
x
—
9z
2
2
—
a
x
x
a
)
3ax
2
b)
y
y
( x + y)
2
2
—
a
x
(x
2
—
x
2
—
c
x
+b
a
?
(x + 5)
+ ax
2
—
x
2
2
+ 8y
2y+
bc + bx
13
—
x
2
3
i
y
+ 2 xy
x
y + xy
+h
a
2
2
x+ z
2
( a + b)
2
—
x
1
2
y
—
x
2
3
E
1 94
a
+3
a
2
a
a
o
x
+x
b
+b
2
z
+ xy
— x2
(a
b)
a
— 25
+ 6a
—
a
2
Z
a
2
-
a
.
a
+x
a
2
x+ y
—
a
9
—
b
1
b+ l
a
—
2
LE M E N TARY ALG E BRA
a
d
3ax
z—
a
b
b
2
2
—
2
— 36
5a
+ ax
— 9x2
2
b +ab
2
2
2
—
4b
a
2 a b+ a
?
—
( cz x)
r—
2
2
2
(e
—
a
2x
a
(x
4
3
+4
—
x
-
x + 2a
2x
x
2
—
—
x
3x 4
2
—
n
—
6 n
.
1+
2
a
3
+x
a
3
—
a
x
x
21
x
3
+x
2
z—
ax
33
2
+ 21
—
2+ a
a
a
y
a
— 6— x
2
—9
a
3
—
x
y
3
2
x
a
27
.
+ 2 xy + y
—
a
fi
—
25
a
2
—
a + 6a
7
2
29
.
+3
x
2
z—
+x
2
2
2
2
2
+ 2 xy + y
sci/
+x
— x
2
.
2
4 + 2a
a
a
+x
2
3
a
2
+ 6a + 8
3
—4
—
n
2n
—
—
2 n
n
a
2
2n + u
2
3
+8
—
a
x
2
—
a
x
2
2
x
2
17
x
— 8 x3
3
8 d + ax
x
)
2 ?
(x
2
2
2
a
z—
xy
x+y
z
—
+ a 12
5a + a
d
2
+a
z
-
42
F RA C TIO NS
32
1 95
4
.
a
33
1
.
34
E x er c i s e 1 1 1 — Te s t
An swer al l
1
.
yo u
c a n o ra
S how t h at
divi s or O f a ny m u l t i pl e
2
by
.
3
.
and
G iv e
c om m on
—
2a
R ev i ew
an d
ll y
of e
of
di v i s o r
it h e r
of
t hem
a n a l g e b r a ic e x pr ess io n
Wr i t e
+b
a
Q u e s tion s
c o mm o n
a
—1
+
a
.
2
two
n
u m b e rs is
a
.
t h a t is
exa c t
l y divi si b l e
3b
.
so m e
a
e x pre s s o n s o f
lg e b r a i c
m ul t i pl e is 2 a
4
i
wh i c h t h e
l o w e st
4
2x
.
Ho w d o yo u d e t er m i ne w h e t h e r a b i n o m i a l is t h e pro
du c t o f t h e s um a n d di ffe r e n ce o f t w o n um b e rs ?
4
.
5
.
Sh o w t h a t
di v i so r
6
a re
.
of
t hei r
a
r oo t s o f
divi s o r o f t w o n u m b e rs is
o f t h e i r di ff er e n c e
s um a n d a so
l
R ec a lli n g t h e
t he
c om m on
t he
.
so ut o n O f e q u a t o n s
l
i
i
i
e qu at o n , x
2
by
fa ct o r i n g
,
what
0?
5x + 6
S h o w h o w m u c h t h e s q u a r e o f t h e s um o f t w o
e x cee d s t h e p r o d uc t o f t h e i r s um a n d di ff ere n ce
7
a
.
n
u m be rs
.
8
.
.
s qua r n
(30
b e t we e n
9
i g t h e bi n o m i a l giv e
a nd (30
7 ) ( 30 7 )
W i t h o ut
,
Ho w m u c h d oes t h e s q ua r e o f
of
t he
exc ee
di ffe r e n c e
d t he
pr o d u c t
4
( 0
Fi n d t h e v a l ue
b = 2,
10
11
c
=3 d=4
,
,
.
cd
.
m
2
3
m
e
8a b
2
3
of
the
f o ll ow i n g
=0 m=
d m
e’
‘
2
when
i
e xp re s s o n s
n
9c X 2 b
2
2
2 2 2
a c e
6a dm
2
3
+c d
2
2
+
n
bc d
2 2
2
n
b m + a Xd
4
5
3
a
1,
LE M E N TARY ALG E BRA
E
1 96
the
Fi n d
c
6a d
.
3
2
— 5 c3 2
y x 3b x
9b d + 8 a d
2
14
.
t h e fo l l ow mg
i
2
2
3
+
50
2
13
2
15
x
2—
8b y + a (2d
3
.
3
2c d
3
.
a
r
}
,
s um
.
2 d) + 5 x
2
5 d) x
( 2y
4
3
Sh o w h o w m u c h t h e s q u a r e O f t h e
e x c ee d s t h e s q u a r e o f t h e i r di ff e r e n c e
16
when
e x pr e s s o n s
=1 d=4
,
,
2
12
of
v a l ue
of
t wo
4y
2
-~
n
u m b ers
.
di ffe re n c e b e t w een
Wi t h o u t s q u a ri n g e i t h e r b i n o m i a l
17
G iv e t h e
.
and
( 20
.
18
W h a t is m ea n t
.
i fi es
s at s
19
sm a
the
2
x
t h ei r
2
+ xy+ y ,
22
of
.
an d
G iv e
re c
h
Fi n d
c
i
a c er t a n n
um b e r
.
su m o f a n y
i pr o c a l s is
o
f
e qua
the
n
the
u m b ers divid e d by
.
the
an d
2
—
x
xy + y
l
two
giv e
l
.
c
.
m
of
.
4
x
2
2
4
+x y +y
,
2
r esu
W h at is t h e
f ac t o r s in
b= 1,
a
.
How m u c h d o es t h e s q u ar e
—
50 4 ?
23
al l
.
the
Fi n d
.
t h at
m ul t i pli e d t o
fr a c t i o n b e
po ss i b l e i n t e g r a l pr o d u c t ?
s um o f
21
id
i
Sh o w t o w h a t t h e
.
sa
a n e q ua t o n ?
B y w h a t mu st
.
ll e st
20
w h e n it is
lt
r es u
of
e x c ee
w i t h o u t s qu ar i n g
lt
the
d e n o m i n at o r
f o ll o w i n g
=4 d=3 n= 5
,
,
,
y
—
a
2
b
b +y
2
'
a
2
+y
d the
sq
u are
.
in m u l t i p li ca t i o n
n um er at o r a n d
v al ue
Of
of
fra c t io ns
c a n c el ?
e x pr es s
io n s
,
w h en
wh e n
a
2
C H A P T E R X VI I I
L I TE R A L
AND FRA CT I O N AL E
S OLUT I O N O F
LITE RAL
2 54
t wo
.
or
A l i te ral
m o re
A ND
Q UATI O N S
F OR M ULA S
FRACTIO NAL
o is a n
ge n e ra l n u m b e r s
e qua ti n
eq
.
Q UATIO N S
E
u at i o n in
w h i c h t h er e
ar e
.
lvi n g s u c h e q u a t i o n s t h e v a lu e o f a n y l ett e r m a y b e
fo u n d b ut o n l y in t e rm s o f t h e o t h e r l e t t e rs
In
so
,
.
,
S o lv e
for
A ddi n g
a
2
—
ax
a
x,
2
u ni t i ng t he
b
bx t o b ot h m e m b e rs
a nd
t er m s
c
o n t ai n in g
(
B y the
bx
2
a
divi si o n a x io m
-
x,
this
eq
u at io n
a a
an d
w e h ave
,
a
+b
—
b( a + b) b o r a b a b
( + b)
T o so lv e a li t e r a l e q u a t i o n fo r a ny l e tt e r in it is
t h e v a lu e of t h a t l e t t e r in t e r m s o f t h e o t he rs
C hec ki n g :
,
2—
2
=
b
b) x a
x
— a
of
2
‘
2
t o fi nd
.
.
E x er c i s e 1 1 2
fo ll o w i n g e q u a t i o n s in t h e l e ft c o lu m n fo r
in t h e r ig h t c o lu m n fo r y a n d c h e c k :
So lv e t h e
t h o se
1
3
5
7
.
—
—
=
4a x 4 b
.
—
5n x = 4 n
.
—
—
=
bx
3 a x 2a
.
s
—
—
=
n
nx
ax
a
bx
+ nx
il
4a = 4 b
2
4
6
8
.
.
.
.
x , an d
2 b + 6y = 3 c + a y
—
ay
ab=
3y
—
by =
5a
—
ay
5b
-
3b
—
—
=
2a 9y a y 1 8
LITE RAL
2 55
.
S pe c ia l
i l d ev i c es
s pe c a
c ear o f
l
F R A C TIO NAL
AN D
w ill b e w e ll t o
D e v i c e s It
fo r c l e a r i n g
.
—
2x
fr a c t i o n s
9
5
+
t h e m on om i a l
of
of
i
e q u at o n s
2x + 6
—
4x
1
10
3x
M ul t i p l yi n g b o t h m e m b e r s b y 1 0
m u l t i pl e
E Q UATIO NS
—
—
4x
18
here
fr ac t i o n s
s
om e
T h us
.
,
8
the
,
d e n o m i n a t o rs
n ot e
99
we
,
—
4x
c o mm o n
l o w est
h ave
8
it e o t h e r m o n o m i a l s c l e a r
o f f r a c t i o n s a n d c o m pl et e t h e s o l ut i o n
In so m e e x a m pl e s it s i m plifi e s t h e s ol ut i o n t o c o m b i n e
fr a c t io ns b e fo r e c l e a r i n g o f fr a c t i o n s T h us fr o m
S u b t r a c t 4 x in
m em b e r ,
ea c h
un
,
.
,
,
.
1
we
o b ta n ,
i
a
c
a
a
a
+
c
1
a
+
c
+
a
two
t h e ir
fra c t i o n s
e q ua l
d e n o m i n at o rs
2
ea c h
x
C h e c k by
s
ub s t i t u t i n g in
x
a
C
x
m em b er , w e h a v e
2
x
h av e t he
l
c
20
c
a re e q u a
+
x
c
2c
If
a
x
f r a c t i o n s in
C o m bi n i n g t h e
1
c
sam e
H ence
.
a
2
0
n
u m e rato r
,
no t
,
2
i
e qua t on
(1)
E x erc i s e 1 1 3
So lv e t h e
3x + 8
12
—
2x
5
fo ll owi ng
—
4x
i
e q ua t o n s
—2
3
x
3x + 4
4
Ul Z
—2
y
5x + 8
y+ 3
—
1
v
15
—
l
v
y+ 1
—
a
l
x
+
—
3 30 2 3
—
5x 4
3x + 8
10
2x + 5
x
2
y
—
5x
a
2
a
—1
- -
a
+1
0,
.
E
2 00
a:
L E M E N TARY ALG E BRA
—
a l
8
3x
x
.
1
—
x
4
2 l
-
-
—
E
i
l
ab
q
+
a
c
c
x
ab
—
5x
0
ac
4
x
10
2x + 2
2
2x + a
2x + 1
—
3x 2
3x + 2
-
)
2
a)
2
—
x
2a
—
2x a
5
-
cx
ex
5x + 4
50
( 2x
( 2x
c
—
x
a
5
41 C
—
x
6x
1
4
x+
6
.
+
x
—
2a c
x
1
—
x
3
bx
x
5
+
72
8x + 3
—
x
2
l
2x
4
15
523+ 6
4
—
2
x
4 b+ x
4
6x + 3
—
x
—1
—
2b x
2
x
x
—9
"
3b
a cx
+2
3+x
x+3
—
x
3
6x + 4
x+ 6
6
9
8
3x + 2
5
x+ 2
2x + 4
8x + 3
4x + 8
2x + 5
2
°
4
3x + 6
2
2x + 1
3x
—
2x
1
—
4x
1
8
6
°
—
x
2
—
l 0x
5
4 6x + 8
—
x
3
4
5
—
4
x
—
x
5
1
1
(x
1
+
c
3
2
2
—
4 x
4x + a + 2 c
—
c
a
x
x + 4a + c
+a+
.
1 + 2x
2
9x + 7
x
2x + 2 x
2
x+ 2
2x
1 2x + 1 1
°
— x
(
—
x
4
)(
—
iv
3
)
x+
2
—
2 x
E
202
LE M E N TARY ALG E BRA
E xe rc i s e 1 1 4
Solv e t h e
fo llo w i n g e qu a t i o n s
2
x+
°
x
2
-
—
x
°
4
—
x
5
+
—
x
+
x+5
c
—
x
5x + 6
8
—
x
—
x
9
—
x
x+4
x+ 2
x
x+ 4
x
2
+x
2
—x
7
—
x
2
— x
c
2
c
-
x
2
x + 2a
x + 3a
x+ 5 a
—
l 0x
8
—
l
x
—
x
6
—
x
4
2
— c2
2
x+ a
—
x
7
—
13
x
—
15
x
—
16
x
5
+x
—
6x 44
64
'
4a cx + 2 a
x+ a
—
5x
3
—
—
a
x
—
2
x
x+
+ —
8 x 6
6
a
+
5
—
x
4
—
x + 2a
8
7
—
x
+ —
2 x 3
—
x
x + 3a
—
5x
3x + 2
-
2
+ —
4
l
x
—
x
a
2x
—
x
3
l
x+
°
x+4
2
—
4x 55
—
x
14
7
+
9
4
—
3x a
5
—
2x a
5
v
'
v
y
2a
—
—
2+y
6
3 (y
-
—
3 53 4
—
4x
2
45
—
2x 2 a
—
2x a
3
— 1
y
1
— 2
y
z
2
6
—
x
5
y
—
2 (y 1 )
—
3
—
3x 4
8x + 5
3
—
x
—
2x 5
15
y
8
—
2x
11
9x + 4
'
—
x
3)
—
9y 1
— 2
)
3 (y
3
1
2
—
3)
2+y
7 y + 86
—2
—4
9
)
LITE RAL A N D F R A C TIO N AL
x+ 2
+
x+ 3
—
{It
+
x+3
—
x
x+
—
4
x
7
—
x
5
x
.
°
—
x
1
x+4
6
+
—
4
x
x
5
—
4
x
—
x
10
x+ 5
x+4
x+
x+
—
x
1
+2
5
—
3
—
x
9
10
x
+9
2c
2x + 3 c
3x + 6 0
x+ 4 c
x+ c
x+2c
E x e rc i s e 1 1 5
P r ob l e m s in
—
2 03
6
—
x
x
E Q UATIO NS
Sim pl e
E quation s
fo ll o wi n g pro b l em s
1 Se p a r a t e 5 9 int o t w o s u c h pa rt s t h at 4 t i m e s t h e s m a ll e r
s h a ll e x c ee d t w i c e t h e l arg e r b y 2 6
Solv e t h e
.
.
2
.
3
.
From
Fi n d t h e
will b e 64 2
4
.
5
.
twi ce
6
7
t he
.
.
.
9
.
l
e a ps e
10
.
n
n
u m b e r m u st
umber
to which
n um
b
er
if
s
ub t ra c t e d
3 2 9 be
a
t o g e t 2 73 ?
dd e d
,
t he
s um
m u st b e m ul t i pli e d by 3 7 t o O b t a i n 999?
A is 3 t i m e s a s o l d
as O l d
Fi n d
.
t he
as
B b ut in 20
,
ag e of e ac h
W h at n u m b er m u st
o ne
2 20
ars
e
y
h e will b e
on
ly
.
divid e by 2 3 t o O b t a i n
t h a t t h e q uo t i e n t
is 4 a n d t h e re m a i n d er 2 0
Divid e
so
'
o f o ne
pa rt
1 63 ?
divid e d by
.
W h at
A
1 3 5 be
.
W h at
o th er
8
w h at
n um b e r
is 5 3 ye a rs
d sin c e A
Wh at
ol d a n d
w as l
n um b e r
m u st b e
é
t i m es
m u st
a
dd e d
B is 3 3
t o 378 t o
.
.
as O l d as
giv e
.
65 ?
Ho w m a ny yea rs h a v e
B?
o n e s ubt r a c t
f ro m
33 t o g et
Divi d e 3 1 5 i n t o t wo part s so t h at t h ere a re t wi c e a s
m a ny di m e s in t h e fi rs t pa rt as t h ere a r e 5 c e nt pi e c es in t h e
s ec o n d pa rt
11
.
-
.
E
2 04
a
12
.
13
.
B y wh at
n
L E M E N TARY ALG E BRA
u m b e r m u st
m ul t i pl y
one
i
obt a n
to
di ff er e n c e b et w e e n t w o n u m b er s is 1 7 ; a n d if 4 is
t h e l arg e r n u m b er t h e s um is 4 t i m e s t h e sm a ll e r
Fi n d t h e n u m b er s
Th e
dd e d
to
num b er
,
.
.
u c h t h a t t h e i n t er est o n
t h e g r e at e r part fo r 2 ye ar s a t 6% s h a ll b e e q u a l t o t h e i n t er est
14
on
Divid e 39000 i n t o
.
the
15
.
r es ul t
16
.
ot h e r
ar
p t
num
W h at
92
as
a
dd e d
fo r 3 yea r s
b er
s
b er
s
at
u b t r a c t e d fro m
to the
n um
O f what
t w o pa rt s
giv es
1 64
.
if t h e
-
The
.
.
an d
is 5 4 t h e t h r ee t e n t h s pa rt ?
,
18
s am e
n um b er ?
di ffe r e n c e b e t w ee n t w o n u m b e rs is
g rea t er Is divid e d b y t h e l es s t h e q u o t i e n t is
Fi n d t h e n u m b ers
m ain d er 4
17
the
32 ;
5
an d
t he
re
.
B y w h at
n
1
u m b e r m u st o n e divid e 3 3;
t o g et 5
T h r e e m e n e a rn e d a c e r t a i n s um o f m o n ey A a n d B
ea rn e d $ 1 8 0 ; A a n d C e a r n e d $ 1 90 ; a n d B a n d C e a r n e d $ 2 00
How m u c h did t h e y a l l e arn ?
19
.
.
.
20
.
Wh at
n
u m ber
is
as
m u c h u n d er 7 7}as it is ov er
l e n gt h o f a r e c t a n gl e is 1 % t i m e s it s wid t h I f
e a c h di m e n s i o n w e r e 3 i n c h es l e s s t h e a r e a w o u ld b e di m i n
Fi n d t h e l e n gt h
is h e d 2 7 9 s q uare i n c h e s
21
.
Th e
.
,
.
22
.
1
.
W h at n u m b e r l i es m id wa y b e t w ee n 3 2;
and
o a t fo r $3 6 payi n g for it in 2 —d o ll ar
bill s a n d 5 0 c e n t pi e c es givi n g t w i c e as m a n y b ill s a s c o i n s
HO W m a n y b ill s did h e giv e ?
23
.
A
m a n b ou g ht
a c
,
-
.
,
24
.
25
.
m uc h
O f what
A
m an
at
was $ 7 65
.
n
u m b e r do e s
i n v e st e d
a
t he
i
c e rt a n
d o ubl e
s um
exc ee
d by
9 it s
5%
a nd
t wi c e
at
as
His a n n u a l i n c o m e fr o m b o th i n v e s t m e n t s
H ow m uc h did h e i n v es t ?
LE M E N TARY ALG E BRA
E
206
35
A
.
do
can
a
pi ec e
of
w o rk in 1 2
’
d ays a n d w i t h C s h e l p t h e y
m a ny d ays c a n C do t h e w o r k ?
18
,
36
can
of
If
.
A
c an
d o t he wh o l e
of
it in 1 5
e ce
,
wh i c h
d o it in 4
c an
pi
of a
d o h a lf
d ays
d a ys
w o r k in 1 0
of
B
c an
In h o w
.
d a ys
d ays in h o w m a ny d ays
,
d o in
and
can
B
b o th
t h e m d o it w o r ki n g t o g e t h e r ?
A
ul at o r b oug h t t w o pi e c e s o f l a n d a t t h e s a me
H
n
n
f
r
i
e
s
old
e
i
c
e
a
t
a
ro
fi
t
1
c
e
o
e
O
7
00
a
d t he o th e r a t
3
p
p
p
a l o s s o f $ 90 0 r e c e ivi n g t w i c e a s m u c h fo r o n e pi e c e as fo r
t h e o t h er
Ho w m u c h did e a c h pi e c e c os t him ?
37
.
s pe c
.
,
.
38
in
At w h a t
.
one
ra t e
a nn
p er
r
= th e
3500
What
.
w ill 33 600
giv e $ 2 7 0 i nt e r es t
8 m o nt h s ?
r
ea
y
Le t
39
um
s um
m u st b e
t
ra e
p er
r
5
1 00
3
—
—
i n v e st e d
a nn um
.
2 70
5%
at
to
‘
giv e
a
u
r
l
a
r
t
e
q
y
i n c o m e of 3 1 0 5 ?
40
am
W h at
.
at
i nt e r est
at
5%
a nn
r
e
p
um
w ill
.
A f at h e r
.
liv e in
both
p ut
t o $ 6000 in 1 ye ar 9 m o n t h s ?
ou nt
41
s um
,
.
is 4 2 ye a rs o l d , an d h is s o n is
h o w m a ny ye a r s w ill t h e s o n b e
3
as ol d
.
a s o l d as
If
h is
f at h er ?
42
.
S e pa r a t e t h e
e x c e ss o f
50
ov er
the
n
g re at er
t h e s m a ll e r
u m b er 1 4 5 i n t o t w o pa rt s s o t h at t h e
ov e r 50 sh a ll b e 4 t i m es t h e e xc e s s o f
.
n
n
h
i
i
l
i
s
i
v
es
t
e
d
a
t
5
a
d
t
e
a
%
p p
r em a i n d er a t
t h e a nn u a l i n c o m e fr o m b o t h i n v es t m e nt s
is $ 660
Fi n d t h e w h ol e sum i n v est e d
43
.
If
o f a c ert a n
i
.
.
44
.
s q u a re
o m is o f it s l e ngt h
fe e t l e ss a n d t h e w id t h 4 fe e t m o r e t h e
Fi n d t h e di m e ns i o n s o f t h e ro o m
Th e w id t h
w e re 4
r nc
of
a ro
.
,
.
.
I f t h e l eng t h
ro
om
w ould b e
LITE RAL
45
A
.
m an
rem a n
i d er
i n v es t e d
th e
How m u c h w as
46
A
.
in 5 %
al l
bonds
at
am o
m an
pa r
,
the
on
.
i der
b o n d s,
Of
it
b o n d s , b uy i n g t h e m
fr o m t h e w h o l e i n v e st e nt
in 6%
l i n c om e
Fi n d his w h o l e i n v e st m e nt
.
th e
fo r 2 ye a rs is
2 y ea rs 6 m o n t h s
h is m o n e y in 4 %
Of
a nn u a
u n t s t o 32 550
an d
5%
at
20 7
for m e r
fo r
l at t er
re m a n
the
it
of
t he
on
Q UATIO N S
a t ea c h r a t e ?
I n v e st e d i
His
.
int eres t
i n t e r est
investe d
and
E
pa r t
The
at
s am e a s
t he
F RA C TIO N AL
AND
m
.
GE N E RA L PRO BLE MS
2 58
A
.
m
ro
l
b
e
p
r
e
n
e
a
l
g
is
pro bl e m
a
al l o f
the
n
u m b e rs in
g e n e ra l n u m b e rs
It is t h ere f o re e vid e n t t h a t t h e s o l ut i o n o f a g e n era l pro b
F or e x a m pl e
l e m i n v o lv es a li t e r a l e q u at i o n
T h e s u m o f t w o n u m b e rs is m a n d t h e l a rg er n u m b er is n
Fi n d t h e n um b e rs
t i m es t h e s m a ll e r
whi c h
a re
.
.
,
.
.
Le t
th e
x
a n d nx
= the l
x
So lv i n g
Th e
ul a
fi
nd
fi nd
T h e se
m
a nd
2 59
.
the
s
ma l l er
r a tio o
f
.
and n x
n umber ,
the two
the l a rger
a re
t he
l m is
r
b
e
o
p
t hei r
r a tio a re
kn o w n
G e n e r a liz atio n in
and
a
'
a
fo rm
the
n u mber s
f
o
the
su m
.
a ny
two
n
u m be rs
wh e n t he i r
.
lg eb r a is
'
of
f
.
of s o lvi n g
f o rm u l a o b t a i n e d a s a
t h e p ro c es s
int e r pre t in g t h e
pr o b l e m s
su m o
di vi de the produ ct
the n u mber s
f
fi n di n g
r a ti o o
r ul e s for
div ide the
n u mbe r s
n u mber ,
by 1 p l us the
g e n e r a l pr o b l e m
r ul e fo r s o lvi n g a l l
a
ber
=m
x
,
n um
.
a nd r a ti o
su
+ nx
g er
n u m b er ,
ul t o b t a i n e d in s o lvi n g a g e ne ra l
s olvi n g a l l pr o b l e m s of tha t type
by 1 pl u s the
To
ar
ller
r es
fo r
To
sm a
t h a t t ype
.
E
20 8
LE M E N TARY ALG E BRA
E xe rc i s e 1 1 6
1
.
t h ei r
Th e
l a rg e r o f
two
s um
is 1 4 8 8
Fi n d
.
n
7 t i m es t h e
u m b e rs is
n u m b ers
the
m
1 488
2
.
3
.
ll er of t wo n u m b er s is
21
Fi n d t h e n um b er s
Th e
sm a
s um
is
their
If
two
.
n
.
Th e
s am e n
of
th e
o t h er
s um o f
dd e d t h e r e s ul t is
Fi n d t h e t w o n u m b ers
a re a
.
u m b e rs is
d
,
u m b e r s is s a nd
Fi n d t h e n um b e rs
t wo
n
.
1 16
.
s um o f
8
two
s
,
-
x
1d
)
-
an d
2
.
=d
s
—x
2
n
u m b ers is
768 ,
an d
t wo
—
7 68 — 1 1 6
m an
u m b ers
.
t heir
di ff ere n c e is
7 68 + 1 1 6
s
d
:
7
A
n
.
i
+d
a n d o ne
,
x
Fi n d t h e n u m b e r s
s
,
and
di ffer e n c e of t h e
R ea d t h ese fo rm ul as a s r ul e s fo r fi n di ng
wh e n t h e i r s um a n d t h e i r di ff er e n c e a r e k n o w n
Th e
l ar g e r
2 7 69,
.
.
8
.
l a r g er n um b er
s m a ll er n um b er
= th e
x
(
S olv i n g
the
,
— x = the
an d s
6
an d
.
Let
5
,
.
u m b e rs
m
h
i
e
is 8 2
e
s
t
t
;
4
-
l +n
l +n
ll e r
.
m"
8
4
sma
2
2
2
of
l a n d fo r 36800 a n d g a i n e d t h e
if h e h a d s o ld it fo r 35 200
s am e s um h e w o uld h a v e l o s t
Ho w m u c h did h e p a y fo r t h e l a n d ?
7
.
so
ld
p i e ce
a
.
,
8
.
e qua
Th e
l
to
n
u m b e r s is a
Fi n d
t h e l ar ge r
s um o f
t im e s
t wo
n
.
and
,
t he
m t i m e s t h e s m a ll e r is
n
u m b ers
l a r g er n um b er
t h e s m a ll e r n um b e r
=t h e
Le t L =
and a
am
-
S olv in g
,
L
,
mL = nL
L
am
m +n
a nd
a—L
.
.
E
2 10
L E M E N TARY ALG E BRA
S OLUTIO N O F
2 60
.
Th e
st
O ft e n fi n d it
p h ys i c s
ud e nt o f
to
n ec e ss a ry
so
F O RM ULA S
h ig h e r m a t h e m at i c s w ill
and
lv e fo rm u l as
Fo r
.
e xa m p e :
l
di s t a n c e pa ss e d o v er b y a n y b o d y m o vi n g wi t h a
u n ifo r m v e l o c i t y in a ny nu m ber o f u n i t s of t i m e is t h e pr o du c t
o f t he v e l o c i t y a nd t h e t i m e
Th e
.
T h i s l aw
e x pr esse
d in
a
fo r m ul a is
d
So lvi n g t h i se q u a t i o n fo r
v and
—
=
d
v
t
W h a t is
t he
a
vt
t, w e h a v e
—
=
t d
v
and
of
v era g e v e lo c i t y
if it ru ns
44 8
t h e pr i n c i pa l , t h e
ra te
t ra i n ,
a
m il es in 1 6 h ou rs ?
.
d
4 48
t
16
i n t e r est is t h e pr o du c t o f
e xpr es s e d as h u n d re d t h s a n d t h e t i m e
2 61
.
Th e
,
.
i = prt
d t h at r in t h i s fo rm ul a re pr ese n t s t h e
a n n u m a n d t t h e n u m ber of yea r s
It m u st b e
r a te
r
e
p
r em e m b er e
.
fo rm ul a o r li t er a l
foll o wi n g fo r m u l as :
S olvi n g t h i s
we h ave th e
,
= i —r t
p
r
~
1
.
Wha t
s um
will yi e ld 360 in
p ut
at
i n t e r est
At w h at
r at e
n
n
er
a
p
——
1
um
1 ye a r 4 m o n t h s 24 d ays ?
—
i pt = 9 1
r , a nd
t,
—
=
t i pr
6%
at
~
—
U
5 3
fo r 1 yea r 4 m e ht h s
1 3 00
.
—
0
7 50
will 3 1 300
am
ou n t
t o 3 1 39 1 in
-5 0 4
3
In h ow m a ny ye ars , m o n t h s ,
?
n
a m ou nt t o 3 2 345 a t 5 %
pe r a n u m
3
p
,
,
i n t er est ?
—
i r r i = 60
.
u at io n fo r
=i — t
p
-
2
eq
a nd
yr
.
d ays
w ill
$2 2 00
3 mo 1 8 da
.
’
.
S OLUTIO N O F
2 62
The
.
of
i
r at o
t he
F ORM ULAS
i u m fe r e n c e of
c rc
21 1
a ny
t o it s
i l
c rc e
T h e e x a c t v a l u e is
di am e t er is approxi m at e l y
s e nt e d by
T h e fo rm ul as for t h e c i rc umf e re n c e o f a c i r c l e a r e
c
in w h i c h
0
= 7r d
a nd c
= 2 7rr ,
i um fe ren c e d
is t h e
c rc
,
the
di am e te r
diu s
=
=
n
2 7rr fo r r a n d
lv
f
d
d
c
a
So e c 7rd or
a s ru l e s fo r fi n di n g it a n d r
2 63 D e n o t i n g t h e a re a b y A t h e b as e by b
t ud e b y h t h e fo rm ul as fo r t h e a r e a o f a t r i a n gl e
ra
,
.
,
,
,
,
The
the
f
a rea o
d
a l ti tu e ,
the bas e
re a
d
th e
r
an d
r es u
the
a ny
a n d a l ti tud e
lti
1
tri a n g l e i s the produ c t
a l ti tude a nd
the
a
lt s
a re :
b
h
f
and
.
,
o
re pr e
f
o
ha lf the base ,
the ba se
or
a nd
ha lf
ha lf the produ ct
.
of t h e a b o v e fo rm ul as fo r b a n d h a n d re a d
t h e r es ul t s as r u l e s fo r fi n di n g t h o s e di m e n s i o n s
2 64 P rim e s an d S ub s c ri pts
Diff e re n t b ut r e l a te d n um
b e r s in a f o r m u l a a r e o ft e n d e n o t e d by t h e s a m e l e t t e r w i t h
di ff e re nt pri mes o r su bs cri pts
Pr i m es ar e a cc en t m ar ks wr itt e n a t th e r ig h t a n d a b o v e t h e n um b e r ;
So lv e
eac h
,
.
.
.
.
s ub s c r
For
ip ts
ar e s m a
l
e xa m p e , a
T h e se
’
,
a
ll figur es
”
a re r e a
one , n su b
two ,
no, n l
,
d
wr itt e n
a
n s ub
r
i
p
,
me ,
thr ee ,
at
n2 , n 3
the
r
igh t
and
be lo w
the
n um
b er
.
.
thir d ,
a s ec on d , a
r e s pe c t
iv e l y
n sub zer o , n s u b
.
fo rm u l a fo r t h e a re a O f a t ra pe z oid we sh al l fi n d
t h e t w o pa r a ll e l b as es d e n o t e d by b1 a n d C2
2 65 D e n o t i n g t he a re a b y A t h e t w o p ar a ll e l s id e s o r
b a s es b y b1 a n d b2 a n d t h e a l t i t u d e b y h t h e fo rm ul a for t h e
a r ea of a n y t r a pe z o id is :
In t h e
.
.
,
,
,
E
2 12
The
tr a pezoi d i s the produ ct
a n d the a l ti tu d e
f
a rea o
two ba s es
per
.
2
.
l
a s r u es
,
So lv e t h e
fo r m ul a 8
A
ld
c ent
3
.
o
v e fo rm u l a fo r bl b2 a n d
fo r fi n di n g t h o s e di m e n s i o n s
abo
E xer c i s e 1 1 7
1
ha lf the
f
a
su m o
the
d
the
f
.
Solv e t h e
resu l t s
LE M E N TARY ALG E BRA
m an
so
a
I
(l
i
il
p i e ce
,
of
and
re a
an d
g am e d
.
Fo m ul as
G e n eral
-
h,
r
fo r
a , r , an
l a n d fo r
n
d l
.
d o ll ars
a
Ho w m u c h did h e pa y fo r it ?
.
"
fo r m u l a
So lv e t h e
s
m l)
;
-
and n
fo r a l
.
.
4
.
yi e ld
a
5
.
6
.
.
m u st b e
i n v e st e d
q u a r t e r l y i n c o m e o f a doll a r s ?
dzwz fo r
Solv e d l wl
By
Fi n d t h e
7
s um
Wh at
se
lli n g s il k
c os t
So lv e
at
p e r ya r d
v2 t = v 1 t
m
eac h
at
n
%
per
g en e r a l n u m b e r
c en t s a
ya r d ,
a nn
l o s t b%
.
+ n fo r
v 1 , v2 , a n d
t
.
Th e
am
.
Wh at
o u nt
10
.
11
.
.
13
.
to m
14
to m
At w h at
in
c
fo rm ul a
t
ra e
per
.
w o uld
i n t er e st
at r
pe r c e n t
nn
r
e
a
p
um
will
yea r s ?
1
—
3
for
5
5
a nn u m
will
q p,
a
,
and
doll a rs
f
.
yi e ld b
d o ll ars
yea rs ?
fo rm ul a F
Solv e t h e
fo r C
In h ow m a n y ye a rs w ill t h e
d o ll a rs
.
put a t
d o ll a rs in n
So lv e t h e
I nt ere st
12
s um
w id t h
.
.
9
.
.
l en gt h O f a r e c t a n g u l a r fi e ld is m t im e s it s
I n c r eas i n g it s l e n g t h a r o d s a n d it s w id t h b r od s
i n c r ease it s a r ea n s q u a r e r o d s Fi n d t h e di m e n s i o n s
8
to
.
m e r c h an t
a
um
at r
So lv e t h e
%
pe r
.
i n tere st o n a d o ll a rs a m o u nt
a n n um ?
fo rm ul a
“
l
h+ l
for
a,
h
g, ,
a nd
l
.
E
2 14
LE M E N TARY ALG E BRA
li m i n at i n g o ne u n kn o wn n u m b er fro m a syst e m
of f ra c t io na l e q u at i o ns it is o ft e n b est t o pro c ee d wi t h o u t
c l e a r i n g t h e e q u at io n s o f f r a c t i o n s
2 67
.
In
e
,
.
2x
5
3
3x
3y _
4
5
M ul t i pl yi n g ( 1 ) by 3
se c o n
d
re s
ul t fro m
the
( 2 ) by 2
and
fi rst
1 5y
3
,
_
( 2)
33
an d s
we hav e
6y _
_
4
Fro m t h i s e q u at io n t h e v a lu e of y is fou n d
s t it ut in g t h i s v a lu e in ( l ) t h e v a lu e of x is 4 0
,
ub t r a c t i n g t h e
t o be 1 2
.
.
Sub
E
LI M I N ATIO N BY
Sys t e m s
C O M P ARI S O N
2 15
of
fr a c t i o n a l e q u a t i o ns h a vi ng t h e u n kn o w n
n u m b e r s in t h e d e n o m i n a t o r s t h o u g h n o t Si m pl e e q u a t i o n s
m ay b e so lv e d as s u c h fo r s o m e o f t h e i r roo t s
In s o lvi n g s u c h e q u a t i o n s o n e o f t h e u n k n o w n n u m b e rs
Th u s
s h o uld b e e li m i n a t e d w i t h o u t c l e a r i n g o f fr ac t i o n s
2 68
.
,
,
.
,
.
v
a:
M ul t i pl yi ng ( l ) b y 3
s e co n
d
re s
ul t fr o m
the
( 2 ) by
we have
a nd
fi rs t
,
2
2
U
fr o m
wh i c h y
a nd s
ub st i t u t i n g in
x
C h e c k by
s
ub s t i t u t i n g in ( 1 )
=
4
a nd
E x er c i s e 1 1 9
S o lv e t h e
fo ll o wi n g
c he c k
( 2)
and
s
u b t ra c t i n g
,
E
216
LE M E N TARY ALG E BRA
fo ll o w i n g sys t e m s fi rst m ul t i pl y ea c h e q u a t i o n
t h ro ug h b y t h e l c m O f t h e k n o w n f a c t o rs in t h e d e n o m i
In t h e
,
.
.
.
n a t e rs
E x er c i s e 1 2 0
a nd c h e ck, e
S o lv e
1 65 ,
a nd
266)
4y
2x
16
5x
3y
44
6x + 5 y = 2 7
5 x + 6y = 2 8
—
2x
§
x
%y
= 36
=
2
+ y 56
5x
2y _
3
5
2:v
5v
_
_
3
3
2
2b
4d
y
2b
0
40
y
d
q
x
2a
x
+
5 y + 6x = 4 7
4x + 3 y = 3 5
li m i n a t i n g b y
a ny
m eth o d
( s ee
ms
E
ay
+ bx
LE M E N TARY ALG E BRA
2a b
a
2
2
=
by+ a x a + b
ax
(a + n )
( o + u)
t
=
a y + bx
a
dx
1
5131
1
5131
-
by
1
_
35
+w
m
_
a
=
b
m+ u
+
+a
2
-
'
2
—
by l b
—
n
x(a
) + an
—
y( a n ) + 3 a n
—
=
b
ax
by
2 =
a x + by
(a + b
—
a
2
2
3 x + 4y + 6
x2
—
2x 3 y+ 1
1
—
4x + 5 y 2
332
—
3x
a+
3y
—
5
6
2
n
_
8
—
n
3
_
+7
a
+9
3
9
l
1
~
1
+
p
1
+
q
3
2
1
+
p
+
1
+
q
8
7
n+5
—
s
4
12
2
n
+5
\
—
s
4
6
—
l
x
—l
y
5
1
—
l
x
— l
y
—
a
—
a
d
c
x
b— d
—
b c
E
C O M P ARI S O N
LI M I N ATIO N BY
x
y
n
219
s
f
x
n
—
n
s
+s
l
y
—
n
s
n+s
n+s
P RO BLE MS I N S I MU LTA N EO US S I MPLE E QUATI O NS
2 69
.
M a ny
m
r
e
s,
b
o
p
l
ll y c o n t a i n t w o o r m ore
s o lv e d b y t h e u se o f a Si n gl e
which
u n kn o w n n um b e r s , a r e e as
r ea
il y
e q u a t i o n c o n t a i n i n g b ut o n e u n k n o w n n u m b e r
T hi s m e t h o d is a dvi sab l e o n l y w h e n t h e r e l at i o n s b e t w ee n
t h e u n k n o w n n u m b e r s a r e s o s i m p l e t h at a l l o f t h e m c a n b e
e x pr esse d in t e r m s o f a s i n gl e u n k n o w n
In o t h e r pr o b l e m s it is b e t t er t o i n t r o du c e as m a n y e q u a
t i o n s as t h e re a r e u n k n o wn n um b e rs
Wh e n us i n g a syst e m o f t w o o r m o r e e q u a ti o n s t o so lv e
pro b l em s e n o u g h c o n di t i o n s m u s t b e e x presse d in t h e pr o b
l e m t o fu rn i sh a s m a n y i n d e pe n d e nt e q ua t i o n s a s t h e r e a r e
u nk n o w n n u m b e rs t o b e fo u n d
.
.
.
,
.
E x e rci s e 1 2 1
So l v e th e
t wo o r m o r e
1
38
.
Fi n d
.
2
.
3
.
c
A
oi n s
Of
e x cee d s
4
5
pa rt
.
s um o f
t he
n
is 3 24
m an
.
in Tw o Unk n own s
fo llo w i n g pro bl e m s
u n kn own n um b ers :
.
tw o
n um b e rs
u m b e rs
T h e l a rg e r
sum
t he i r
111
T he
l
— P r ob e m s
i
u sin g
is 1 4 8 ,
i
a nd
differe n c e is
t h e ir
.
u m b e rs is 3 % t i m es
Fi n d t h e n u m b e rs
of
two
n
t he
sm a
ll er
,
and
.
g e d 37 i nt o di m es a n d n i c ke l s
H o w m a n y o f e a c h did h e h a v e ?
ch an
o n se c u t i v e n um be rs
Fi n d
o f t h e l ar g e r b y 6
two
i n vo lvi n g
e q u at o n s
c
,
.
of
t he
t he
sm a
n u m b e rs
Divid e 1 1 8 i n t o t w o pa rt s s o t h at 7 t i m es
s h a ll e x c e e d 3 t i m e s t h e l a r g e r b y 1 00
.
.
,
re c e
ll er
ivi n g
n um
b er
sm a
ll e r
.
t he
LE M E N TARY ALG E BRA
E
6
I f the
.
of a
p u pil s
l
c ass
s ea t e
a re
5 pu pil s m u s t s t a n d
I f 4 a r e put
H ow m a n y pu pil s
is n o t o c c u pi e d
diff ere n c e is
8
sum o f
H a lf t h e
Th e
.
1 28
tw o
the
Fi n d
.
l e n gt h of
it s p er i m e t er is 1 1 6
n um
n
b ench
in t h e c l ass ?
a re
.
.
o n e ach
3
o n e a ch
.
7
d
b e rs is 7 3
u m b ers
4 t i m es t h e ir
.
gl e ex c ee d s its Widt h by
Fi n d t h e di m e n s i o n s
a rec t a n
.
'
14 ,
and
.
u m b e r s w h o se s um is 5 0 t h e fi r st b e i n g
g rea t e r a n d t h e se c o n d 1 5 g re at e r t h a n t h e t h i r d
9
Fi n d
.
th ree
,
o ne seat
,
and
,
b enc h
n
20
,
.
10
y
=2
11
.
;
eq
and
—
=
=
10, y
wh e n x
3
,
of
33 2
so
.
of
.
T hre e
t on s
of
sa m e
fi rst
of
n
fi rst
um b e rs
,
T he
.
t h e t h i r d is 7
17
.
x
=4
,
re c e
c oa
l
a nd
2 t o ns
of
h a rd
t h e pr i c e p e r t o n
n
u m b e rs is
and
se c
o n d is
t he
g r e a t er t h a n t h e
iv e
d ail y
a
d a y m o re t h a n
a
o f so
c oa
of
re c e
l
ft
coa
and
ea c h
l
c os t
6 t o ns
of
.
tw i c e t h e t h i r d , t h e Se c o n d
s um o f
t h e t h r ee
num
b er s is
55
.
.
A m a n i n v est s part of 33 200
at
6%
and
the
r est a t
a nn
On e
8
.
p r i ces , 2 t o n s
t h re e
is 59
u a l i n c o m e fro m t h e t w o a m ou nt s is 3 1 8 0
a m ou n t of e ac h i n v est m e n t ?
I f the
th e
,
wh e n
ea c h
ives 2 0 c e n t s
d a il y w a g e of e a c h ?
hard
Fi n d
Th e
Fi n d t h e
.
in
7 m aso n s t o g e t h er
m as o n
w h a t is t h e
is 5 l es s t h a n t h e
16
if
.
n um b er s
,
ft c o s t
15
t h r ee
and a
At t h e
.
b,
a and
,
c a r pe n t e r s a n d
If 3
.
s po k es
fi rst a n d
Fi n d t h e n um b ers
a c ar pe n t e r ,
14
4 m o re
then the
.
w a ge
s um
Th e
.
ax
fr o nt t h a n in e a c h
w a g o n a n d in t h e 4 w h e e l s t h ere a re 1 1 2
H ow m a n y s po kes a r e in e a c h w h ee l ?
.
g rea t er
sec o n d
13
if
u at i o n
T h ere are
whe e l o f a
.
r e ar
s po kes
12
=
b
+ y 32 , find
In t h e
di m e n si o n of
a re c t a n
gl e is
8,
a n d o ne
,
w h a t is
di m e n sio n
E
222
Oi
plu s
2 1 is
a
the
LE M E N TARY ALG E BRA
se c on d
to the
dd e d
pl us
'
fi rst
,
of t h e t h ird is 92; a n d if
t h e s um is tw i c e t h e t h ird
Le t
and
and
3
wh e n
.
b er
s = t h e s e c o n d n um b e r
t = t h e t h i r d n um b er
f
= th e
firs t
n um
,
,
.
f +s +t
2f
.
1 68
g qt f — g z
f
‘
t
fl
"
2
5
=
2
1
2t
f+
v e r a g e w e ig h t o f 3 pe rs o n s is 1 64 lb Th e a v e r
a n d o f t h e se c o n d
a g e w e ig h t o f t h e fi r s t a n d s e c o n d is 1 59 l b
Fi n d t h e w e ig h t of e a c h
a n d t h i r d 1 65 lb
25
Th e
.
a
.
.
.
26
In
.
ch
than
p e rs o n s
27
.
a c om pan y
ild re n
2 9 pe r s o n s t h er e w e r e 1 5 m o re a dul t s
an d 4 m ore m en t ha n w om en
H ow m a n y
of
.
of each
H o w m a n y bu s h e l s
.
b us h e l
a nd Ol d
m i xt u r e
of
2 00
c o m pa n y ?
ki n d w e r e t h er e in t h e
w h eat
ea c h
of
w h eat
n ew
at
a
85 2 a b u s h e l m a y b e m i xe d t o m a ke
w o rt h 90 ¢ a b u sh e l ?
at
bu sh e l s
u m e ra t o r
a
l a r g e r o f t w o fr ac t io n s is 8 a n d
Th e s u m of t h e
t h e n u m e r a t o r o f t h e s m a ll e r f r a c t i o n is 5
fr ac t i o n s is 1 } a n d if t h e n um e r at o rs are i n t er c h a n ge d
Fi n d t h e fra c t i o n s
t h e ir s um is 1 3
28
The
.
n
of
the
,
.
,
.
.
29
T he
.
s um
s um o f
of
t h e t h re e
tw i ce t he
an
the
of
a
t ri a n gl e is
d s t he th ird
a n d t h e s u m o f t h e fi r s t a n d t wi c e t h e t h i r d e xc e e d s
by
Fi n d t h e th ree a n gl es of t h e t ri a n gl e
t w i c e t h e se c o n d by
The
fi rs t
and
gl e s
se c o n
d
exc ee
.
gl e of p a pe r w e r e 4 in s h o rt e r a n d 3 in
I f a s t ri p 2
w id e r t h e a r ea w o u ld b e 2 s q in l e s s t h a n it is
in w id e is c ut o ff o n a l l Sid es t h e a re a is dimi n i s h e d 1 84
Fi n d t h e di m e n si o n s
s q in
30
If
.
a re ct a n
.
.
,
.
.
.
,
.
.
.
31
.
b= 5 ;
.
In t h e
a nd
if
eq
u at i o n
w hen
a
ax
by = 2 0 , fi n d
= 8 b= 3
,
%
.
x a nd
y if
w he n
a
=7
,
E
32
s um o f
T he
.
LI M I N ATIO N BY C O M P ARI S O N
t h e t h r ee
2 23
o f a n u m b er
digit s
is 1 5
The
.
h
h
f
h
i
a
t
e
s
u
m
l
e
s
lf
o
t
e o t h e r t w o ; a n d if
a
c
p
1 98 is s ub t r a c t e d f r o m t h e n u m b er t h e fi r s t a n d l a st digi t s
Fi n d t h e n u m b e r
a re i n t e r c h a n g e d
digi t in
t ens
’
,
.
.
3 3 A m a n h as 34 9 in d o ll a r b ill s h a lf d o ll a r s a n d q u ar t e r s
H a lf o f t h e d o ll a r s a n d g o f t h e h a lf d o ll a r s a re w o rt h
-
,
.
,
.
-
How m any
34
t h e q u a rt e r s
w o rt h 35
a re
.
h as h e ?
i
c o ns
A and B
.
of
and
t h e h a lf- d o ll ars
of
a re
8 m il es
a pa rt
I f t h ey
.
s e t o ut a t
t he
s am e
di re c t i o n A w ill o v e rt a ke B in 4
I f t h ey t r a v e l t ow a r d e a c h o t h er t h e y will m ee t in
ho ur s
1
1 33 h o u rs
At w h a t r a t e d o e s e a c h t r a v e l ?
ti m e
s am e
t r a v e l in t h e
an d
,
,
.
.
A
At 35 l e s s pe r a c r e , h e
c o u ld h a v e b o u g h t 4 0 a c res m o re fo r t h e m o n e y ; at 34 m o r e
2
r
f
r
h
h
h
0
a
c
es
ess
t
h
e
m
n
ld
v
ug
l
a
r
e
u
a
e
b
o
t
o
o
e
c
c
er
e
o
p
y
,
35
.
m a n b o u g ht
a
n
f
a
e
i
c
e
o
l
d
p
.
.
Fi n d
t he
n
u m b er
b o ug h t
o f a c res
and
t h e pr i c e p e r
‘
a c re
.
for 7 l b O f c o ffe e a n d 5 l b
On e w o m a n pa id
o f c o ff e e a n d 1 0 l b
s u g a r ; a n o t h e r p a id 32 0 5 fo r 3 l b
36
.
.
.
.
.
fo r 7 l b o f s ug ar
i ; a n o t h er pa id
Fi n d t h e u n ifo rm pr i c e o f e ac h p er pou n d
r ce
.
and
6 1b
.
.
of
of
.
of r ce
i
.
.
37
a nd
.
A
h a rv es t h a n d
A ugu
e ac h
fo r h is b e a r d
k
e
e
d ay
w
T he t erm
re c e
st ,
.
14, 1 5 ,
39
.
he
of ser
ive d 3 12 3
38
Th e
a nd
s
en
.
did
vi c e
.
fo r
a nd
w o rk h e
i
c ont a ne
d
J ul y
e ac h w o r k d a y
Fo r
fo r fe i t e d 5 0 ¢ fo r his b o a r d
-
.
.
8 Su n d a ys ;
t h e t h r ee pa i rs
of s id es
ea c h s id e ?
-
Ho w l o n g is
A c l assro o m h as 3 6 d es ks
T he s eat i n g
c a pa c
o f ea ch
a re
ki n d
t o wo rk t wo m o n t h s ,
At
l m en t
s e tt e
he
How m a n y d ays did h e w o r k?
u m s of
17
not
ga ged
i ty
of
t h e re ?
the
so m e s n
of a
t ri a n gl e
a re
i gl e a n d s o m e d o ub l e
ro o m is 4 2
Ho w m a n y d e s k s
,
.
.
E
2 24
40
.
A
so
ld
1 0 m ore t h an
at
s h ee p
35
sa m e n
h ad t he
B
LE M E NTARY ALG E BRA
u m ber
B an d 2 5 t o C
T h ey e a c h t h e n
B e fore A m a d e t h e se s a l e s h e h a d
to
.
,
.
,
C t o get h er
and
H ow m a n y di d
.
fi rst ?
41
.
A
b oy
b o ug h t
s om e
p e a c h es
the
at
of
r at e
2 fo r 5 4
3 fo r 5 58, pa yi n g 36 fo r a l l O f t h e m
He
t h e m a l l a t 4 0 ¢ a do z e n a n d m a d e a p r o fi t o f 34
Ho w
a nd som e ot h ers a t
so
h av e
each
ld
.
m any
42
.
did
A
A
and
.
C
’
h e b uy
at ea c h
r
p
i ce ?
h a s h is m o n e y
’
s
and
s
B
’
i n v est e d a t
a n n u a l i n t er e st
s
t o g e t h e r is
an d
H ow m u c h m o n ey h a s
B
and
at
t o g e t h er is 33 98 ;
A
’
s
and
ea c h o n e
C
’
C
at
B
’
s
t og e t h er is
s
i n v e st e d ?
gu l a r s h e e t o f pa pe r is 6 i n c h e s
g r ea t er t h a n h a lf it s l e n gt h
I f a s t ri p 3 i n c h es w id e w e r e
c u t o ff o n t h e f o u r Sid es it w ould c o n t a i n 360 s q u ar e in c h es
Fi n d t he dim e n s i o n s of t h e p a per
43
.
Th e w id t h
of
a rectan
.
,
.
.
um b ers and 2
1}
o f t h e s e c o n d is 1 1 8 t h e s um o f g O f t h e se c o n d a n d 7 o f t h e
2
O i t h e t h i r d a n d 7 O f t h e fi rs t is
t h i r d is 93 a n d t h e s u m Of
1 1 2 w h at a r e t h e n u m b e r s ?
44
.
I f the
s um
of
Of
the
fi rst
of
t hre e
n
,
,
,
45
.
A
sea t s a r e a l l
p a sse n ge rs
.
The
does
s um
di ff e re n c e is 1 4
f e et Fi n d t h e
.
47
.
.
4
l on g
s ea t s
.
Wh e n t h e
n
r
n
r
a
h
c
5
6
e
s
o
s
a
e
se
t
e
d
ea
l
o
g “s e a t
,
,
p
n
n
h
n
c
h
h
e
H o w m a ny
a
e
g
er
s
a
ea
s
o
r
t
o
ss
t
p
.
each
of
s e at a c co mm o
d ate ?
,
.
'
.
of m o ney a t s i m pl e i nt er e st a m oun t s
in 1 4 m o n t h s it a m o unt s t o
s am e r a t e
s um i n v e s t e d a n d t h e ra t e
a s um
At t h e
Fi n d
ki n d
of t w o s id es O f a t ri a n gl e is 5 8 f ee t a n d t h e
f eet T h e perime t e r of t h e t r i a n gl e is 10 3
l e n gt h of eac h sid e
In 8 m o n t h s
t o 37 8 0
sh o rt a n d
o c c u pi e d
h o ldi n g 6 m o r e
46
h as 1 2
s t r ee t c ar
t he
,
.
E
22 6
L E M E N TARY ALG E BRA
T HREE OR M ORE U N KN O WN N U M B ER S
2 70
.
To fi n d t h e
v a lu e s
s i mu l t a n e o u s
of
thr ee
u nkn o w n
n u m b e rs ,
three
eq ua t i on s
n e c e s s a ry
i n d e pe n d e n t
a re
In
g e n e ra l t h e re m us t b e a s m a n y i n d e pe n d e n t Si m ul t a n e o u s
e q u a t i o n s as t h e r e a r e u n k n o w n n u m b e r s t o b e f o u n d
,
.
In
be st
lvi n g syst e m s o f s e v era l Si m ul t a n e o u s
t o e li m i n a t e b y a ddi t i o n o r s ub t r a c t i o n
i
e qua t on s ,
so
T h us
.
it is
,
6x + 4 y + 2 z = 3 2
To
so
2x + 3 y+ 3 z = 2 5
lv e
2g
5z
22
6x + 9g
9z
75
6x + 4 y+ 2 2
32
5g
7z
43
4 x + 6y
6z
50
4x
5z
22
4x
M ul t i p l yi n g ( 2 ) by 3
fro m ( 4 )
Su b t r a c t i n g ( 1 )
M ul t i pl yi n g ( 2 ) by 2
M ul t i pl yi n g ( 7 ) b y 7
Dividi n g ( 9) b y 5 1
S u b st i tu t i n g t h e
fo u n d
t o be 4
t he
or
C he c k by
in
2g
8 y+
,
5 1y
1 53
y
v a lu e
in
of
v a lu e o f x is fo u n d
s u b st
(5)
it u t i n g
the
l
28
1 96
,
S ub s t i t u t i n g t h e
.
z
56y + 7 2
5g 7 z
f ro m
S ub t r a c t i n g ( 5 )
is
,
fro m ( 6)
Sub t ra c t i n g ( 3 )
z
,
or
in
v a l ues
43
3
the
v a lu e of
o f y a n d z in
to b e 2
l
c a c u at e
d v a lu es of
x,
y,
a nd
a nd
—
—
6x 3 y 2 s = 1 5
—
5x 2
9z = 13
+ y
4x + 3 a = 33
C o m bi ne
i
(1)
ne w e q u at o n
and
( 2)
li m i n a t e y T h e n c o m bi n e
n
m
n
t
h
r
r
2
d
e
li
i
a
t
e
e
i
e
x
o
a
3
( )
a nd e
fou n d w i t h
.
.
z
E
27 1
al l
On e
.
th e
or
LI M I N ATIO N BY
m ore
u n kn o w n
C O M P ARI S O N
o f a sy s t e m o f e q u a t o n s
i
n um b ers
227
m a y n ot
i
c o nt a n
.
3 x + 3 y = 33
So lv e
6y
—
2x = 3 2
62
M ul t i pl yi n g ( 1 ) by
6x + 6y = 66
2,
-
Su bt ra c t i n g ( 2 )
6x + 3 8 = 5 1
f ro m
M ul t i pl yi n g ( 3 ) b y 3
3 z + 6y = 1 5
—
,
6x + 1 8 2 = 96
A ddi n g ( 5 ) a n d
S u b s t i t ut i n g t h e
2 1 2 = 14 7
v a l ue o f
z
in ( 2 )
a n d so
lvi ng
6
y
Sub st i t u t i n g t h e
v alu e o f y in ( 1 )
and so
lvi n g
s
u b sti tu t i n g
x
=5
= 6 a nd
,
,
y
,
5
x
C h ec k by
an d (3 )
,
z
7 in
(2)
E x er c i s e 1 2 2
So lv e t h e
1
.
f o llo w i n g
sy s t e m s o f e q u a t o n s
i
x + y= 18
—
2x + 3y 4 2 = 1 6
z = 19
—
4x 2 y + 3 z = 4 5
—
—
8 x 3 y 4 z = 28
y+
x+ s
=l7
2
.
2x + 3 y
—
3x
=
2
z
3y+
30
—
4 x 6y+ 5 z = 4 5
=
2
3x
5
8
3
+ y+
2 x + 4 y+ 24 = 2 z
4x + 2y
14
52
E
228
LE M E N TARY ALG E BRA
—
=
2
z
3x 1 5 +
4y
—
=
3z
1
5
x
3 y+ 2
5y
x+
=
3
+ y 23
10
.
=
—
22
2
x
4y 3 8 +
4g + 5 z +
13
.
.
3y+
—
3z 3
z
y
14
+ 2x
16
18
—
=
15
2
2
4
5x + y
= 38
=
29
z
4
3x + y + 2
—
=
2
5 2 + 4x
y
3x
4y = 1 2
—
=
10
4
x
4
z
3y
+
2x +
12
3x + 3y
.
z
—
—
=
14
2
z
2
x
3y+
—
=
27
x
3
5 y+
62
11
2 2 + 4 z = 5x
3x + 3
=
z
35
3
2x + 4y +
2z
—
=
z
35
4
3 y+ 5 x +
=
5
4x + 3 a + y 3 5
14
.
4 y+
3z
=
27
x
2
+
=
4 x + 3 z + 2 y 33
=
a
x+y
15
17
.
x
+
z
= b
.
C
E
2 30
L E M E N TARY ALG E BRA
1
1
2
10
a
—
a x l 3
(
2§
2 73
.
-
1
)
M ea s urin g is R atioin g
m a g n i t ud e is t o fi n d it s
16
2
—
m
n
.
To m eas ure
.
m
2
—
ki n d
a ny
2
n
of
s o m e st a n
d a r d u n it o f t h e
M e a su re d m a g n it ud e s
ki n d o f m a g n i t ud e b e i n g m e a s u re d
a re e x p r e s se d b y s o c a ll e d c o n c re t e n u m b e r s s u c h as 6 in
i
r at o
to
.
-
.
,
1 0 ft
.
,
The
4 lb
.
,
20
5 days , e t c
ra t i o o f t w o s u c h
a c r es ,
v alu e o f
a
,
.
n um
b e r s is
l
l
c a c u at e
d by
fi rst expres sin g both nu mbers in a common u nit a n d t h e n
fi n di n g t h e v a lu e o f t h e r at i o of t h e s e e q uiv a l e n t s T h u s
1—2
—
b ut 1 2 in t o 36 in
t h e r at io o f 1 2 in t o 3 ft is n o t 3
,
.
.
or
g
.
.
,
.
,
i
r
o
,
g
I f the
t w o m a g n i t ud e s
uni t it is
,
w i t h ou t m e a n i n g t o
1
.
18
12
4
.
1 mi
.
7
.
1 00 15
fo ll o w i n g
d
s pe a k
of
r at o s
t he i r
660 ft
.
.
1 t on
i
r at os
2
.
5
.
24 31
:
7 da
“
(H
8y
7 hr
.
.
”
-l
x+v
a
2
b
2
_
a
3
z—
a
+b
ab
—
x
2
x
a c omm o n
ex pr e ss e
E x er c i s e 1 2 3
Si m plify t h e
in
ot b e
c a nn
-
3
+b
2
5
—
10
3x
i
.
RATIO
274
A ratio
.
g r e ate r in e qu ali ty is a r a t i o in w h i c h t h e
g re at e r t h a n t h e c o n se q u e n t T h u s , 7 5 is a
of
d e n t is
r a t i o o f g r e a t e r i n e q u a li t y
a n t ec e
27 5
'
A
.
r at
23 1
io
of
.
.
l e s s i n e qu a li ty
d e nt is l es s t h a n
l ess i n e q u a lit y
the
ce
is
c o n se q u e n t
i
a ra t o
Th us
.
in w h i c h t h e
,
5
7 is
a n te
i
a rat o o f
.
2 7 6 Th e o r e m
.
f
be
a
(a )
.
i
ra t o
be 5
dd e d t o bo t h t e
r
:
ms
or
i
5+n
n
A
b
)
(
st
ude nt
t he
2”
fo r
a rat o o f
i
to
.
Divid e
fin d
2
x,
.
one
3x ,
3
.
3
p ar
an d
Di v id e
ot h e r a s
as
a sc a nt
e ac h o t h e r as
C a ll
2
Divid
5
.
n
5 +n
n um b e r , n ,
3 +n ,
5+n
or
3 +n
'
(T,
is
l e ss
th a n
5’
3
°
l e s s i ne q u alit y is l e ft
t o the
.
E x e rc i s e 1 24
1
i
3 +n
illu st ra t i o n
t h e po s i t iv e
r at o
5 +n
3
3 +n
and
.
quotien t + r em a in der
divi s or
51
s a me
the
g rea t e r i n e q ua li t y
3 + n) 5 +
divi den d
s n ce
g
,
givi n g
a d d in g
.
i
3,
in equ a l i ty i s dimi n is hed ,
is in cr ea sed , by
a rat o of
F or
T h e n dividi n g :
Or ,
r
r
a
t
e
e
g
to both i ts ter ms
r
u
m
b
e
i
i
v
n
os
t
e
p
Le t t h e
f
r a tio o
i n equ a l i ty
l ess
a nd a r a tio o
I llu s tr a tio n
A
.
3
e
t
5x
3
3x
1 6 ft
li n g
5
and
P r ob l e m s
l o n g i nt o
.
t w o p a rt s t h a t
a re
th a t
and
.
the
.
o
t h er
5x
.
N o te
3x + 5x = l 6,
.
t he
n
um ber
80
i nt o
t w o pa rt s t h a t
a re
to
eac h
.
an
1 8 - fo o t
sc a n t
li n g i n t o
t w o pa rt s t h a t
a re
E
2 32
4
as
S e pa r a t e 1 2 1
.
8
t w o p a rt s t h a t
i nt o
3
:
5
Th e v a l ue o f a fra c t i o n is 2
.
I f bo th
.
v a ue o f
l
i n a l fr a c t i o n
ea c h
to
ot h er
2
3
the
r es u
l t i n g fr a c t i o n
t e rm s
is
i n cr ea se d
Fi n d t h e o rig
2
.
a re
.
N o ti c e t h a t t h e
as
are
.
by 2 , t h e
t or
L E M E N TARY AL G E BRA
t
n um e r a or o f
the
or
igi n a l fr a c ti o n
is t o t h e d en o m i
.
is t o t h e
d e n o m i n a t o r o f a f ra c t i o n as
I f t h e n u m e r a t o r is i n c r e a se d a n d t h e d e n o m i na t o r
3 :4
di mi n i s h e d b y 5 t h e v a l u e o f t h e res ul t i n g frac t i o n is 2
Fi n d t h e o r igi n a l fra c t i o n
6
The
.
nu m e r a t o r
.
,
.
.
7
.
The
v a lu e
b o t h t e rm s t h e
of a
fra c t i
r e su l t i n g fr a c t i o n
o rigi n a l fr ac t i o n
8
.
Th e
i
ra t o
fi e ld is 2 5 a c r es
9
—
o n is 1 0
u b t r a c t e d fro m
Fi n d t h e
t h e v a lu e 2
I f 4 is
h as
s
.
.
fi e ld s is 2
Fi n d t h e a r e a o f t h e s m a ll e r fi e ld
of
.
the
a re a s o f
two
The
.
l a rg e r
.
fi e ld s o f t h e sa m e s h a pe a re as t h e s q u a res
H o w d o t h e are as o f t w o fi e ld s
o f t h e i r c o rr e s po n di n g Sid e s
13 ?
c o m p a r e if a p a i r o f c o rr e s po n di n g Sid e s a re a s 7
9
.
T he
a r e as o f
.
P R O P O R TIO N
277
.
A proportio n
E xa m pl es :
an d
Fou r n u m b e r s
o r p ropor ti on a l ,
278
o
p
.
rt io n
279
.
,
Th e t e rm s
a
i
5x
as a ,
if
a n e q u a t o n o f ra t o s
is
b,
b=
i
4x = 5
4
c , a nd
d,
c
of t h e
d,
or
a re s a
a
c
b
d
i
r a t o s a re c a
id
.
t o be in
p
te rm s
of
ll e d
rop ortion ,
t h e pro
'
.
E xtr e m e s
an d
M e an s
t h e p r o p o rt i o n a r e t h e
t e rm s a r e t h e m e an s
of
.
.
e xtr e
fo u rt h t e rm s
a n d t h e sec o n d a n d t h i r d
Th e
me s
fir st
and
E LE M E N T A R Y A L G E B R A
2 34
21
1
1
m+ a
282
M e an Pr o po rtio n a l
.
of a
p r o po r t i o n
this
n um b e r .
e xt r e m es
se c o n
s a m e n um b e r , a s
t he
t h e 1 5 , is
a :x
,
b,
a re
I f the
.
d
t h i r d t erm s
and
in 5
:
45,
m e an pro portio na l b e t w e e n t h e
a
.
T h u s in
a an d
'
3a
=
a:
is
w hence
11:
b,
x
a
m e a n p ro po rt i o n a l b e tw e e n
w e have :
and
x
2
=
a b,
WT
)
A m ea n pr op or tion a l between two
of thei r pr odu c t
,
o r,
in w o r d s
n u mber s
i s the
s qua r e r oot
.
Th ird P ro por tio n al In t h e
t h e n um b e r 0 is a thir d pro port io n al
283
.
T h us
in
2 84
A Fourth P roportio n a l
t h re e
a
A
.
.
7 6
80 is
,
num b ers , a ,
c
It is t h e
d
c o m pl e t e s
b
T
gg
5
t h ir d
an d c ,
n um b e r ,
f o ur
a
b,
a
7
—
h u s in 1 5
-
i
p ro po r t o n a
a and
to
pr o p o rti o n a l t o 5
.
an d
: c,
b
.
20
.
pr o po rt i o n a l t o t h e
n u m b er
d in t h e pr o po rt io n
is t h e
t er m e d p ro po r t i o n
f o ur t h
b= b
A fo urt h
w h i c h w i t h t h e t h re e
3 9 is t h e
:
giv en
n um b e rs ,
.
pr o por ti on a l t o 7 , 1 3 ,
an d
21
.
E xer c i s e 1 2 7
Fi n d m ea n
pr o p
1
.
3
and
27
3
.
1
and
81
5
7
.
.
i
j
-
ah
3
and
o rt i o n a l s b e t w e en
1
§ 5
25
8
.
a
+b
and
9 ( a + b)
3
P RO P ORTIO N
Fi n d
9
11
13
23 5
fo ll o w i n g
t h i r d pro po r t i o n a l s
.
2
and
6
10
.
4
a nd
9
.
2
a nd
22
12
.
7
1
and
2
.
—
113
—
8
and
w
o
14
h
.
H
16
I
.
31
and x
+ y and
x
+y
-
a;
2
y
Q
Fi n d f o u rt h
17
.
19
.
21
23
4, 8,
an d
12
5 , 6,
and
12
m,
.
n, and
m+ n,
.
i
fo ll owi n g
to the
l
n
r
a s
r
o
o
t
o
p p
%
.
20
.
22
p
—
m n , an d
.
25
18
.
x+
m
2
24
.
and
a , x, a nd
a, a
x
,
,
x
y
and a
2
7
4
1
4
and x
9
,
—
x
l
a nd
1
.
12, 3,
P RI NC IP LE S OF PRO P ORTI O N
28 5
.
Si n c e
eac h o f
wri t e
Us i n g
t he
f o ll o w i n g
pr o d uc t s is 2 4 , w e m a y
2
fo u r n u m b e r s o f t h e se t w o p ro d u c t s w e
m a y wr i t e t h e t w o c o l u m n s b e l o w t h e fi r s t b e i n g pr o po rt i o n s
a n d t h e s e c o n d n o t pr o po r t i o n s
T es t by § 2 8 1 t h e e x pre ss i o n s o f b o t h c o l um n s a n d sh o w
t h at t h e e x pr e s s i o n s o f t h e fi r s t c o l um n m e e t t h e t e s t w h il e
t h o s e in t h e s e c o n d c o lu m n d o n o t
on
ly
t he
,
,
.
,
,
.
P R O P OR TI O N S
E
X PR E SS I O N S
1
.
2
.
5
.
2
3
NOT
P R O P O R TI O N S
E
2 36
LE M E N TARY ALG E BRA
N o ti c e t h a t in t h e firs t c o l um n t h e pr o p o r ti o n s ar e m ad e b y us i n g
bo th f a c t o r s o f o n e o f t h e pr o d u c t s a s m ea ns a n d b o t h fa c t ors o f t h e o t h er
In t h e s e c o n d c o lum n n o ti c e t h a t t h i s p l a n is n o t
pro d u c t a s ex trem es
o b s er v e d a n d t h a t t h e e xp r e ss i o n s o bt a i n e d ar e no t p r o p o r ti o n s
,
.
.
,
If the p r odu c t of two n u mbers equ a l s the
other n u m ber s , the fa c tor s of ei ther p r odu c t m a y
28 6 P rin c ipl e
.
r
o
d
u
c
t
o
p
f
two
.
be ma d e the m ea n s
f
o
r
i
n
r
o
o
t
o
p p
a
Suppo se a
T O pr o v e
Proof
a nd
s
d ==b
-
bd
z
’
i m plify
b
d
c
etc
’
Yo u
the 4
the
ex tr em es
both
s
of
of
a
'
=
d b c
by
bd ,
$5
.
id es
of
d=b
-
a
~
c
by
pr o v e d
a re
ed ,
s
and
i m il a r l y
t h e 8 p o ss i b l e p r o po rt i o n s yo u
eq
l e tt er s
r
c
o
d
u
t
p
3
O t h e r pr o p o r t i o n s
.
other
m em b er s
bo th
u at i o n
sh ould b e a b l e
the
the
,
c
i i g
Se e h o w m a n y
fr o m
=
a nd
obta n n
,
f
o
c
Divid e
.
thes e
.
A l so divid e
a
a nd
a
b
d
ob t a n
i
.
c an
wr i t e
0
t o w r i t e two , b e gi nn i n g w i t h
a ny o n e o f
.
E x e rc i s e 1 2 8
Write
eq
al l
t h e pr o po r t i o n s you
c an
fr o m
the
follo w i n g
u at i o n s :
1
2
9
.
3
10
.
4
.
m q
-
.
1
=n-
p
i
m a y b e d e r iv e d f r o m o t h e r e q u a
t i o n s s o m a y pr o p o r t i o n s b e d e r iv e d f r o m o t h e r pr o port i o n s
T h e pr i n c i pl e s fo r d e r ivi n g pr o po rt i o n s fro m pr o po r t i o n s
a r e n o w t o b e e st a b li s h e d
287
.
J us t
a s e q u at o n s
.
.
E
238
a
_
—
o
g a
,
( I)
a
( 1)
t h en
T o pr o v e
L E M E N TARY ALG E BRA
+b
c
b
pr o c e e d by
a nd
+d
d
thus
a n a l ysi s ,
A NA L Y SI S
A ss um e
a
( 1)
+b
c
b
+d
d
R e du c e t h e i m pr o per fra c t i on s t o m i xe d n u m b er s
or
a
Whenc e ,
m ay
j u s t giv e n
n o w c o n str
kn o w t h a t
A dd
1 t o b ot h
b
d
a
c
a
c
b
d
OOF
t h e pr oo f , by
uct
s
9
b
R e du c i n g
s t e ps
d
'
9+ 1 = f + 1
+b
c
+d
W he n e i t h er o f t h e l a st
t
he
Pr opo r ti o n
by
gd
z
’
d
w e hav e
a
+b
c
+d
b
c
a
c
b
c
i m pr o p e r fr a c t i o n s
to
a
t io n
th e
v er s i n g
id es o f t h e e q u a t io n s
a
-
re
d
an d
from
d
.
We
C
b
c
PR
We
=f+1
d
t w o pr o po r t i o n s is
is
p r o p o rt i o n ,
sa
id
i n ferr e d di r e c t l y
t o b e t a ke n b y
ad d
i
.
2 91
.
is
O ft
en
c a ll e d
P ro por tio n b y S ub tra c tio n
pr opor ti on ,
the
pr o p o r ti o n by
.
If
four
com pos i ti o n
n u mber s a r e
they wil l be in p rop or tion by s u b tra c tio n
the difi eren ce
ei ther
add i ti o n
f
o
the ter ms
a n tece den ts or
the
f
o
ea ch r a ti o
form
f
consequen ts o
-
the
a
.
.
in
Tha t is ,
o
r
i
r
o
t
on
p p
ra ti os
.
with
P RO P ORTIO N
a
—
a
o
—
c
b
d
b
-
a
_
b
d
’
P r o po r ti o n by
.
—
c
d
c
a
Use t h e m e t h o d o f a n a l ysi s Jus t
2 92
2 39
s u btr ac ti o n
Pr oportion by
is
o
as
it w as
ft e n c a ll ed
Add ition
d
us e
ab o
ve
pro po r ti o n by d i vi si on
S ub tr a c tio n
an d
If
.
nu m ber s a re
tio n
If
.
i n p r opor tion they wi l l be i n proporti on by
a nd s u b trac tio n
.
four
ad d i
.
a
c
b
d
t hen
’
C om b i n e t h e
P r o po rti o n
composi tion
by
a nd
a
+b
a
—
r es u
b
lts
of
c
+d
c
—
d
'
t h e pr i n c i p l e s
a dd ition a nd s u btr a
d i vis i on
ction
is
o
of
a nd
2 90
ft e n c a ll ed
291
.
pro por ti o n by
.
E x e rc i s e 1 2 9
F ro m e ac h of t h e fo ll ow i n g w r i t e a pr o po rt i o n ( 1 ) b y
a l t er n a t i o n
( 2 ) b y i nv ers i o n ( 3 ) b y a ddi t i o n (4 ) by s ub
t ra c t i o n a n d ( 5 ) b y a ddi t i o n a n d s u b t r a c t i o n
1
.
,
,
,
,
,
2
6
'
g
3
—1 1
77
3
—
m n
F r o m e a c h o f t h e f o ll ow i n g e q u at i o ns w r i t e a pro po rt i o n
c o m m e n c i n g w i t h e a c h o f t h e f o u r f a c t o r s ; t h e n t a ke e a c h
n
n
r
n
r
1
r
i
b
a
l
t
e
a
t
i
o
o
o
t
o
( ) y
( 2 ) b y i nv e rs i o n a n d ( 3 ) b y
p p
a ddi t i o n a n d s ub t r a c t io n
2
.
,
,
1
3
3
.
2
.
o
a
M
o
.
s
Fi n d v al ue s
1
—
sis
.
,
7
7
—
:c
2
x+ 2
s
g
4
-
o f a:
in t h e
f o llow i n g
12
.
o9
-
.
.
i
n
r
s
o
or
t
o
p p
6
x+ 5
5
4
11:
3
1
7
3r + 5
11
11
5x
5
5
E
24 0
LE M E N TARY ALG E BRA
x+4
—
x
3
°
4
.
5
.
6
.
1
Divid e
91
x
°
8
i nt o
t w o pa rt s t h at
+6
—
x
1
6
5
to
a re
e ach
o t h er
as
2%
Divid e m i n t o
T he
other
as a
t w o pa rt s t h a t
diffe r e n c e b e t w ee n tw o
b is d
Fi n d t h em
.
,
are
n
to
ea ch ot h er a s a
um b er s
t h at
a re
to
b
.
ea ch
.
b e r m us t b e a dd e d t o e a c h t e rm of 3 6 = 4 8
t o giv e a n o t h e r pr o po rt i o n ?
8 B y w h a t n u m b e r m u st ea c h f a c t o r o f t h e p r o du c t s
2 5 5 1 a n d 3 1 4 0 b e r e d u c e d t h a t t h e pr o d u c t s m a y b e e q u a l ?
7
W ha t
.
n um
.
9
t o m ake
.
s um s
m u st
e a ch
re
W h at
wh i ch
n
u m b e r m u st b e a dd e d
a re
Wh at
to
ea c h ot h e r a s a
of
fac t or
of
duc e d a n d e ac h fac t or
t h e pr o d u c t s e q u a l ?
be
10
n u m b er
B y w h at
.
to
t h e p rodu c t
be
b ot h m
i n c r e as e d
an d n t o
,
giv e
b?
dd e d t o m a n d s ub t r a c t e d fro m n
give s n u m b e rs t o e a c h o t h e r a s a b?
I n c r easi n g n u m er at or
1 2 T h e v a lu e o f a f ra c t i o n is
a n d d e n o m i n at or b y 2 giv es a f r a c t io n w h o s e v a lu e is
W h at is t h e fr a c t i o n ?
11
.
n
u m b er
a
.
d e n o m i n at o r o f a fra c t i o n is
n u m e ra t o r
R e du c i n g b o t h t e rm s b y
Fi n d t h e fr a c t i o n
w h o se v a l ue is
13
.
The
.
6
1
gre at e r t h a n t h e
giv es a fr a c t io n
.
I f t h e d e n o m i n at o r
of a
fr a c t i o n w h ose v a lu e is
i n c rease d a n d t h e n um erat o r d e c r e a se d b y 3 t h e v a lu e of
r es ul t i n g f r ac t i o n is
Fi n d t h e fra c t io n
14
.
,
is
t he
.
15
to
.
um b er m u st b o t h
3
—
fra c t io n w h o se v a lu e is 1 0
B y w h at
giv e
a
n
t erm s
of
e
b
3
—
i n c r e a se d
fra ct i o n is
I f 7 is a dd e d t o t h e
n u m er a t o r a n d 2 t o t h e d e n o m i n a t o r t h e r e c i pr oc a l v a lu e o f
Fi n d t h e o rigi n a l f ra c t i o n
t h e o rigi n a l f r a c t i o n is o b t a i n e d
16
.
Th e
v a lu e of
a
,
.
.
LE M E N TARY ALG E BRA
E
242
E x ercis e 1 30
1
as
A ss u m e
.
t h e t i m e , t,
l aw fo r t h e
A ns
2
of
t he
am
i
am o
u nt
of
w,
,
w a t e r in
t h e in - fl o w
b e ga n
o f w a t e r in t h e b a r r e l
s n ce
w a t e r in t h e
3 6 = q 3,
°
pas s e
at
t he
va r y
g e n era l
time , t
.
ft er 3 m i n u t es o f flo w t h er e a r e 3 6 qt
b arr e l
Fi n d q a n d st at e t h e l aw d e fi n i t e l y
.
w = 36, a n d t = 3 in t h e
= 1 2 a n d t h e d e n i te or m
fi
f
q
,
or
w
.
W ri t e
.
to
a
S ub s titut e
3
b a rre l
a
w = qt
.
S u pp o s e t h at
.
ou nt
A ft er
2 m in ut e s
d i nt o
t h e b arre l ?
S ub s titut e
l a w , w = qt,
bt a i n i n g
t h en
t h e l a w is
of
o
.
12 t
flo w h o w m a n y q u a rt s will h a v e
of
t = 2 in w = 1 2 t,
g en er a l
.
t i
o b a ni n
w = 12
g
°
o u nt of w a t er in a c i st e rn is a s s um e d t o v a ry as
t h e s q u ar e o f t h e t i m e t s i n c e th e ih flo w t h r ough a t u b e
b e g a n E xpr ess t h e g e n er a l l aw c o nn e c t i n g w a n d t
2
G en er a l l a w :
w qt
4
Th e
.
am
-
,
,
.
.
5
Su ppo s e t h a t
.
i
c st e r n
In
Fi n d
.
q
a
ft e r
5 m i n u t es t h e re
a n d st a t e
2
=
= 225
w
w
t
u
t
q , p
and
t h e l aw
t = 5,
o
of
.
or
q
=9
,
.
.
2
=
In w 9t ,
su
b s tit u t e
t = 3,
gi v i n g
qt
(8 1
in
.
i t
c s er n
A ft er h ow l o n g w ill t h e re b e 900 q t
.
2
900 = 9 t
°
8
in t h e
Fi n d t h e q u a n t it y o f w at e r in t h e c i st e r n a ft er 3 m i n u t e s
.
flo w
7
.
bt a in in g
'
6
d e fi n it e ly
225 = q 25,
2
w = 9t
D e fi n it e l a w ,
2 2 5 qt
are
If
.
y
cc 11:
and
n ec t in g x a n d
We
h av e ,
firs t
n
T h er e fo r e
H en c e
,
,
t = 1 00 ,
an d
= 1 0 w h en
y
x
?
y
y
,
M ki g y = l O
a
,
or
2
an d x
= 5,
= km
.
10 = 5k
k
2
y
2x
.
t = 10
=5
,
.
)
in t h e
( a ft er
c st e n ?
i
1 0 m in
r
.
)
w h a t is t h e l aw
con
VARIATIO N
by
s,
fo r c e F
v ari es as
a
,
for c e
F
,
,
a m o u n t o f s t re t c h ,
t he
th e
E x pr e ss t h e
.
O f st r et c h
di n a ry
10
k,
to
i g
s pr n
a n d e x pr es s
11
.
12
.
t t ch 8
t h e f or c e
Th e
s re
,
,
,
i
v a r es a s
F
.
m ark s
g ra du a t i o n
of
an
b al a n c e ?
t h e l aw
a
H o w m a n y po un d s
e a st r e t c h o f
.
th e
,
d efi n it el y
How m u c h w o uld
giv
l aw
by t h e
sh ow n
W h en t h e fo r c e is 2 0 l b
.
the
.
Ho w is t h i s l aw
or
gt h of
g e n er a l
st r e n
10
i
is 5
i n c h es
.
Fi n d
.
fo r ce
of
s t re t c h
of
fo r c e
32 l b
.
s t ret c h
t he
w o u ld h a v e t o b e
i g?
s pr n
e x e rt e
d
n c h es ?
v a r i es a s t h e s q u ar e o f a s id e 3
W h e n s = 5 A 2 5 Fi n d k a n d e x pr es s t h e l aw c o n n e c t i n g
H a v e yo u m e t t h i s l a w b e f o r e ?
A a n d s in d e fi n i t e fo r m
13
Th e
.
A,
a re a ,
o f a s q uare
.
,
.
,
,
.
the
l t it ud e o f a re c t a n gl e is c o n st a n t t h e a r e a A
Wr i t e t h e g e n e r a l l aw
re c t a n gl e v a r i e s a s t h e b as e x
15
I f t h e b as e is
14
of
I f th e
.
a
,
.
d e fi n it e fo rm
16
Th e
.
a rea ,
r
g e n e r a l f o rm
.
,
.
id e
A
,
of
.
Th e w o r k ,
r
20
.
.
an
of
e qu
il at e r a l
.
.
E xpr es s t h e l aw in
t r i a n gl e
v ari es as t h e
c o nn e c t i n g A a n d s in
t h e t ri a n gl e is 6 t h e
a re a
is
d e fi n i t e fo rm
w o f a m ac h i n e v a r i e s as t h e n u m b er o f
r u ns
Wri t e t h e g e n e r a l l a w o f w o r k fo r
t h e l a w in
.
,
.
.
W o r ki n g 3 h o u r s
wo k
is 96
E x pre s s t h e l aw
.
a n d e x pr ess
t h e ma c h i ne
a re a
.
,
.
h o u r s , h , t h a t it
19
s
,o f
W h e n t h e s id e
Fi n d k
18
1 2 , t he
,
.
squa e o f a s
17
,
,
t h e m ac h i n e
E x pr ess t h e l aw
of
t he
does
m a c h i ne
foo t t o ns
d e fi n i t e f o rm
-
in
.
Ho w m u c h wo rk w ould t h e m a c h i n e do in 1 m i n u t e ,
1 of an
5 6
hou r ?
C H A P T E R XX I
P O WE R S
RO O TS
.
INV OLUTIO N
296
In
.
m o nom i a l s t o
n
m
a
s
ol
i
l
o
p y
.
2 97
.
r
o
e
w
p
.
an d
l e a r n e d h ow t o r a i s e
a n y p o w e r a l s o h o w t o s q u ar e bi n o mi a l s a n d
T h o se s e c t i o n s sh o uld b e re vi e w e d h e r e
140 , 1 83 , 1 85 ,
1 87 w e
,
.
Involu ti o n
t h e pr o c e s s
of
i i g
w h o s e ex po n e nt is a pos i t iv e i nt e g er
Is
ra s n
a
n
um b er
a n e x po n e n t , a n d
the
i n di c a t e s h o w m a ny t i m e s t h e n um b e r is
fac t o r is c a ll e d t h e e xpon e n t of th e po w e r T h u s
wh i c h
.
(a
is
T h e b as e
.
i d
r a se
to
a
of a
po w er
x
p o w er
in
)
a
2
p ow er
xa
Th e
of
the
= a3
a
e x pr es s o n o f
by
.
a ny
divi so r
T h us
It h a s
b e en
t ake n
as a
,
3 b)
3
i n v o lu t i on is t h e n um b e r wh i c h
b as e , t h e
2
>< a
m
sh own
Xa
"
an
ypow e r o f a b ase b y
e x p o n e n t s a re a d d e d .
3
t h i s l a w in
a
2 99
o nent
.
sa m e
i
ex p
( 2a
4
It h as b ee n s h ow n t h a t t o m u l t i pl y
a ny
a
.
I n v o lu t i o n is i n di c at e d b y
298
to
Th u s
,
5
g e n e r a l n u m b ers is
a
t h at t o
m
“
divid e
any
po w e r
of a
b ase
l ow er po w er o f t h e sam e b a s e th e e x po n e nt of t h e
is s u b tr a c te d f r o m t h e e xpo n e n t of t h e divid e n d
,
.
4=
—
a za
a
5
a
2
= a4
ELE M E N TARY ALG E BRA
24 6
PO WER
A P RO D UC T
or
repr esent a ny t w o n um b ers a nd n a ny
T h e n ( a b) w
ill r e pre s e nt a n y pow er o f t h e
po s it iv e i nt e g e r
B y d e fi n i t io n of a po w e r :
pr odu c t of a ny t w o n um b e r s
-a b a b a b -a b
t o n fa c t o rs
t o n f ac t o r s ) ( bbb
t o n fa c t o r s )
(aaa
302
Le t
.
b
a and
"
.
.
~
°
,
,
a
The
Th e
b
"
power of the pr od u ct of two or mor e
the nth p ower s of the n u mber s
nth
pr oduc t
n
f
o
nu mbers
.
exp e ss o n o f
r
i
t h i s l a w in
( ab )
"
g e n e ra l n u m b e r s
a
n
b
is
"
.
im il a r m a nn er it m ay b e sh ow n t h a t t h e l aw
Th u s
t he p rodu c t of a ny n u m b e r o f f a c t o rs
In
is the
a s
.
( 2a
2
b
n
c
)
3
2
3
a
b
6 3" 3
c
h o ld s fo r
,
8a b
6 3n 8
c
E xe rc i s e 1 3 2
Writ e t h e po w er o f ea c h of t h e f o llo w i n g
1
5
( 2a )
3 4
x
3
( )
2 3
.
.
2
6
3
.
.
(4
3
7
P O WER
30 3
.
We h a v e
( ar b )
2 3
(x y )
t
.
n
2
"
.
A FRA C TI O N
or
t h at
se e n
g
to
-
n
f ac t o r s
n
f a c t o rs
,
’
a
°
a
to
-
a
b b b
°
a
to
~
n
fa c t o rs
,
n
b
n
The
nth
divided by the
The
e
m
r
a
t
o
w
r
t
h
n
u
e
r
f
a
n
i
s
t
h
e
n
t
o
e
o
i
h
c
t
o
r
f
p
f
o
er
o
w
p
n th
e xpre s s
a
f
o
r
w
e
o
p
t
h
e de n omina tor
_
io n of t h i s l a w in ge n e r a l n u m b e rs is
n
IN VOL UTIO N
24 7
E x e r c i s e 1 33
o f e ac h o f
G iv e t h e po w e r
t he
fo ll o wi n g
PO WER S OF BI N O M IAL S
B y m ul t i pli c at io n , t h e
b m ay b e
r
(a + b )
2
3
—
—
a + 3a b l 3ab + b
b)
3
—
a
(a + b )
4
-
3
(a + b )
5
(a
5
b)
2
3
2—
8
—
—
3 a b l 3 ab
b
2
“
a
4
—
(a b )
—4a b 3 -i- b 4
a
s
5a h
a
5
5a b
4
f
1 0a b + 1 0a b
3
5 ab + b
5
1 0a b
3
5 ah
b
5
3
2
2
3
2
1 0a b
2
4
F ro m a n exam i na t i o n o f t he se po we rs
c o n s id e r i n g it t o r e pres e n t t h e e xpo n e nt o f t h e
l o w i n g l a w s h o ld in e ac h e xpa n s i o n :
305
1
.
.
E
te rm
the
f
o
a nd ever y ter m , ex cept
2
.
The
n u mbe r o
f
i t is 1 grea ter tha n the
ex pa n s ion , except
the fi rs t,
c on ta i n s
te r m s in the
f
ex pon e n t o
4
o r e x a ns io ns ,
p
,
ver y
b
o bt ai n e d :
3
(a
f
w
e
s
a
o
o
+
p
fo ll o w i n g
r
e
w
o
,
p
the l a s t,
t he
fol
c on ta in s a
;
b
.
ex pa n s ion
the power
.
is
n
+ 1 ; tha t is ,
E
24 8
3
the
.
If both terms
exp a n s ion a r e
4
.
If the
si on a r e
5
.
1,
.
the
t
i
s
i
o
ve
p
the
even
f
a
n ega tiv e ,
ter m s
f
i n the
it in creas es by 1 in
the
od d
fi rs t
The
f
o
ea c h su cceed in g
te r m
ter m
ea c h su cceedi n g
the
ter m i s
foun d
f
o
the
ex p a n
expa n s ion
is
n,
expa ns ion
is
.
the
f
o
ter m
co e
s u cceedi n g
f
o
.
term
s eco nd
ter m s
fiicien t of the fi r s t ter m of the
coefiic i en t of the s eco n d ter m i s n ; a n d the
.
the ter ms
i
i
os
t
ve , a l l
p
n ega ti v e
b in the
ex p on en t o
a re
.
ter m is
ex p on e n t o
The
a nd
7
f
i t d imi n i s hes by 1 i n
a nd
6
The
the bi nomi a l
o
s econd
o
s
i
t
i
v
e
p
,
LE M E N TARY AL G E BRA
.
ex pa n s ion
is 1 ;
fiicient of a n y
coefiic ien t of the
coe
by m ul tip l ying the
p recedi n g ter m by the exp one n t of a i n tha t ter m , a n d divi ding
the pr odu ct by a n u mber 1 grea ter tha n the expone n t of b i n
tha t ter m
Th e
.
st a t e m e n t o f
t h ese
l aw s c o n st i t u t es w h at is c all e d t h e
b in o mial th e or e m
T h e t h e o re m is t r u e o f a l l t h e e x am p l e s
giv e n We sh a ll t a ke it fo r g ra n t e d t h at it is t r ue fo r a n y
posi ti ve in teg ra l power o f a bi n o mi a l b ut a g e n e r a l proo f li e s
b e yo n d t h e sc o pe o f t h i s b o o k
St u d e n t s w ill fi n d it h e l pful t o m em o ri ze t h e c o e ffi c i e n t s
of t h e 1 st 2 d 3 d 4t h 5 t h a n d 6t h po we r s
.
.
,
.
_
,
,
,
,
.
,
T h ese c o e ffi c i e nt s m ay b e a rr a ng e d in a t a b l e fo rm i n g
’
w h a t is k no w n as P as c a l s T ri ang l e , as f o llo w s :
C oe ffi c i e n ts o f l s t po w e r
1
C oe ffi c i e n ts o f 2d po w e r
5
C o e ffi c i e n t s o f 3 d po w e r
3
1
9
4
1
C oe ffi c i e n t s o f 4t h po w er
6
10
1
10
5
C oe ffi c i en t s o f 5t h po w er
0
15
C o e ffic i e n t s o f 6t h po w e r
15
M
0
20
6
306
.
—
—
—
—
—
t
r
1
i
3
b
‘
0
0
b
A
r
i
l
1
3
ffi c i e n t is t h e s u m o f t h e n um b e r a b o v e it a n d t h e
n u m b e r t o t h e l e ft o f t h e l a t t e r
Th e c oe ffi c i e nt s o f t w o t erm s e q u a ll y di s t a n t f ro m t h e
firs t a n d l as t t e rm s o f t h e ex pa n si o n a re e q u a l
E ach
coe
.
.
E
50
L E M E N TARY ALG E BRA
E VO LUTIO N
root o f a n um b e r is
pro du c t is t h e n u m b e r
308
o ne o f
A
.
t he
equa l
“
w ho s e
f ac t o rs
.
Th u s
,
2 is
a
roo t o f 8
,
1 6 , 3 2 , 64 ,
3 is
a r oo t o f
9, 2 7 , 8 1 , 24 3 ,
5 is
a r o t of
o
2 5 , 1 2 5 , 62 5 ,
a re n a
m e d fr o m t h e
R o o ts
m ak e t h e
n
u m b er
See t w o
.
etc
etc
.
.
et c
u m b e r of e q u a l fa c t o rs
d e fi n i t i o n s 1 90
n
th a t
.
Wh at ro o t of 1 6 is 2 ? Wh at r oot of 1 6 is 4 ? W h at r o o t
W h a t r oo t o f 64 is 4 ? W h at r o o t o f 8 1 is 3 ?
o f 64 is 2 ?
E v ol ution is t h e pro c e s s
e q u a l f ac t o r s , of a n u m b e r
30 9
t he
of
.
fi n di n g
a r o ot , or o n e o f
.
E volu t io n is
i n di c at e d b y
pl ac e d b e fore t h e n u m b e r
the
r ad i c a l
s
is
ig n
.
di c a l s ig n a lo n e i n d ic a t es t he s qu ar e r o o t I f a ny
o t h e r r o o t is r e q ui r e d i t i s i n di c at e d b y a s m a ll fig u re c a ll e d
t h e i nd ex of the r oot w ri t t e n in t h e \/ o f t h e r a di c a l s ig n
t hus :
The
ra
.
’
,
,
A
pa rt
sym b
ol o f a gg re ga t io n w i t h t h e r a di c a l s ig n i n di c at es t h e
of th e expre s si o n t h a t is affec t e d by t h e s ig n
.
m ea n s t h e
Thus V
,
m e ans t h e
Th e
sq
l o ng b a r abov e is
Any
roo t of
c a ll e d a r a d i c al
S in ce
is
,
a
of
r oo t
l m
a v in c u u
of
.
the
See
and
su m
65
of
24 ,
25
W h il e
a nd
24
.
.
u m b er i n
d i cat e d b y t h e r a di c al
s
ig n is
.
ev ol u tion
a n um ber
n
u a re
s um
is the
f
r ev ers e o
the nth power
f
o
i n vol u tion , the
whi ch i s
a
.
nth root o
f
a
E
VOL UTIO N
ROO T
3 10
.
f
nth r oot o
the power by n
A PO WER
"
n th ro
e xt r ac t n
The
f
(a )
i g th e
by
o
Si n c e
m
or
25 1
a
of
ot
is
w
r
o
e
p
b o t h m em b e rs ,
obta ined
by divi ding the
expo nen t
.
E x e rc i s e 1 3 7
1
H o w w o uld yo u fi nd t h e
.
Th e
?
T h e c ub e r oo t
2
G iv e t h e
.
f o ur t h
i n di ca t e d
sq
u a re
Th e
ro o t ?
of
r oo t
fift h
a
po w e r ?
r oo t ?
t h e fo l l o w m g
r o ot of ea c h o f
ROO T OF A P RO D UC T
31 1
.
t he n
Si n c e
V
The
a
f
f
the nth
= ab
Wh y
.
the p r odu ct
nth r oot o
u
ro
d
c
t
o
p
n
n
f
o
f the fa cto rs
r oo t o
"
two
or
more
factors
is the
.
E x er c i s e 1 38
Fi n d t h e i n di c a t e d
2
5
WW
.
9
11
the
.
C/2 h
V
4 9a
3
8
3
7
t he
.
.
fo ll o w i n g
{ 7 1 6m
W813?
.
10
.
.
12
.
B y t he
by
6
,
.
roo t o f e a c h o f
sa m e
p r i n c i pl e ,
lvi n g t h e
fo ll o w i n g :
re so
a n y r oo t o f a n um b e r
numb e r
i nt o
it s pr i m e
5
-
“
m ay b e
fa c t o rs
.
I
fo u n d
Ob se r v e
E
2 52
li ke
In
m a nn er ,
L E M E N TARY ALG E BRA
so
lve
3
15
O b ser v e
2 1 95 2
16
4 7 54 8 7 2
l
a so
,
60
So lv e t h e
17
fo ll o w i n g
18
.
21
(x
2
.
2 —
—
+ x 2) (x
V
.
2—
—
x
6) ( x
4x + 3 )
ROO T OF A FRA C TI O N
312
.
Fr om
a
t h e l aw ,
b
mn
a
b
The
f
nth root o
d ivided by the
a
mn
fra ction
f
nth r oot o
we h a ve
m
m
i s the
nth r oot o
f
the den omina tor
the
n umer a tor
.
E x e r c i s e 1 39
G iv e t h e
313
ber ;
.
fo ll o w i ng i n di c a t e d r o o t s
A ro ot is c all e d an odd roo t if it s i n d ex is a n odd num
,
an e
ve n roo t , if it s i n d ex is
an e
v e n n u m b er
.
N U M B ER OF ROO T S
314
—
.
8) X
Si n c e 8 X 8
—
8)
64 , t h e
s q u a r e r oo t o f
64 , t h e s q u a re
r oo t o f
64 is
64 is 8 ,
l
a so
8
.
and s nc e
i
E
2 54
L E M E N TARY ALG E BRA
T o i n di c at e t h a t a ro o t is po s it iv e o r n e g a t iv e
s ig n r e a d p l u s or min u s is g e n e r a ll y u s e d :
,
,
x/E
‘
321
i
a
V
2
Si n c e
.
i n v o lv e d
,
(V
C
«8 1
?
i
x
s am e s
sa me s ign a s
ha ve the
Z
Z
/
,
53
5
r
h
h
a
o
w
e
s
v
e
t
e
p
odd
odd r oots
er
al u
‘
S
S
-
x
V st
3
,
s
fl
ig n
V
a,
t he
as
n u mber
the
Th e prin c ipa l ro o t o f a n um b e r is t h e
t h e sam e Sig n a s t h e n u m b er i t se lf
3 22
h as
dou b l e
th e
,
u mb e r
Thu s ,
.
— 2 4 3 65
rea
.
n
;
l
—
3b
wh ic h
r oo t
.
T h e pr i n c i p a l
r oo t o f 1 2 5 is 5 ;
s
q u ar e
of
r oo
t
4 9 is 7 ,
of
— 1 2 5 is — 5
—7
no t
Th e p r i n c i pa l
.
c ub e
.
T O F I N D T HE REAL ROO T S OF M O N O M I AL S
323
.
R ul e
divide the
Give
f
r oots o
.
r equ ir ed
Fin d the
f
ea c h
the
s ig n
expone n t o
od d r oots
m
r
i
v
u
b
e
s
s
i
t
e
n
o
p
f
the
r oot o
l etter by the i ndex
f
the
o
s ig n
the
i tself,
n u mber
the dou bl e
f
o
ffi
co e
ci en t,
and
root
.
and
i
v
g e
ev e n
.
E xer c i s e 1 40
G iv e t h e
fo ll o w i n g
—
1
.
5
.
9
325
15
.
W
0
8 12
2
.
6
12
10
ro
V
ot s :
5
m
.
56;
.
SQUARE ROO T OF A P O LY N O M I AL
3 24
.
A s we h a v e
of a l l
po l yn o m i a l
We sh a ll
ol
o
i
l
n
m
a
p y
roo t
l e ar n e d
t ri n o m i a l
sq
u a re s m ay
,
Si n c e
the
s q ua re
—
—
a
b
the
s q u are r
so m e
.
h ow t o
l
1 93 ,
o o t of
b e d e t e rm i ne d b y i n s pe c t i o n
u a re by t he
(
,
s q u a r es , a n d
n ow sho w
sq
and
ex t r a c t
u se o f
)
2 —
2
a
*
the
t he
s q u are r o o t o f a n y
f o ll o w i n g fo rm ul a
2
—
—
+ 2 ab i b
t he
s q ua re
ro o t
of
t h e t ri
E VOL UTIO N
i l s q ua re is a + b:
t i t y w i t h it s s q ua r e ro o t
C o m pari n g
no m a
1
The fi rs t ter m
.
a rra n ged
f the
o
the
f
o
r
o
w
e
p
o b se r
we
,
root
25 5
a
2
2
—
—
—
—
l 2 a h l b in t h i s i d e n
ve :
is the s quare
the fir s t term
.
If the s qua r e of the fi r s t ter m of the
2
the p ower , the r ema in d er i s 2 a b + b
2
f
r oot o
r oo t
.
is
s u btr a c te d
fr om
.
firs t
Th e
t erm
3
.
te r m
of
th e
The
f
o
t erm
roo
t
of
t he
an d
th e
secon d
the
re
is t h e pr o d uc t
i
r e m a n d er
se
ter m
-
c o n d t er m
f
the
o
ma in der by 2 a
r oot
t he
firs t
by d ivi d in g the
fi rs t
Th ere for e
.
is
fou nd
tw i c e
of
,
.
2 a b+ b = ( 2 a + b) b
2
4
the
.
su m
If we mu l tipl y the
res u l t
fr om 2a b+ b
2
,
f
o
2a
a nd
ma in der i s 0
the
re
b by b
a nd s u btr a c t
.
d m e m b er o f t h i s f o rm u l a re pr e se nt s t h e sq u a r e
o f a ny b i n o m i a l ; b u t s i n c e t h e t e r m s o f a n y p ol yn o m i a l m ay
2
2—
b e g r o u p e d s o a s t o f o r m a b i n o m i a l a l 2 a b+ b m a y a l s o
r e p r es e n t t h e s q u are o f a n y po l yn o m i a l
I f t h e ro o t c o n t a i n s t hr e e t e r m s a r e pr esen ts th e s qu ar e o f a bi n o m i a l
a n d 2a b r e p r e s e n t s t w i c e t h e p ro d u c t o f a bi n o m i a l b y a m o n o m i a l ; if
t h e r o o t c o n t a i ns f o ur t e r m s a r e pr e s e n t s t h e s q u ar e o f a tri no mi a l
a nd 2 a b r e pres e n t s t w i c e t h e p r o d uc t o f a tri no m ial b y a m o n o m i a l
The
s ec on
-
,
.
2
,
,
2
,
,
.
32 5
.
T he
i g
e xt r a c t n
e xa m p e
fo ll ow i n g
t h e s quare
r
o ot o f a
9a + l 2 a
3
1 2a
3
l 2a
3
6
6a
s
2x
2
ill ust ra t e s t h e
t r i n o m i a l s q ua r e
l
2
x
x
x
r
ss
o
c
e
p
of
.
+ 4x
4
3a
3
2x
2
2
2
4
+ 4 36
fi rs t t er m o f t h e r o o t is 3 a t h e s q uar e r oo t o f 9a w h i c h w e p l a c e
a t t h e r ig h t o f t h e t r i n o m i a l s q ua r e
3
S ubt r a c tin g t h e s qua r e o f 3a fr o m t h e tr in o m i a l t h e r e r e m a i ns a p a r t
2
t h a t is r e pre s e n t ed in t h e fo r m ul a b y 2ab + b
D i v i d i n g t h e fi r s t t erm o f t h e r e m a i n d e r b y 6a w e o b t a i n t h e
s e co n d t e r m o f t h e r o o t w h i c h is 2x
3
The
6
,
,
.
,
.
3
,
,
E
256
M ulti pl yi n g
res u
F
lt fr o m
ro m
pr od u
3a
3
of
3
x
2
+ 4x
3a
an d
3
l 2x is t h e
s
b y 2x
t h er e
4
,
tr i no m i a l
2
- -
2
—
6a i 2x
3
1 2a
the
ct
LE M E N TARY ALG E BRA
2x
the
2
,
s
+ 1 2a
x
2
+ 4x
s ub r a
quar e
of
2x
t
th e
qu ar e
of
t her e
is
b y 2a
3a
no
tw i ce t he
rem a in d e r
3
,
.
and
2a + b
ar e
~
com p l e te
di v is or
c o m p ar e
an d
th e
.
r e pr es e n ed
an d
s
and
,
s ub r a
.
the
2
q ua re r o o t o f t h e t r i n o m i a l
,
3
t c t ed
we h av e
t h i s w o r k t h e n um b e r s
r es pe c ti v e l y t h e pa r ti a l d ivi s or
C h e ck : C a l c ul a te
In
i
n o r e m a n d er
is
t c ti n g
and
2
c a ll e d
.
th e
r e su
lt
w it h
9a
6
4
We
.
v e t h a t in t h e e xt r a c t i o n o f t h e s q u a r e
r o o t o f a po l yn o m i a l su btr a cti on is a n es sen ti a l pr ocess ; t h a t is
t h e p r o c es s c o n s i s t s in t h e s ub t r a c t io n f r o m t h e po l yn o m i a l
Th e
o f t h e p a r t s o f w h i c h t h e pol yn o m i a l is c o m p o se d
fi r st p art s ub t r a c t e d is t h e sq u a r e o f t h e fi rst t erm o f t h e r oo t
whi ch t he
a n d t h e se c o n d p a rt s u b t ra c t e d is a pr od u c t
r e ma i n d e r is k n o w n t o c o n t a i n
326
.
o b se r
,
.
,
,
.
3 27
a pp
li e s
to
m o r e t h a n t w o t e rm s
i
r o o t co n t a n s
If
m et ho d
sam e
The
.
po l yn o m i a l w h o se
a ny
.
t c o n ta i n s 3 t e r m s t h e s ub t r a c ti o n o f t h e s quar e o f t h e
fi rs t t e rm of t h e ro o t w hic h is a bi no mi a l is c o m p le t ed w it h t h e s e c o n d
I f t h e r oo t c o n ta i ns 4 t erm s t h e s ub tr a c ti o n o f th e
s ubt r a c ti o n
is c o m p l e t e d
s q u ar e o f th e fi r s t t er m o f t h e r o o t whi c h i s a tri no mi a l
w it h t h e t h i r d s ubt r a c ti o n ; a n d s o o n
the
r oo
,
,
,
.
,
,
,
.
first pa rt i a l divi so r is t w i c e a m o no m i a l ;
a b i n o m i al ; t h e t h i r d t w i c e a t ri no m i a l
Th e
t wi c e
1 0a
2
,
25a
4
2 5a
4
4 0a
3
x
l
- -
1 0a
“
4 6a x
’
2
2
2
3
24 a x
4
9x
5a
2
4a x
2
3x
3x
2
24 a x
s
,
x
3
t he
d
4d x
8d x
We
secon
.
1 6a
2
t he
fin d t h e
fir s t
c
a nd s e o nd
t er m s
q u a r e r o o t o f a t r i n o m i a l s q ua r e
M ulti plyi g
n
the
fi t te r m
rs
of
of
the
4
9x
ro o
t
as
if
w e w e re
g etti n g
.
th e
ro o
t ( 5a
,
2
— 4ax
) by
,
2 , we ha v e
L E M E N TARY ALG E BRA
E
2 58
9
3 —-
.
4 x l 4 0x
4
64 a
4x
5
4
—
—
1 6x l 2 5 x
6
7
3
1 92a + 64 a + 1 2 0 a + 2 5
2
2
6
2 5 x + 9x + 1 + l 0 x
9a
6a b
4 7a b
3
4
6x
3
2
1 6a b
2
64 b
3
4
4
5
—
2x + 5x
x
6
8x + 4
1 98 a
4 —-
2
3 6b + 2 5 6
4
1 6x
2 ——
l 7 6x y f 60 xy + 4 8 x y l 2 5 y
4
—
1 03 a
8x + 1 6
2
l 4 2a
x
—
305 6
—
2 —-
3
3
l 0x
4x + l 7 x + l 0 x
40 a b
4
60 a
S6C
4a
4
4a
“
b
2a
x
a
'
27
4
b
4
“
x
‘
3
2
b
b
3
2x
2a
+
2
c
55
2
x
5
4
3
4x
1 0a
2
2
x
af
2a
b
b
2
26
c
7
(
2
3
+
c
_
2d
+
1 6x + 1
+x
9
16
2
2
5x +
25
71
4
4 0 bc
1 6a c
+ 2a x + 2 +
2
64 x
7 0 a x + 4 9x
6
3a
3x
4
48a x
x
2
-
2
3
9 ——
_
+
i9
5
a
2 2 __ —
—
a
x
+
+x
2
4{
a
.
+
2
s
2x
-
_
3
4
1 1a
x
i’
2
2
25a
c
x
x
3 65 6 + 695 y
1 2x
1 6b
3
4
-
2
2 2
3
2
4
- -
3
2 0a b
l 6o
4
2
3
4
—
—
x l 4 9a
2
2
6
3
3
-
25 b
4a
2
25a
3
4
5 6bx
E
VOL UTIO N
259
SQUARE ROO T OF NUM B ER S
329
o ne ,
1
2
u a re s of t h e sm a ll est a nd l ar g est
a n d t h r ee figu r es a re a s f o llo w s :
The
.
t wo ,
10
2
1
:
n
sq
1 00 =
2
1 00
:
u m b er s
1 00 00
80 0 1
01
The
n
um ber
s q u a re r o o t o f
It
two
fol l ows
fig u res
at
the
th e
n
the
u m b er a t th e
tha t if
ea c h,
of
l e ft
a n y s qu a r e
begin ni n g
sa me a s
of
ig n
r ig h t
in
s
is
id e nt it y is
t he
.
s epara ted
the
a t u n its ,
n u mber o
ea c h
i n to period s of
n u m ber of fi gu res i n
r
i
e
od s
p
f
Wh e n t h e nu m b er o f fig ur es in t h e s q uar e is od d th e l e ft h a nd p e r i o d
i s I n c o m p l e t e c o n t a i ni n g o n l y o n e fig ure
I f a re pres e nt s t h e t e ns a n d b t h e u n i t s in t h e s q ua r e r oot
a + b r e pr esen t s t he
o f a ny s q u a re of t h r e e o r f o u r fig u r es
2—
2
—
Then
s q ua r e r oo t
a n d a l 2 a b l b r e pr es e n t s t h e s q u a r e
t h e fo r m u l a e x pr e sse s t h i s p r i n c i pl e :
A n y s qua r e of three or fou r fi g ur es i s equa l to the s qu a re
of the ten s of i ts s qu a re r oot p l u s twice the p r odu ct of the ten s
by the u ni ts p l u s the squa re of the u n its
the
r oot
is the
the
.
-
,
.
,
,
-
-
.
,
,
,
Fo r
.
exam p e ,
l
57
2
( 50
a
X7) + 7
2
2
3 249
49 00
2 a = 1 40
5 76
2a + b = 1 4 4
5 76
S e pa r a ti n g t h e n um b e r i n to p e r i o d s o f t w o fi g ur e s e a c h w e fi n d t h a t
t h e r oo t c o n t a i n s t w o figu r es un it s a n d t e ns
T h e s q ua r e o f t h e n u mber o f t e n s in t h e roo t is f o un d w h o ll y in 5 4
T h e l a r g es t s q ua r e in 5 4 is 49 w h o s e s q uare ro o t is 7
H e n c e t h er e a r e
n o t m o r e t h a n 7 t e n s i n t h e roo t
Si nc e t h er e are 7 t en s in t he r o o t a = 70 an d a 2 = 4 900 S ub t r a c ti n g
2
a
w h i c h in t h i s e xa m p l e is t h e s q ua r e o f 7 0 o r 4 900 f r o m t h e n u m b er
w e h a v e a r e m a i n d er o f 5 7 6
Th i s r e m ai nd er is t h e prod uc t o f tw o f a c to rs r e pr ese n t e d b y
T h e pa rt i a l d i v i s o r 2a is t w i c e 7 0 o r 1 40
,
,
.
.
,
.
,
.
o
,
,
,
.
,
,
.
,
,
,
,
.
,
E
2 60
L E M E N TARY ALG E BRA
D i v i d i n g 5 7 6 b y 1 40 , t h e q uo ti en t is 4 , w h i c h is pr o b a bly t h e u n it s
fig ure o f t h e r oo t The c o m pl e t e d i v i s or , 2a + b, is 1 4 4
u ltip l yin g 1 4 4 b y 4 , a n d s ubt r a c ti n g t h e pro d uc t f r o m 5 7 6, t h e r e is
’
.
.
M
i
n o r e m a n d er
H e n c e , 7 0 + 4,
.
or
7 4 is t h e
r oo
t
.
We m ay a bb re v i a t e a n d s i m plify t h e w o rk s om e wh a t
t h e c i ph ers a n d c o n d e n s i n g t h e o t h e r p a r t s as f o ll o w s
by
om
itti n g
,
9 6 0 4 98
A t fi r s t w e wr ite o n ly
t he
1 4, 8 ,
i
r e m a n d e r , excl usi v e o
If
,
on
the
f
m u lti p l yi n g
a ny
a nd
18
of
r i g ht- ha nd
t h e p a r ti a l d i v i sors ,
fi g ur e
a nd
d iv id e
.
c o m pl e te d iv i sor b y t h e l as t figur e o f t h e r o o t
c
t h e pro d u t is l a r g e r t h a n th e r e m a i n d er , t h e
t oo l ar g e a n d m us t b e d i m i n i sh ed b y 1
l as t fi gur e
of
the
roo
t
,
is
.
A ft er
d e t e r m ini n g t h e
p ar ti a l d i v
isor
to
form
it s fig ur e o f t h e
c o m pl e t e d i v iso r
’
un
the
roo
t
,
we
a nn e x
it
t o th e
.
E x e rc i s e 1 42
Fi n d
the
sq
u ar e
r oo t o f
1
.
2304
2
5
.
4 624
6
fo ll o w i n g
.
34 8 1
3
.
5 1 84
4
.
4761
.
7 3 96
7
.
57 76
8
.
7 5 69
m et h o d
li es t o a ny n u m b er wh o se r oo t
It is o nl y n e c e s s a ry
is expre sse d b y m o r e t h a n t wo fig u r es
t o c o n s id e r a l l t h e r o o t a l re a d y fou n d as t e ns
3 30
.
The
s am e
the
a pp
.
.
1
5 7 1 5 36 7 5 6
44 95 7 0 2 5 670 5
1 3 40 5
Wh en t h e
pa rt i a l d i v i s o r is
c o n t a i ne d
l i
o f t h e r igh t h a n d fig ur e a nn e x a c i ph e r t o t h e r o o t a n d al so to t h e d i v i sor
In th e se c on d exa m pl e
a n d a nn e x t h e n e x t pe r i od t o th e d i v i d e n d
a b ov e 1 3 4 is n o t c o n t a i n ed in 67
not
in t h e d i v i d e n d ,
e xc us v e
-
,
,
.
,
.
E
2 62
L E M E N TARY ALG E B RA
S epa ra te the decima l i n to peri ods
ea ch, begin n in g a t ten ths
The proces s i s the s a me a s w ith whol e n umber s
333
.
Rul e
.
f
o
two
figu res
.
.
F rom the right of the root poin t ofi
a s there a r e peri ods of deci ma l p l a ces
ma n y de cima l p l a ces
as
.
fi
m
E a c h per i od o f a d ec i a l m us t h a v e t wo g ur es
I f we Wish t h e
s q ua r e r o o t o f a d e c i m a l t o 2 p l a c es , w e s h o ul d h av e 4 d e c im a l p l a c es in
t h e n um b e r ; if w e w i s h t o c a rry t h e w o r k to 3 p l a c es , w e s h o ul d h a v e
.
-
6 de
c i m a l pl a c es
p l a c e s m ay b e
in th e
i n c r eas ed
n um be r ; a n d so o n
by
i g c i ph ers
a nn e x n
T he
.
b
n um er o f
d ec i m a l
.
E x e rc i s e 1 44
Fi n d
the
o i m at e
a ppr x
sq
u a re r o o t o f
the
foll ow i n g
1
.
2
.
.
4
3
.
.
0 36
5
.
6
.
.
8
7
.
.
0 64
T O F I N D T HE SQUARE ROO T OF A C O M M O N FRAC TI O N
3 34
.
R ul e
f
s qua r e root o
If both terms
.
If ei ther term is
a nd
fi
nd
ter m
ea c h
f
ra
c
t
i
on
f
o
a
ua re
f
r oot o
the decima l
r oo t s o f
2
3
.
the
.
'
the
.
E x e rc i s e 1 4 5
sq
fi
nd
s epa r a te l y
n ot a s qu a re , reduce
the s qu a re
a r e s qua r es ,
3
t he
fo ll o w i n g
3.
fra ction to a
deci ma l ,
C H A P T E R XX I I
E XP O NE NTS
RAD I CAL S
.
E XP O NE NT S
33 5
.
Fun d am e nta l
foll owi n g l a w s
1
.
3
.
a
La w s
h a v e b ee n
i
rest ri c t io ns
l y divi s i b l e by
and
gre a t e r t h a n
n
.
a
m
.
t h at m
a re
i i
d
2
in l aw 2 t ha t m is
e xac t
est a bl s h e
i
c e r t a n res t r c t o n s
m
5
The
Un d e r
.
po si t ive i n te g e rs ;
a n d in l a w 4 t h a t m is
n
n,
a re
.
fra c t i o n a l o r ne g a t iv e n u m
be rs T he o l d d e fi n i t i o n t h at a n expo ne n t i n di c at e s h o w
m an y t i m e s a n u m be r is t a ke n as a fa c to r c a n h a v e n o m e a n i ng
We n o w e xt e nd t h e no t i o n o f ex po ne nt
fo r s u c h ex po n e nts
t o giv e m ea n i n g s t o t h ese ne w fo rm s o f e xpo n e nt b ut it is
c o n v e ni e n t t o d o t h i s in s u c h w a y t h a t the fi ve l a ws a bov e
s ha ll hold for the n ew for ms o f e xpo ne nt
33 6 B ut m
.
a nd n
m ig h t be 0 ,
,
.
.
,
.
3 37
.
D e finition of
°
a
.
In l aw 2 ,
if m b ec o mes
we h av e :
B ut ,
°
a
The re fore
.
a
,
°=
al s
o
a
eq
u al
to
n,
"
5
7
4 1’
5
1,
( no t its elf 0 ) with a n expon ent 0 equa l s 1
%
f
D
e
fi
n
i
t
i
n
o
a
o
In l a w 4 if m is n o t a m u l t i pl e o f n
3 38
B y t h e l aw o f ex po n e n t s fo r
a fr ac t i o n a l e xpo n e n t a r i se s
e v o lu t io n w e h a v e
A n y n umber
.
.
.
,
.
g
x/d ,
a
i
r:
a
ir
m
4
a
3
,
a nd
ge ne rall y
,
aW =
Wa
m
E
2 64
L E M E N TARY ALG E BRA
A posi tiv e fra ction al expon en t in dica tes a r oot of a p ower of
The d en omin a tor i s the i ndex of the r oot a n d the n um
the ba s e
.
i s the
c r a tor
f
exp on en t o
the power
.
D e finitio n of
In l aw 2 if n 1 8 g rea t e r t h a n m
t h e q u o t i e n t h as a n e g a t iv e e x po n e n t
Si n ce l a w 1 is t o h old fo r t h e n e w fo r m s o f e x po n e nt
a =a
w e h ave :
a
a
a
T h ere fo re
1
33 9
.
,
.
,
”
"
n
"
‘
"
,
B y t he
ax o m ,
divi s i o n
i
A n y n u m ber with
f the n u mber
o
T h us
2a
,
“
1
wi th
=
2
2
a
—
a
n
a n ega tiv e ex pon en t
i s equa l to the reci p r oca l
n u mer i c a l l y equ a l posi tiv e exponen t
a
—2
15,
l
"
‘
.
G
a
2
z
2
-
2
a
0
—2 2 —3
a h
x y
3
27
2
3
.
by
"
3
RA D I CAL S
3 40
A ra d ic al is a n I n di c at e d r o o t of a n u m be r
.
i n di c at e d b y t h e r a di c a l
Th us
1
x/d l x 5
s
ig n
,
W
- -
,
a r e a l l r oo t s
,
by
or
,
a
i
,
R o o ts a r e
fr ac t i o n a l expo n en t s
,
and
VE
T
.
r ad i c an d is t h e n u m b e r w ho s e I n di c a t e d
f o u n d T h u s t h e ra di c a n d O f x/fi is 1 5 ; o f
The
V
In this
the
u n de rs tood
tha t
341
.
.
cha p ter , excep t
in
§
3 7 5 , i t is to be
m ea n s the pos itiv e
s ig n
The
i n d ex o f t h e
Thus
or d e r
r oo t
,
is t o be
and
.
—
—
a
x , it is a
x
r oo t
it is 9a ,
“
of
.
or
d e gr e e ,
s qu a r e
of a ra
f
root o
the
r a d i ca n d
.
di c al is d e t erm
i n e d by t h e
.
ond ord e r , or s e c on d d e g r e e ;
WE is a ra di c al o f t he th ird ord eir o r th ir di d eg re e
W h at is t h e d e g re e o f t h e ro ot a ? O f x ?
3 42 A r ation al n u mb e r is a po s i t iv e o r n e g at ive i n t e g e r
o r a f r a c t i o n w h o s e t e r m s a re i nt e g e rs
,
is
of
t he
sec
.
.
.
.
E
2 66
34 7
.
Surd s
illu st ra t e
a
L E M E N TARY ALG E BRA
rise in
l ulat i ng
ca c
t he
as
,
fo llo wi n g 3 exa m pl es
.
E x e rcis e 1 4 6
1
s
C al c ul a t e t h e
id es a re 1
Le tti n g x d en o t e
x
2
of a sq
di a go n a l
un it l ong
.
2
=2
th e
C a l c ul at e t h e
.
L e tti n g x
a
d en o t e t h e
l e n gt h
of
t h e d i a go n a l , w e h av e
whic h is
x
wh i ch
gi v e s
a
a s ur d
l en gt h
of
t he
2
or x
a
x
a
a
§
2
m i xed
ltit ud e
a
.
of
.
t rue w e igh t b e d e no ted b y w
Wh en t h e bo d y is p la c ed in o n e pan sup
po s e 1 0 lb ih t h e o t h e r p a n j us t b a l a n c e it
B Y t h e pr i n c i p l e o f t h e l e v e r
.
.
.
Wh e n t h e bo d y is p l a c ed in t h e o th er
1 2x
,
D i v i d in g
w e h av e
( 1) by
p an
wy
ll
w = 1 20 ,
2
We
m ay
l
a so
fi nd t h e
ra
ti o
of
or
w
a nd
d iv i d in g ( 1 ) b y
o
s uppo se
12
)
V 1 20
q
lb j us t b a l an c e
.
( 2)
an d
t h e un e ua l
wy
( 1)
.
10
” 011
wx
l oy
wx
Th e n
w e h av e
,
Le t t h e
.
c l e a ri n g
,
.
ar ms ,
b y wr iti n g
( 2 ) t h us
( 3)
1 2x
bt a in i n g
f
M ulti plyin g th ro ugh by
,
y
E x t ra c ti n g
s
.
of
.
,
it
,
a
2
s ur d
D et e rm i ne t h e t r u e w e ig ht
a bod y b y m ea ns o f a b a l a n c e
u ne q u a l a rm s x a n d y
3
.
l t i t ud e o f a n e q uil at e ra l t ri a ngl e of s id e
a
2
w h ose
.
or x
,
u are
q ua re roo t s
,
wh i c h
l
a so
is
a s ur d
.
,
RA D I C ALS
S I MPLIF I C ATI O N
2 67
RADI C ALS
or
e xa m p es
j us t giv e n s h o w t h e n e e d for su r d s in
c a l c u l a t i n g t h at t h ey a r i s e j u s t as o t h e r n u m b e rs ar i se in
h
a
e
a
e
b
re
t
h
e
o
l
lvi
n
g
n
t
t
r
t
o
g
ar
d
e
d
a
d
as n u mbers
r
b
e
m
so
y
p
348
.
l
Th e
,
-
,
.
R e d uc tio n of r ad i cal s is t h e
form wi t h o ut c h a n gi n g t h e i r v a l ue
34 9
.
pro c es s
o f c h an
gi n g t h e i r
.
to g e t t hem
im plifi e d
fo rm fo r c a l c ul a t i n g
R a dic a l s
are s
i nt o m o s t
c on
v e n i en t
.
A ra di c a l is not in its s im pl est fo rm fo r c a l c ul at in g
1 If the ra d ica nd ha s a fac to r tha t is a power of the
‘
.
d eno ted by the ind ex
2
a ny
3
If the
.
f
o
is i ts elf
rad ica nd
factor of the index of
If there i s
.
rad ica l
in
a ny
A
ra
ra d ica l
the
;
power
the ra d ica l ;
a
de no mina tor
a
denominator
f the deg ree denoted
.
by
o
u nde r
the
ra dica l
sign , or
a
.
di c a l m a y b e s i m plifi e d wh en
fa c t o r w h o s e i n di c a t e d roo t c a n b e fo u n d
3 50
de gree
t he
ra
di c a n d h a s
a
.
E y t h e l aw
In
al l
a re c o n s
351
.
of
wo rk in
id e re d
Rul e
V
V
i m plifyi n g s urd s
Take
.
write i t a s
s
,
7
4 5
,
:
V
on
4
V
ly t he
5 = 2 V 5,
r
i
n
c
i
a
p
p l
l
a so
r oots
.
whos e ind ica te d
a nd
§
3l l
20 =
f
ou t o
r oot ca n
be
the
r a d ica nd
fou nd
the l a r ges t
fa ctor
Fi nd this ind ica ted
.
fiic ien t of the other fa ctor
a coe
root
.
E x e rc i s e 1 47
Si m plify t h e
fo ll o w i n g
2
.
s ur
ds
V I2 5
8
—
11
.
{ VS 1
°
.
x/
W
E
2 68
Th e
by t h e
r oo t o f
c oe
l f a c t o r w h e n fo u nd is m ul t i pli e d
o f t he m i xe d s u r d
t he
ffi c ie nt
LE M E N TARY ALG E BRA
i
ra t o n a
27
.
18
.
22
.
2m
15
T
W
19
.
3
63
23
.
26
.
28
.
m
7
272
( c + a x/
.
1 2 5 47
8
a
.
is
(H u
.
,
.
14
25
,
m
.
“
me
b
16
sa
W
20
.
24
.
5a
m
W
W
F r o m t h e pr i n c i pl e of 34 6 m a n y roo t s c a n b e
c ul a t e d a ppr o x i m a t e l y f ro m a fe w giv e n v a lu e s
F o r e x a m pl e giv e n :
W
V3
a nd V 3
t o c a l c ul a t e o t h e r r o o ts a ppr o x i m a t e l y
352
.
c al
.
,
:
,
a s x/ 1 2 8 = 8
V
=
2 8 1 4l 4 =
a nd
.
et c
.
E x e r c i s e 1 48
F ro m
and
I
.
3
V
the
of
giv e n v a lu es
t he
sq
u a re
a nd c
ub e ro o t s
of
2
l ul a t e :
ca c
S
14
{ VI
.
ra di c a n d is i t se lf
d e n ot e d b y a fa ctor of t h e i n d ex o f
s h o w n in t h e fo llo w i n g exa m p l e :
353
.
W h en
3
V
V 25
V 7a b c
2 3
t he
75
(
—
2
5
V
b C V Ttl C
'
.
,
a nd
4
a
p o we r
the
t he
d e gr e e
r a di c a l pr o c e e d a s
,
\/4 9a b
2
of
4
c
6
2 z 4 6
/
\ V 7 a bc
E
LE M E N TARY ALG E HR A
22
25
V
.
fi g
.
1
26
.
V
24
6)
?
27
.
E x e rc i s e 1 51
wi t h t h e
2
6
To
A
ra di c a l sig n
(4 a )
.
.
x
a nd s
2
g f
y
RED U C E A M IXED NUM B ER
m ix e d
i m plify
TO
AN
r d m a y b e r e d uc e d t o
re v e rs i n g t h e pr o c e ss o f s i m plifyi n g s u r d s
3 57
.
:
su
E N TI RE S URD
r
a n e nti e
.
Thus
r d by
su
,
E x e r c i s e 1 52
E xpr ess t he
1
5
9
13
follo wi n g
2 x7 7
.
5V
.
2
§
6
.
i
a s e nt r e s
2
7
V
§
3
7
.
.
.
urd s
2a x/5 a
22
4
0 5 25
8
12
.
7
4
5 3
-
.
14
é V
l §xV
l
.
.
18
.
( a + x)
a
Sa
z
x
3
9x y
2 é a x/ 8 a x
—
—
.
15
.
-
‘
2
1
5
2
5 ax
2
l
é
8 be
3
a
x+4
x+4
—
4
x
x—
4
ADD ITI O N AN D SUBT RA C TI O N OF SUR D S
3 58 S urd s
dd e d o r
t rac t i n g t h e i r c o e ffi c i e n t s
.
a re
a
.
s u b t ra c te
d
by
a
ddi n g
or
sub
RA D I C ALS
r d s a re s u r d s whi c h in t he i r s i m pl est fo r m
of t h e sam e d e g re e a nd ha v e t h e s am e ra di c a n d s u c h as
35 9
a re
S imil ar
27 1
.
su
2 V 5 , 4 V 5,
a
-
5;
a
V
,
—
b
az,
3 x7 7 , 5 x
7 7 , 9 x7 7 ,
Tw o
s
or
m o re
u b t ra c t i o n
on
su r
ds
be
c an
et c
u nit e d i nt o
a r e s i m il ar
w h e n t h ey
ly
a nd
,
.
W
I0 W
W
—
W
W
— 3
W
is
as
—3
a
ddit i o n
sh ow n
or
he re :
1 25
—
—
5
by
on e
7
= 6 \ 2x2
E xe r c i s e 1 5 3
S i m plify t h e
1
2
.
3
4
.
.
45 + 2 V
—
.
4V
.
12
.
—
48 4
5 x/ 1
ev
es
—
I28
s
.
.
—
27 3
4
V
a
3
.
m
_
—3
15 2
—4
2 V 1 6x
2
V
5
3 V 1T2 + 6 \/45
/375 7 2 0
e
t c
.
9
10
4V
7 x/T75
.
e
7
follo wi n g :
—
iv
'
V
M
—
V
54 x
3
— —
—
l V
2 5 0x
3
-
20 + 2 V 1 2
4
M
+2
W
_
2
W
—
g o+ s v r — ev
m
g rem v soe
—5
2 V 48 6
V 54 + 4 m
3 V 1 62
11
—
.
w ere
—a
_
5 x/ l 6 +
-
T O RE D UC E SURD S T O T HE SA ME ORD ER
3 60
by
.
S ur d s
express ing
ing the
common
ra
o f d ifi er e n t
the
fra ctiona l
r a d i ca l s
.
as
ex pone n ts
denomin a tor ,
di ca l sign s
or d e r s are c h a n g e d t o t h e sa m e o r d er
T h us
and
fra ction a l
to
expon en ts , a nd r edu c
equ i va l en t
the n
,
a nd
fra ction s
expr ess in g
the
ha vin g
su r d s
a
wi th
L E M E N TARY AL G E BRA
E
272
T he
l o w e st
Th en
2 = x/ 8
s
o n d e no m i n at o r o f
%
2
,
V
Si n c e
c o mm
2
5
V
and
V8
{75
{7 2
3
3
V
,
the
e x po n e n t s
is
and
.
9, we
at o n c e s ay
c an
th at
T h i s pr i n c i pl e e n a b l es u s t o co m pare r a di c a l s o f di ffe re nt
o rd e rs a s t o r el a tiv e m a gn i tu de
T h e s i g n s o f i n e qua l ity a r e
T h e s ig n
and
m e a n s gr ea ter
.
,
m ean s l es s tha n
tha n ;
.
E x e rc i s e 1 54
C o m pa r e t h e
1
5
.
and
V
7
2
5
7
.
f
r
ra
a
s
o
p
foll o w i n g
i
5
5
.
3
3
.
di c a l s
an d
V
and
\/
5
2
3
.
6
6
.
A rra n g e In o rde r o f v a l ue V 7 V
,
,
6,
2V 3
and
2V 5
an d
V
and
2
.
MULTI PLI C ATI O N OF S UR D S
T h e pr o d u c t o f t w o
is f o u n d b y l a w 5 , 3 3 5
3 61
.
or
m o re
s ur
ds of
t he
s am e o r
der
.
Fo r fr a c tion a l
ex pon en ts
N ot i c e t h at t hi s
sa m e o r
d er
.
Th us
a pp
t h i s l a w t a ke s t h e
li e s
on
ly
w he n t h e
o rm
f
sur
ds
a re o f
t he
,
E xer c i s e 1 55
M ul t i pl y a s i n di c a t e d
1
4 x/3 3 x/5
-
.
2
5
7
.
8
2 x/ 7 s x/ 7
o
.
4V 5 5V 5
~
.
S V E 2 x/8
-
.
3
2 x/5 3 x/1 5
-
.
E
2 74
L E M E N TARY ALG E BRA
E xer ci s e 1 57
M ul t i pl y t h e follo wi ng
1
2
.
3
.
4
V
3
.
by 4 + 2 V
d
by
—
3 3
d
v
2
1 2 + 3 x/5 b y 4
3a
.
-
3V
d
by 2a + 2 V
—
Ct
5
6
.
—
7
.
8
.
9
—
x/5 b y x/ é
s /é
‘
«5 + n / 7
3
.
W
—
2V
l z x/fi
-
by
3+ 4 \
G by 4 V
—
7 3
V
—
3
V
5
M ul t i pl y by i nspe c t io n :
10
.
12
.
14
.
16
—2
)
(x
—x
.
)
11
.
15
.
17
.
19
.
V 5)
(V
-
i5+
D IVI S I O N OF S URD S
3 64
.
by t he
Th e q uo t i e nt of t wo s u r d s o f t h e s am e o r d e r is
i
n
h
l
e
f
v
o
l
u
t
i
o
a
s
t
a
d
t
fo
r
m
ul
a
n
e
r
i
n
c
i
o
e
s
t
e
p
p
V2
V 5?
fo u n d
RA D I C AL S
h e r e , t h i s pr i n c i pl e
Th us,
A s in m ul t i pli c at i o n ,
s
urd s o f t h e
s a me or der
27 5
so
.
a pp
li es o nl y
to
E x er c i s e 1 5 8
G iv e t hese q uo t i e nt s b y
i ns pec t io n :
6
.
RATI O N AL I Z I N G SURD S
Ra tio n a liz in g is t h e pro c ess of m ul t i pl yi n g a s u r d
O b se rv e t h e
b y a n u m be r t h a t gi v es a r at i o n a l pr od u c t
fo ll o wi n g :
3 65
.
.
7
r ational iz in g f ac tor is t h e fa c t o r b y wh i c h a s urd is
m ul t i pli e d t o giv e a r a t i o n a l pr o du c t
Whe n t he pr O d l l Ct o f t wo s urd s is r a t i o n a l e i t h er s u rd is
t h e r a t i o n a li z i n g f a c t o r o f t h e o t h er
N am e a r a t i o n a li z i n g fa c to r o f e ac h of t he foll o w i n g
s u r d s a n d gi v e t h e p ro du c t s :
Th e
.
,
.
5
.
V
3 66
.
t e rm s
8
A
8
b in o mial
are s
u rd s
.
su
rd is
Thu s
,
a
4+
bi no m i a l
V
5,
V
one
or
—
3 2 , a nd
3
.
4 \/1 6xy
both o
V
f
6+
wh ose
V
7
.
E
2 73
3 67
s ur
d
.
A
b in o m i a l qu ad r ati c s ur d is
s u r d s , a r e O f t h e s econ d or der
or
,
LE M E N TARY ALG E BRA
a
b i no m i a l
s ur
d
.
urd s a r e t w o b i n o m i a l q u a d ra t i c
t h a t di ff e r o n l y in t h e s ig n o f o n e o f t h e t e rm s
For exa m pl e a + x/ b a n d a V b as a l so
3 68
.
Con j ug ate
who se
s
s
u rd s
.
,
V
V 5 and V
7
7+
V
5
o nj ug a t e s u rd s
—
Si n c e c o nj ug a t e s u r d s a re o f fo r m s a + b a n d a b the
u
a
t
r
d
i
s
ra
t
i
a
o
a
n
t
w
o
co
n
e
s
u
s
n
l
u
r
o
d
c
t
o
f
y
j
g
p
He n ce it f o ll o w s t h a t a n y bin omia l qu a dr a tic s u rd m a y be
r a tiona l ized by mu l tip l yi n g i t by i ts conj ug a te
T hu s
a re c
.
,
.
.
‘
,
and
E x erc i s e 1 5 9
N am e
a nd
giv e
i
li z i n g fa c t o r
p r odu c t s :
a r at o n a
the
2
4
a
.
of eac h of
—
2 x/5
.
3 x/é
+ 2 \/b
t he
3
fo ll o w i n g s u rd s
.
6 x
.
s
—
70 3
.
x/6
9
.
E x e r c i s e 1 60
Rat io n ali z e t h e d en o m i n at o rs o f th e foll o wi n g
4
°
3
3
—
v
x/ 2
3 + x/ 2
8
V
—
6
x/i s
0
O
Vm
—
c
V
m
ii
—
m
+
V
v
5
1
6 7
V
—
l
-‘
—
ct
E
2 78
7
10
12
LE M E N TARY AL G EBRA
12 +4 V 5
.
9
15+ 3 V 6
.
9
—
30 6
.
V
20
—
2x + 3 y 2 V 6xy
.
—
h
2a +
2V
.
a
2
+ab
13
APP ROXIM ATE
VAL UE S
or
a
.
2
+ b+ 2 a V b
S URD S
roximate valu e of a surd is fo u n d by ext r a c t
in g t h e i n di c a t e d r oo t t o t h e r e q ui re d d e g re e of a c c u r a c y
It
is fre q u e nt l y n e c e ssa ry t o fi n d t h e v a l ue of a f r a c t io n wi t h a
r a di c a l d e n o m i n a t o r
In s u c h w o rk m u c h l a bo r is s av e d b y fi rst r at io n a li z i n g t h e
divis or o r den omin a tor
Thus
372
Th e
.
app
.
.
.
,
Si m plify
,
3
3V 5
3 2 23 60 7
V5
5
5
of t h e f o ll o w i n g divi s i o n s fi n di ng t h e n u m e r
i c a l v a l l ie c o rre c t t o 5 d e c i m a l p l a c e s h a vi n g giv e n t h at
e ac h
,
,
V
2
V
:
3=
and
,
V
5 = 2 23 60 7
.
.
E xe rc i s e 1 62
5
.
18
I RRATI O N AL E Q UATI O N S I N O N E U N KN O WN
3 73
.
t a i n i ng
An irr atio n al ,
an
irratio n al
ra d i c al e qua tio n is
root of t h e unkn o wn
or
u at io n c o n
num b e r
Thus
an eq
.
,
olv e a n irrat io na l e q u at io n t h e first st ep is t o free t h e
T h i s is do ne b y r ai si n g bo t h m em b e rs
e q u at io n of r a di c a l s
of t h e e q u at io n t o t h e s am e po we r
To
s
.
.
P o we r
equa l
.
Axiom
.
The
s a me
powers
f
o
equ a l
n umbers
a re
RA D I C ALS
T o s o lv e
Sq u a ri n g
2 79
x/2x
,
—
/5x
7
5x
7
x
5
,
9,
2 55 + 1 7
.
u ar i n g in t h es e t wo ex am pl e s ar e si m pl e
e q u a t i o n s a n d are s olv e d a s s u c h
Ra di c a l e q u a ti o n s c o nt a i ni n g m o re th a n on e r a di c a l m a y
h a v e t o be s q u a re d m o re t h an o n c e
Th u s t o so l v e
The
r e su
lt s o f
sq
.
,
.
,
V
S ub t r a c t i n g fl
,
S q u a ri n g ,
a:
5
a
:
5
\/x
5
5
i:
x/a
l 0 x/5 f
—- x
x
Un i t i n g t erm s ,
Dividi n g b y 1 0 ,
S q u ari n g ,
1 0 x/E
30
3
x
9
.
di c a l eq u at io ns it is
s ign sh a ll d e no t e o n l y p r in ci pa l r oots
5
Ve rifyin g
V9 5
37 4
.
W it h
ra
a
gree d t h at t he ra di ca l
.
,
5
5
5
ub st i t u t io n of 9
id ent ity 9 sa t i s fi e s t he
fo r
Si n c e t h e
giv es an
s
,
a:
e quat
in t h e
io n
origi n a l
eq
u at i o n
.
E x erc i s e 1 63
So lv e
1
v e r ify t he follo w i ng
=
E
E
9
/
T
x
a nd
-
.
3
.
5
.
m
x/m
x/
7
z
t
a
=a
w yn
:
.
Vfi
«
t
-
Afi fl
ii
x/E
T
9 x/5
.
V
1
5
2
E
2 80
19
.
37 5
eq
l x/a l
—
/ds
x
.
A
wh i c h ,
t h e posi tive
we
,
x
st atem e nt
u at io n
T h us
- -
L E M E N TARY ALG E BRA
u n d er t he
ass u m pt o n
i
sq
of
th at
u a r e roo t c a nn o t b e s at i s fi e d
s o lvi n g b y t h e us u a l m e t h o d
,
i rr a t i o n a l
s h a ll m e a n
an
.
,
ob t a i n
a:
9
A t t e m pt i n g t o v er ify we h a ve
.
2
w h i c h is
for m
m ay b e in t h e
n ot a n
5
3
i de n tity
.
lli n g t h a t x/d: m ay
b e e i t h e r t h e pos itive o r nega tive roo t a s the con ditio ns of the
n
n
n
n
r
r
r
re
t
a
b
h
s
i
n
r
n
i
i
g
o
ig
v
if
i
o
bl
e
m
e
i
e
a
d
t
s
e
u
,
p
y g we
q
h ave
Se t t i n g
as
id e t h e
a ss u m pt o n a n d r e c a
i
,
,
,
=
t 2
i
3+5
.
O f t h ese po s s i b ili t i e s a s t o s ig n , w e c a n g e t a n id e n t i t y by
—
2
It is w o r t h n o t i n g
us i n g + 2 for V 5 5 a n d
3 fo r x/cv
.
of t h i ng s
ing had b e e n o m i tt e d
In sq u a ri n g a r a di c a l
du ced w h i c h t h e giv en
t hat t h is
w ould
s t at e
no t
h a ve
be e n fo u nd if v e r ify
.
free d of
ra
and
di c a ls
so
u a t i o n a r oo t is som e t i m e s in tro
e q u a t i o n did n o t c o n t a i n
T hu s
eq
,
.
lv e d b y
a:
2
the
an d x
u su al
l ea ds
pr o c e ss ,
,
to
6
Ver ifyi n g fo r 2
3 3 O T h i s c h e c ks
—
Ve rifyi n g for 6
2 T h i s d o es n ot c h e ck
H e n c e , 2 sa ti s fi e s t h e e q uat i o n un d e r t h e a ss um pt i o n t h a t
.
,
.
,
'
i n di c ate s o n l y t h e
v
r
r
i
t
i
s
u
a
t
os
e
e
oo
,
p
q
wh il e 6
do e s
not
C HA P T E R XX III
Q UAD R ATI C
3 77
.
A
qua d r atic e qua tio n is
t h e u n kn ow n n um b er
d e g r e e in
2—
2 0 , 4x
2
F533
q u ad ra t i c
eq
u at i o n s
io n O f
e x a m pl e
Fo r
.
d et erm i n i n g t h e d e g ree o f a n e qu a t i o n it is
t h e e q u a t i o n is fi rs t r e du c e d t o it s s i m pl es t fo r m
t h at
.
ass u m e
doe s
d
t h at
.
on s tan t te rm in a q u a d r at i c e q u a t i o n is t h e
n o t c o n t a i n t h e u n k n o w n n u m ber
Th e
d
5a
In
378
se c o n
,
and
3 6,
t he
e q u at
an
.
15
a re a l l
E Q UATIO N S
c
t erm
.
S o m e q ua d r a ti c e qu a ti o n s c o n ta i n o n ly t h e s quar e O f t h e un k n o w n
n um b e r ; o t h e r s c o n t a i n b o t h t h e s q u a r e a n d t h e fi r s t p o w er o f it
H e n ce
t h ere a r e t w o k i nd s of qu a d ra ti c e qu a ti o n s
.
.
.
379
not c
.
A
r
o nt a i n t h e fi rst
3x
380
.
An
i
c o nt a n s
n u m be r
.
2
:
is
u
a
d
a
t
i
c
e
u
a
t
i
n
q
q
r
u
e
p
o
r
o
w
e
p
of
t he
a n eq
u at i o n
n u m b er
u n kn o w n
— 1 6 4 m2 = 36a
,
1 08 ,
afi e c t e d
d o es
Thus
t h at
.
,
.
u
a
d
a
t
e
u
t
i
n
c
a
q
q
o is a n e q u at i o n t h at
an d se c o n d po w e rs O f t h e u n kn o w n
r i
bo t h t h e fi rs t
Thus
'
,
3x
2
=
x
15,
5
+
—
x
2
4x = 8
,
x
z—
ar
L
-
b
.
quad r a ti cs ar e a l so c a ll ed in c om pl e te qua d ratic s
q ua d ra ti cs are c all e d com pl e te qua d ra tic s
P ur e
,
and a
ffe c t e d
.
T HE
38 1
pu re,
.
GRAPHI C AL
M E TH O D
in compl ete qu a dra ti c,
.
.
E x e r c i s e 1 64
We sh all
now
g ra ph x
2
-
a,
S O LUTIO N
T h e n ormal form
2—
is 15
a =0
Th e G r aph i cal S olu tio n
or
OF
fo r
282
G raph ing
a
9,
a
4,
a
O,
a nd a
of
Q UA D RATI C
1
—
G r aphing x
a
l ul at e
x
x
o,
:
—9=
2, 3, 4 ,
1,
—8 ,
9,
-
Q UATIO N S
28 3
gra phi n g
or
50
— 9 we
,
2
fi rst
t h e po i n t s
l o c at e
an d
ca c
2
f or a = 9 ,
2
.
E
-
—1
5,
5 , o, 7 , 1 5 ,
-
-
,
s,
-
2,
—3
,
5,
—4
5
,
0, + 7, + 15
D raw a sm oot h c ur v e ( 1 )
R ec a ll t h a t
t h ro ugh t h e se po i n t s
“
2
W h at is a: w h e r e
x
as ks :
.
—
x
t he
c ur
T he
t he
”
is o?
or
9
2
v
t he
,
a nd
+3
in
t ut e d
i fy it
s at s
2
is
rea
—3
—
x
9=
x
-
,
O
z—
a re
a
for
s u b st i
se en
to
=4
or
a
l
we
l
c a c u at e
0,
a:
113
2
.
g ra ph i n g 13 4
pl o t t h e poi n t s :
—4
x
f ro m
.
These
.
2
2
2
s een
of
roots
G r aph ing
.
dil y
—
3
+ 3 and
to b e
figu re
a re
wh e re
x
”
?
e cr os ses the hori zon ta l
a n sw e r
H en ce
Wh at is
and
4,
3,
—4 — 3 0
,
,
,
,
5,
+ 5, + 1 2, + 2 1 ,
-
—
1,
— 2 —
,
3,
3,
—4
,
5
0 , + 5 , + 1 2, + 2 1
d ra w a sm o o t h c u r ve li ke c u rv e
t h ro u g h t h e po i nt s
b ut is s i m pl y
T h i s c u r ve is o f t h e sa m e fo r m a s c urv e
r a i se d u pwa rd 5 u n i t s
T h e x v a lu e s o f t h e c r o s s i n g po i n t s
2—
4=0
2 w h i c h a re th e roo ts of x
a r e h e re + 2 a n d
a nd
.
,
-
.
.
,
z—
g ra ph i n g c ur v e (3 ) for x a fo r a = 0 o r
2
g ra ph i n g t h e c u rv e fo r $0 t h e r e q ui re d c u rv e is d ra w n t h ro ug h
t h e fo llo w i n g c a l c ul at e d a n d p l o tt e d poi n t s
3
.
Si m il ar l y ,
,
,
,
,
1,
2,
3,
4,
5,
— 1
,
m = 0,
1,
4,
9,
1 6,
25,
1,
x=0
2
H e re
t h e re is b ut
—2
,
3,
4,
5
4,
9,
1 6,
25
v a l ue o f t h e
h o ri z o n t a l v iz
on e :c-
t o u c h i n g- po i nt wi t h t h e
-
,
.
i g
c r oss n
0
.
o r r at h e r
E
2 84
t h e re w e re t wo
B e c a u se
u pw a r d
L E M E N TARY ALG E BRA
l on g
so
as
it
i g
c r o ss n
c r ossed
-
p o i nt s
as
the horiz on ta l , w e
two equa l 0 s h er e
In re a li t y t h e r e is
—
0 a re t h e s a m e p o i n t
+ 0 a nd
’
.
v e m ov e d
the
c ur
sa y
o n l y t he
r oo t
g r a ph i n g
2
t h e re
a re
0 , be c a u se
.
4
c
—
G r aphing x
a
2
.
a nd
ul at e
0,
a:
x
2
pl ot
for
—
a
4,
or
x
cal
+ 4 , we
t h e po i n t s :
2,
1,
4,
3,
5,
—
1,
3,
-
-
4,
-
5
=
4
+ 4 , + 5 , + 8 , + 1 3 , + 20 ,
+
d ra w t h e s m o ot h c ur v e (4 ) t h ro ug h t h e m T h e c u rv e
b e i n g 4 u n i ts hig he r tha n cu rv e ( 3 ) d o es n o t t o u c h t h e
T h e re a re n o c ross i n g poi n t s a n d t h e
h o ri zo n t a l a t a l l
a lg e b r a i c w a y o f sa yi n g t h i s is t o s a y t h e r oo t s a r e i m a g in a r y
l
and
We Sh all see l at e r t h a t t h e ro ot s a re
an d
.
-
.
.
.
We
t hen th at
pu r e q u a d r a t i c in
g e n e ra l h a s two
r oots t h a t a r e n u m er i ca ll y equ a l b u t o f op pos ite Sig n s
b ut
t h at if t h e g ra ph o f t h e fi rs t m e m b e r j u s t t o u c h e s t h e h o r i z o n
I f t h e g ra ph d o es n o t c ut
t al t h e r e is b ut on e r oo t v iz 0
t h e h o ri z o n t a l t h e re a re n o r ea l r oots
B ut s i n c e t w o r es ul t s a r e fo u n d b y s o lvi n g
3 82
.
se e
a
,
.
,
.
,
.
,
x
i
.
e
.
,
we
a re
V3
a:
s ay
t h at
two
t he
if
ro o t s ,
ima g in a ry
,
p os itive
— a
a n d a:
g r a ph li e s
on e
2
-
en tir e l y a bov e
and
t he
a,
the hor izo n ta l , t h e r e
other n eg a tive , a n d
both
.
S O LVI N G QUAD RA TI CS B Y F A C T ORI N G
d r a t i c e q u a t i o n s by fa ctorin g
giv e n in 2 1 5 a n d o n pag e 1 64 s h ould be r e vi e w e d h er e
T h i s is n ot a g e n e r a l m e t h o d fo r it is li m i t e d t o t h o s e
e q u a t i o n s t h e fi r st m e m b e rs o f w h i c h ar e r e a dil y f a c t o re d
A pur e q ua d rat i c e qu at i o n whi c h is re d uc ibl e t o t h e f o r m
2
x
a
0 is r e a d il y so lv e d b y f a c t o r i n g
W h e n re du c e d t o t h i s f o r m it is e v id e n t t h a t t h e fi rs t
3 83
.
Th e
s
olu t i o n
of
q ua
,
.
,
,
.
.
E
2 86
L E M E N TARY ALG E BRA
E x e r c i s e 1 66
Solv e t h e
1
3
5
2
x
.
fo llow i ng by f ac t o r i n g a nd v e r ify :
+ 1 1x
z
— 5x
.
2z
.
—
3r
2
2
12
-
z
0
4
7x
So m e
.
e q u at
.
x
3
(
-
4)
(
m= 0 , 4,
a
x
=
3
+ ) 0
the
(
3
v a lu e s of
5
7
9
11
13
15
17
19
—
x + 8x
2
x
3
.
x
3
.
ot s
x
—
—
=
b
a
a
x
b
w
0
+
2
+x
+
— 42x = 0
—
5x
=
m
0
6
2
—
1)
x (x
3
.
x
+ 7x
2
.
.
6x + 3 x
(
—4
x
2
—
4 1 7 515
.
633
.
—
5x
6x
2
2
4
6
8
10
—7 = x
2
12
14
(x
secon d
2
a
r —2
) (
x
+ 2)
2,
(
r
a nd
1
—l
)
=0
.
u nkn o wn in t h e giv e n
fou n d a n d v e rify t h at t h e y
,
.
9x = 0
2
.
x
the
follo w i n g b y f a c t o ri n g
2
3
.
.
-
wh i c h t h e y w ere
c o rr e c t r o
So lv e t h e
— 2 0x
2
—3
a nd
u at io n s fro m
.
6x + 1 1 x
higher degree tha n the
O b se r v e t h e foll o w i n g :
E x e r c i s e 1 67
1
.
x
Sub s t i t u t e t h e se
a re
2
— x2 = 1 2x
x x
eq
.
6
io n s o f
m a y b e so lv e d b y f ac t o r i n g
38 5
—
42
1 23:
9
16
18
'
an d
v e r ify :
—
x + 5x
x=
2
3
.
5
4
.
x
2
+ a x + bx + a b
.
x
.
—
4 93:
633
2
8
—
x +x
30x = 0
2
3
.
—
0
—
—
4)
x (x
3 (x
2
.
2
—
=
x
9
x +
3
.
.
.
x
3
(x
x
4
9x
—
=
x
x
0
5
6
+
2
2
-
x
—x — 2
)
— 1 7 x2 + 1 6 = 0
Q UA D RATI C
Q UA T IO N S
E
287
SQUARE ROO T M E T H O D OF S O LUTI O N
A
386
.
n
ormal
q ua
form , m = a ,
l
so v e
by
d
t a ki n g t h e
a nd
re
du c i n g it
s q ua r e
to t he
of
r oo t
b ot h
.
Axiom
R oot
is
d ra t i c
g
m e m b ers
equ a l
pu re
E qu a l pri n c ip a l
.
f
r oots
o
equ a l
n u mber s a re
.
E xt r a c t i n g t h e s q u a r e
b o t h m e m b e r s fw e h a v e :
ro o t O f
x/ d
x
d o ub l e Sig n b e l o n g s t o t h e u n k n o w n n u m b e r as w e ll as
—
i
x
t o t h e se co n d m e m b e r b u t as
x/a is t h e s a m e a s
i
n
i
h
n
h
e
b
s
u
b
r
h
i
r
d
u
l
Sig
d
fo
d
F
r
s
e
as
o
t
o
e
se
e
e
e
o
t
t
/
x
s e con d me mber o n l y
T he
:
,
.
.
A pu re qua d r atic equ a tion ha s two
one pos itiv e a n d the other nega tive
mer ica l l y equ a l
r oots n u
,
.
exa m p e ,
l
Fo r
= 8,
25 ,
x
=
+
-
5 , h av e th e
—5
5,
t
r oo s
.
g at iv e n um b e r is i m a gi n ary
we o b se rv e t h a t wh e n a is ne g at iv e b o t h ro o t s a re i m a gi n a r y
A ll t h is w as sho wn m o re c l ea rly in 38 1 by t he a id o f
t h e g ra ph s
Si n c e t h e
s q u a r e ro o t o f a n e
,
.
,
.
E xerc i s e 1 68
So lv e b y t h e s q u a re
3
—
2 x
x+
.
387
th e
.
o o t m et ho d
1
1
°
r
°
2+x
5
5
A ny
n or ma l
3
4
—
5
x/ x
c
ompl e te
for m
b,
and c
in te g ra l
or
—
4
x
1_
_
x+ 4
2
a
.
m+ a
qua dratic
o m ay b e re d u c e d
e quati n
to
,
ax
a,
3
deno t i ng
f ra c t i o n a l
2
+ b x+ c
a n y re a
,
l
t h o ug h
n
a
0,
u m b e rs
m ay
,
n ot
pos i t iv e
be 0
.
or
ne
gat iv e
,
E
288
a ny c o m p et e
Si n c e
it is
ca
L E M E N TARY ALG E BRA
q ua d r a t i c m a y b e
g e n e r al qu a d r a tic
l
ll e d t h e
re
t o this
du c e d
f o rm
,
.
l y t h e s qu are root m ethod o f so lu t i o n t h e fi rs t m e m be r
m u st b e m a d e a s q ua r e
Fo r t h i s pu r po s e t h e fo rm Of t h e
e q u a t i o n 18 c h a n g e d t o :
To
a pp
,
.
ax
2
+ bx
T he pro c ess o f m a k i n g t h e fi r s t m e m b e r o f a q u a d r a t i c
e q u a t i o n a s q u ar e is c a ll e d c o m pl e tin g th e s qua r e
3 88
.
.
The
be 1 ,
g e n e r a l q ua d r a t i c
m a y b e a ny n u m b e r g r ea t e r t h a n
it
or
in t h e
of a
v a lu e
T O C O M PLE TE
389
.
C o n s id e r t h e
a r ra n
t h e t e rm s
1
+ bx +
m ay
.
IS
a
0,
1
g e d trin omia l s quar e ,
+ 2 0 35 +
x
of
SQUARE WHE N
T HE
2
Tw o
ax
,
2
are s q
c
g
.
an d
u are s
the
t e r m is t h e
ot h er
prod u c t o f t h r e e f a c t o rs , v iz : T h e f a c t o r 2 , t h e s qu a re
o f t h e fi r s t ter m , a n d t h e s q u a re r o o t o f t h e l a s t te rm
.
roo t
.
The
bi n o m i a l
s t he
s um o f
the
fi rs t a n d
Dividi ng t h e
se c o n d t e r m s o f a n y a r r a n ge d tr in omi a l s qua re
se c o n d t e r m 2 0 x by t wi c e t h e s q u a re r oo t of t h e fi rs t t e r m
i e by 2 x t h e q u o t i e n t is c wh i c h is t h e s q u a r e r o o t of t h e
2
2
m i ss i n g t e rm
A ddi n g 0 t o x + 2 cx w ill t herefo re compl ete
2
x
+ 2 cx
r e pr e s e n t
.
,
.
.
,
,
,
,
,
.
the
s qu a re
3 90
.
.
R ul e
R edu ce the equa tion to the genera l form
the s qua r e of ha lf the coefiici en t of at
.
to both member s
b o t h m e m b er s ,
.
fi t m mb
t hu Obt n n g
To m a ke t h e
rs
s
er o f x
e
ai
r oo
Wh en c e
t
i
ax o
2
— 6rv = 7
a s
q uar e we mus t
,
i
13
B y th e
a n d a dd
?
m,
,
S ub s t i t u t e t h ese in t h e
— 6x + 9
x
—3
x
16
+
= 7,
4
and
—1
giv e n e q u at i o n a n d v e rify
C a r e f ull y o b s erv e t h e f o ll o w i n g i m po rt a nt t r u t h
a dd
9 to
E
2 90
L E M E N TARY ALG E BRA
of t h e ro o t s m u st b e t h e n e g at iv e coefi cien t of a:
2
in t h e e q u a t io n in wh i c h t h e c o e ffi c i e n t o f x is 1
in
a n d t h e pr od u ct of t h e ro o t s m u s t b e t h e con s ta n t t e r m in t h e
s a m e e q u a t io n
T h e s um o f t h e roo t s is 3 t h e c o e ffi c i e n t o f a: w i t h re v e r se d
—
hi
h
i
h
n
Sign ; t h e produ c t is
c
s
a
n
r
w
c
o
t
e
s
t
t
e
m
t
3
The
s um
.
,
.
E x e r c i s e 1 70
S o lv e t h e
—
x
1 68
2
2
350
and
fo ll o w i n g
v e rify
— 2x
2
— 1 02:
4
— 1 20
y
2
7
6
2
—
823 95
.
8
f
9 (c
10
.
2
323
n
2
333
eq
.
14
— 95
16
a
l
e xa m p e ,
2
3x + 4x
.
2x + 7 x
2
.
n
.
2
— 1 1n
2
1
5
+
y
y
.
x
.
2
-
1 3r
2
3x + x
.
— 11
=
2
0
8
+
y
y
2
.
,
l
t o so v e :
M ultiply b y 8
2x2
15
561:
,
D i v i d i n g 5 62: b y t w i c e t h e
S q uar in g 7 a nd a dd i n g ,
4
B y th e
2x + 3x
.
v o id f ra c t i o n s firs t m ul t i pl y b o t h m e m b e r s of t h e
by fou r time s the coefiicien t of
TO
u at io n
F or
12
— 1 1n
2
393
33
-
2
r oo
t axio m
s
quare
5 6x + 49
-
,
0
1 20 ,
ro o
t
'
of
2
1 615
,
t he
q uo ti en t
is 7
.
1 69
7
x = 5 and
W h en c e
I f t h e c o e ffi c i e n t o f x in t h e gi v e n e q ua ti o n is m a d e 1 t h e c o e ffi c i e n t o f
— a n d t h e c o n s t a n t t e r m is
a: is
g
T h e s um o f t h e r o o t s is é t h e c o e ffi c i e n t O f r w it h re v e r s e d Sig n ; t h e
5
k
i
k
s
t
h
or
i
c
i
c
n
t
n
t
t
r
T
i
c
c
w
ro
d
u
c
t
w
h
h
s
t
h
e
s
a
e
m
h
s
h
e
s
o
e
p
,
2
,
.
,
-
.
.
u m b e r a dd e d t o c o m pl e t e t h e
coefil ci ent of a: i n the g iv en equ a tion
O b s er v e t h at t h e
t he
f
s qu a r e o
the
n
.
sq
u a re
is
Q UA D RATI C
E Q UATIO N S
29 1
E x erc i s e 1 7 1
C o m pl e t e t h e s q u a r e ,
1
3
5
7
9
11
13
15
2
s
olv e
an d
v e rify
:
— 711:
.
323
.
—
12
x
2
2
42
.
2
—
4
7 51:
6
m + 6m
2
.
2
.
3x
—
1 4r + 8 = 0
5r
2
.
2
.
3 u + 9u
+ 6x
.
—
2
3t
2
2
2
23 + 7s
2
a
— 2x
— 10
=
2
1
0
y
y+
x
.
2
2x
—
l 0y + 3 = 0
3y
2
.
B
+ 8a
2
— 5x
—5
2n
.
8
.
2
2
— 1 2B
S O L UTI O N B Y FOR MULA
=
x
c
b
0 m a y b e t a ke n t o r e p r e
+
+
se n t , o r t yp ify , a n y qua dr a ti c equ a ti on , in w h i c h a l l t erm s
2h a v e b ee n t ra n s po s e d t o t h e fi rs t m e m b e r , t h e x t e r m s b e i n g
3 94
.
co m b n e
i
eq
u at i o n
d i nt o
s n
The
co n s t a nt
Th e
t e rm s
a
ax
2
t e rm ,
i gl e
th e
l o
as a s
fit-t e r m s ,
a nd
the
.
l i
s o ut o n O f a x
2
=
x
b
c
0
+ +
h a n d l a w fo r w ri t i n g t h e
C o m pl e t i ng t h e
roo t s
o f a ny
s q ua r e a n d s o
fo rm ul a o r sh o rt
e q u a t io n of t h a t f o r m
giv es
lvi n g
a
,
.
,
T h i s is t h e fo r m ul a fo r wr i t i ng t h e r o o t s di r e ct l y wi t h o u t
c o m pl e t i n g t he s q u a re
It is t h e fi n a l res u l t t h at is a l ways
a rr iv e d a t b y c o m pl et i n g t h e s q u a re
a n d it m a y a l w a ys
b e w r i tt e n d o w n a t o n c e
.
,
.
LE M E N TARY ALG E BRA
E
2 92
N o t i ce t h e r e
two
a re
r oo t s , v
iz
2a
fo ll o w i n g b y
So lv e
1
t he
fo rm ul a
—
1 0x
x
2
.
5
a
:
T
2
B y t he
t he
eq
u se o f
u at i o ns
at
2
.
2x
:
x/ 2 5 + 2 4
4.
— 1 3x
=
1
5
0
+
fo r m ul a
w r i t e b y i n s pe c t i o n t h e
e n d o f E xe r c i s e 1 7 1
t h is
th e
im at e
.
O b s er v e t h e
r o ts of
o
.
T O F I N D APPROXI M ATE VAL UE S OF ROO T S
3 95
—2
and
12
fo llo w i n g
QUADRATI C E Q UA TI O NS
OF
pr o c e ss fo r
l ul at i n g
ca c
a ppr o x
ro o t s
( 2)
( 1)
—9x + 1 6 = 0
-
— 1 2x + 2 5 = 0
1
-
71
+
x
— 16
x
2
— 1 2x + 62 = 62
/
g
x 17
x
=4
x
= 6 5 62
-
.
and
x
-
25 = 1 1
1
6 = + 3x/T
x
=6i
x
= 7 65 8 +
1 658 +
.
.
.
a nd
2 43 8 +
O b ser v e in e a c h c a s e wh e t h e r t h e s um of t h e roo t s
t h e c o e ffi c i e nt o f a: w i t h r e v ers e d Sig n
eq
u al s
.
E x erc i s e 1 7 2
Fi n d t h e a ppr o xi m a t e r o o ts
t h e f o llo w i n g :
— 3x
t o t w o pl a c e s
2
d e c i m a l s of
—
x
5x + 3 = 0
2
.
of
,
E
294
L E M E N TARY ALG E BRA
E xe rc i s e 1 7 3
So lv e t h e
1
4
3
.
5
.
.
9
.
13
5x
x+ 6
—
x
2
3
4
fi
4
+ 4x
.
fi
—
So m e
.
I4 = 0
expr es s
io ns
(
+ 2)
(
2
x+ 2
)
.
.
12
.
14
.
x
6
3
+ 2x
4a + 8
in q u a d rat i c fo rm with
su c h fo r ex a m pl e a s ,
ar e
a compound expres s i on ,
x
+x
.
s
2
—
x
5
*
.
5x
x
*
e
2
.
x
.
10
3 97
to
u a t io n s
+ 4x
6-
x
eq
2
4
7
11
x
.
fo llowi n g
12
a nd
x
+3+2v
x+
—
3
u at i o n s m a y b e s olv e d b y f a c t o ri n g
n
f
n
n
h
c
e
s
o
d
x
2
a
d
t
e
e
o r x/ x + 3
o
+
(
)
T h ese
fo r
eq
,
f
r e er en ce
3=0
the
fi rs t
one
.
E x erc i s e 1 7 4
So lv e t h e
1
.
3
.
5
.
—
—
x
8
x/ x
2
x+ 6
3 98
by
f o llo w i n g by f ac t o r i n g
4
(x
.
.
u a t io n s m a y b e put
n u mber t o b o t h m e m b e rs
So m e
a ddin g a
eq
m a y be put in q u a drati c
fo rm by a ddi ng
T h i s is in t h e q u a drat i c fo rm
f a c t o ri n g
S q uar i n g
By
.
w i th
,
1 2 , t hu s
r e fe r en c e t o
x
2
— 4x
,
x
,
2
4x
12
25
an d
16
.
4x
13 0
2 —
4x — 4 = 0
an d x
x
T he
l as t
two
e
q ua ti o n s
2
i
a r e o rd n a r y
=
2
0
77
+
(
2
2—
2—
=
4
x
x
o
s4
8
5
( + )
in t h e q u a dr at i c f o rm
Fo r e xa m pl e
x
.
(x
— 4x2
2
e
.
—3
q uad ra ti c e q ua ti o ns
.
+ 12
.
N
E
U
A
RATI
ATIO
S
C
D
U
Q
Q
E x e rc i s e 1 7 5
fo ll ow i n g e q u a t i o n s in
So lv e t h e
1
3
—
3x + 5x
2
4
.
.
x
i
s=0
2
i
=
0
x
6
5
+
+
3
x
x
q u a drat i c
-
5x
i—
“
4
3
’
5x
f o rm :
7x + l = 5
x
.
x
.
8
.
10
.
—
x
6x
— 24 x = 8
2x
l 8x = 4
( n + s)
i
§—
x
2x
‘
"
~
(
(x
14
x
—
x
ii
.
5 23 +
20 x = 4 8
—
4
x
/
x
2
—
— x
=
5
1 10
+
3
GRAPHI C AL S O LUTI O N OF QUADRATI CS
n
m
k
e
s
s
a
e
u
a
t
r
t
c
a
a
f
u
n
O
g
i
l
l
io
d
i
io
t
u
o
s
a
q
399 T h e r a ph c
q
n
u
s
o
f
t
s
b
t
O
ss
o
h
n
d
t
e
a
i
g
s
i
ili
ol
i
t
r
oo
h
e
of t
y
p
t he m e a n n
so m e w h a t c l e are r
T o so lv e gr a ph i c a ll y t h e e q u a t i o n
.
,
,
.
Fi rs t gr aph
the
fu n c t i o n
—
=
1,
x
0,
2
x
6x + 8 fo r t h e
-
7,
et c
.
,
1 , 0, + 3 , + 8 , + 1 5,
et c
.
,
5,
3, 4,
1 , 2,
— 6x + 8 = + 1 5 , + 8 , + 3 , 0 ,
-
v a l ues
6,
P l ot t i n g t h e s e p o i n t s a n d
c o nn e c t i n g t h e m a s in t h e fig u r e
“
we h a v e t h e
T o a s k fo r
2
giv e x
w h at
a re
— 6x + 8
g r a ph o f x
t h e v a lu es O f
2
is
t he
x— a ues
v l
.
x
t h at
to
as k
of
t he
g r a ph
of t he
c r o s s i n g po i n t s
C l e a rl y
h o ri z o n t a l
o ve r t h e
=
x
v
l
r
e
a
a ue s
t h es e
+ 2 and
-
.
"
2
fia
e r c a spa ce
ma
-1
G r a ph
v
,
ti
Of x
s pa c e
l
2
—6x + 8
'
33
+4
.
curv e of t h e fig ure is
2
w
a
s
a
x
k
x
i
e
r
c
a
a
t
u
d
i
l
+
l
+
y
p
q
q
ll e d a par ab ol a a nd a ny
giv e s a pa ra bol a fo r it s g r a ph
ca
:
Th e
.
E
2 96
L E M E N TARY ALG E BRA
T h is fig ure giv es t h e g ra ph s o f fo ur qua d ra t i c s O b
t a in e d b y kee pi n g t h e c o n s t a nt t e rm e q u a l t o
1 2 an d c han g
ing t h e c o e ffi c i e nt of t h e
x t er m o n l y
2
Th e q u a d r a ti c s x + 7 x + 1 2
2—
7 x + 1 2 giv e t h e s a m e
a nd x
Shape o f c urv e ; e ith er b e i n g
t ur n ed ov e r t h e v er ti c a l a x i s
gi v es t h e o t h er T h e s a m e is
t ru e of th e gra ph s O f x + 8 x
8x + 1 2
+ 1 2 an d x
G r ph
12
f
+
T h i s m ay b e e xpr e s sed b y
s
f
(1
s a y i n g t h a t r e v e rs i n g t h e Sign
3 I?
o f t h e c o e ffi c i en t o f x in th e
_ 7
(4
q ua d r a ti c t urn s t h e gr a ph
S l
2 = 1 v rti l P
o v e r a ro un d t h e v e r t i c a l a x us
400
.
-
.
.
2
_
‘
a
.
o
x
or
a
'
ax
a
ca e
,
e
ca
8
a08
.
oo t s o f s u c h pa i r s o f q u a d ra t i c s are n um e r i c a ll y e q u al
but of o ppo s i t e s ig n s
G iv e t h e r o ot s f r o m t h e figu r e fo r q u a d ra t i c e q u a t i o n s
ma d e b y pu t t i ng e a c h
of
t h e f our q u a d ra t i c
t r i no m i al s e q u a l t o 0
A ll fo u r of t h e g ra ph s
h
n
h
r
h
e
t
ug
oi
t
o
t
o
p
g
+ 1 2 o n t h e v ert i c a l
Th e
r
,
.
.
.
T h i s figu re s ho ws
t he g r a ph s O f qu a d r a t i c s
al l
of w h i c h h a v e t h e
40 1
.
constant
t e rm
C o m pa re t h e
-
12
.
g r a ph s of
t h e p a i rs
p
Gr a h
o
f
x
f or
a
+4
a
-
2
+
A
a
:
+1
a
=
-
Sca
v rt
e
12
ax
l
e
ica
1
l p
s
a ce
E
2 98
L E M E N TARY AL G E BRA
E x erc i s e 1 7 6
So lv e t h e
1
3
5
—
x
2
.
x
z—
.
eq
3x
2
— 5x
2
.
fo llo w i n g q u a d r at i c
x
.
4
.
6
x
u at io ns g raph i c all y
.
2
+ 3x
2
+x
2
=
x
0
5
+
x
x
x
°
CHARA C TER OF T HE ROO T S OF QUADRATI C E QUATI O NS
403
e q u at
The
.
c h a r ac t e r O f
io n is d et erm i n e d
o o t s o f a n y c o m pl e t e
e x a m i n i n g t h e s o l ut io n s
the
by
+ bx + c 0
it is ass u m e d t h a t
ax
In t h i s
d ra t i c
a
u
q
r
of
2
di s c u s s i o n
n u m b ers a is g r e at e r t h a n
t iv e o r n e g at iv e
D e n ot i n g t h e root s b y r l
z ero , a n
,
d b
b,
a,
and
and c a re e
c a re r ea
it h e r
s
o
p
l
i
.
a n d T2 ,
w e have th e
v a l ues :
x/b
b
4a o
2
2a
2a
u re o f t h e t w o r o o t s a s r e a l o r i m a gi n a ry r a t i o n al
2—
or i rr a t i o n a l d e pe n d s o n t h e v a l u e o f b 4 a o
2—
4 ao is c a ll e d t h e d i s c rim in an t o f t h e
T h e e x pr e ss io n b
root s
404 O b se rvi n g t h e fo rm ul a s fo r r l a n d 7 2 it is e vid e nt
The
n at
,
,
.
,
.
.
,
th a t :
1
.
When
the di scr imin a n t i s
r a ti on a l , a n
2
.
When
d
u ne qu a l
3
.
When
the
s qua re
4
.
c ompl ex
Th e
n u m ber s
the discrimin a n t is
n um be r
is
a n um
l n um b e r s
n um b e r s
a + b \/
.
to
rea
the
r
ze o
a re
rea l ,
r oots a re r ea l ,
a
nu mb e r
os itiv e
p
a
r oots a re r ea l a n d conj ug a te su rds
j
d e n o ti n g
equ a l
the discr imina n t i s
When
r oots
.
a r e con u ga te c om p l ex n um bers
A
the
s qua re
.
the di s cr imin a n t i s
r a tion a l , a nd equ a l
a
no t
a
.
neg a tiv e num be
r the
roots
.
b er
of
th e
f orm
a
+bv
— 1,
a
an d
b
.
an d
—
—
V
b
1,
a
‘
ar e
conj u ga te
co mpl ex
Q UA D RATI C
40 5
E
Q UATIO N S
2 99
fo ll o ws t h at we c a n d e t e rm i n e t h e n at u r e o f
a n y q u a d ra t i c e q u at i o n w i t h o ut so lvi n g it
It
.
r oo t s o f
.
t he
For
e x am p e
l
2
3x
In t hi s
ro
ots
are
— 7x + 2 = 0
—
4a c = 25
b
u at i o n
re a l rat i o n a l
eq
,
,
.
a n d u ne q u a
4x
In t h i s
n um b e r ,
e q u at
t he
Si n c e 2 5 is
2
io n
—
b
4a o
B ut
.
,
0
2
— 24
2
l
u are t h e
t a ke t he e q u a t io n
a sq
Sin c e
.
— 24 is a ne g at iv e
r oo t s are c on ug a te co mp l e x n umber s
j
.
E x e rc i s e 1 77
di s c r i m i n a nt d e t erm i n e
o f ea c h o f t h e f o ll o wi n g e q u a t i o n s :
use o f
B y t he
the
r oo t s
1
3
5
7
9
11
13
.
.
.
r
.
e qua
4
5x + 8 x
6
—
=
1
x
5
x
7
+
—
4x
2
.
0
8
4x + 1 = 0
10
4x + 6x
2
.
.
.
.
.
oo t s ?
17
—
4x + 2 = 0
2x
2
ro o t s ?
16
2
2
r oo t s ?
15
7x + 3 = 0
2
r oo t s ?
14
—
4x
2
t he
.
12
Fo r w h a t
v a lu es o f
?
n
I rr a t i o a l r o o t s
F o r w h a t v a l ue
n
Of a
t he
—
x
2
.
.
x
2
n at
of
ure
7x
=
5
x
0
6
+
+
—
x
3x + 5 = 0
2
.
.
2
=
x
5
0
3
+
+
2
— 5x
x
.
x
.
—
x
2
5x + 8 = 0
will 2x + n x + 8 = 0 h a v e
eq
ual
h ave
eq
ual
w ill 3 x + 2n x + 3 = 0 h a v e
eq
ual
2
w ill
—
ax
12 x + 6 = 0
2
I m a gi na ry r oo t s ?
v a lu e s o f
I m a gi na ry root s ?
Fo r w h at
Fo r w h at
n
v a lu es o f c w ill
R e a l ro o t s ? I m a gi na ry
2
—
1 0x + c = 0
5x
ha ve
2
ro
l r oo
u al
ots ?
For w h at
?
ts
eq
v a lu es of n will
Fi n d t h e co rres po n di n g v alu es
h av e
of x
.
E
3 00
406
.
By
L E M E N TARY ALG E BRA
m em b e rs
dividi n g bo t h
=
b
x
c
0 by t h e
+ +
,
be c o m es O f t h e fo rm
eq
uat io n
i
ax
x
s
olu t i o n s o f t h i s
su m
ci en t o
ffi
co e
f
f
o
x
the two
with
,
g at i v e i n t e g r a l
’
or ne
of x
i
by
—
r oo ts
f
2
x
o
r eve r s ed s ign
or
,
.
e q ua t o n a r e ,
r,
The
g en e r a l q u a d ra t i c
2
of x t h e e q u at io n
=
2
x
+ p +q 0
in w h i c h p a n d q a r e po s i t iv e
t io n a l , a n d 2 p is a n y c o e ffi c i e n t
Th e
t he
c oe ffi c e nt
2
2
of
394
403
or
.
q
=
2
x
+ p + q 0 is
— 2
p,
the
.
The p rod u ct of the two roo ts
con s ta n t ter m of the equ a ti on
f
2
x
o
=
2
x
+ p + q 0 is q, the
.
e n ab e
fo re g o i n g pr i n c i pl es
n
n
h
e
r
s
e
u
a
d
r
a
t
i
c
u
a
t
i
o
s
w
i
t
giv
oo
t
q
q
40 7
.
The
l
t wo
us
to
f o rm
.
I f t he
r oo t s
of a
ffi c i e n t o f x is
e q u a t i o n t h e n is
c oe
q ua d r a t i c
a nd
4,
x
It h a s
l
a re a
—
= 0 t he
x
5
(
)
,
dy bee n
2
i
e q u at o n
t he
c
a re
408
.
5 , t he
a nd
—4 5
.
T he
+ 4x
pr o
—9
r o o t s ar e
2 1 5 , 3 84 - 5 , th a t
ve d
a nd
5
if ( x + 9)
.
—5 =0
)
,
r ev er s ed
.
FOR M A QUADRA TI C E QUATI O N WITH G IVE N ROO T S
Rul e
.
S u btr a ct
f
ea ch o
the
from
r oots
f the two r ema in der s equa l to ze ro
—
7
e q u a t i o n w h o se ro o t s a r e 6 a n d
ro
d
u
c
t
o
p
Th e
9
t e r m is
o n st a n t
O b s er v e t h a t t h e k n o w n n u m b e rs in
t h e r o o t s of t h e e q u a t io n wi th their signs
To
—
.
(
=0
x
x
2
+x
42
0
,
.
or
is
x a nd
pl a ce
a re
E
302
L E M E N TARY ALG E BRA
E x er c i s e 1 7 9
the
F a c t or
1
4
7
10
a
.
a
.
a
.
a
.
2
fo llo w i n g :
4a
2
6a
2
1
—
x
3
—
4
x + 4x
—
x + 8x
2
1
5a
—
n
6n + 1 1
2
—
n
2
2a + 4
2
—
8x 2
2
11
—
x
2
.
2
6n + 1 3
— 16
8
3x + l
12
2
.
77
+ 9n + 2 3
PRO BLE MS I N QUA D RA TI C E QUA TI O NS
41 0
Si n c e q u a d r a t i c
.
wh ose
l
i
s o ut o n
i
e q uat o ns
i n v o lv e s s u c h
have two
pr o b e m
l
ro ot s , a
i
a n e q u a t o n a pp are nt
ly
h as t wo
v a l ues o f t h e u n k n o w n n u m b e r o r t w o r oo t s
B o t h r o o t s m a y sa t i s fy t h e e q u at i o n b ut o n l y o n e of t h e m
E s pe c i a ll y is t h i s
m a y s at i s fy t h e c on d i tion s of the pr obl em
t ru e w h e n t h e r o o t s a re s u r d s o r i m a gi n a r y
In so lvi n g pr o b l e m s t h at i n v o lv e q u a d ra t i c s w e s h o uld
e x a m i n e t h e r o o t s o f t h e e q u a t i o n a n d r e j e c t a ny r o o t t h a t
d o es n o t s at i sfy t h e re q u i r em e nt s of t h e pr ob l e m
,
.
,
.
.
,
.
E x e r c i s e 1 80
So lv e t h e
1
416
.
.
2
.
590
.
3
.
The
Fi n d
Th e
A
is wid e
of
l
u m b e rs
n u m b e rs
t wo
t h e t wo
s um
Fi n d
of
t he
n
u a re s O f
n um b e r s
sq
t h e t h re e
gul a r fi e ld o f 4 a c r e s is
W h at a r e t h e di m e n s i o n s ?
.
.
t h re e
If
the
n u m b e rs
s um
of
c o ns e c
u t iv e n um b e rs is
.
.
5
t h e i r pr o du c t is
and
.
Th e q uo t i e nt o f o n e
a n d t h e i r pr o du c t is 2 8 0 0
4
is 42 ,
re ct a n
.
Q ua d rati c s
r
m
:
b
o
e
s
p
foll o w i n g
s um
P r ob l e m s in
the
is 98 0 , w h at
a re
12
ro
d s l o n g er t h a n it
n u m b er
di vid e d b y
Fi n d t h e n u m b e rs
a n ot he
r is
7,
.
s qu a re s
t he
n
of
t h re e
u m b e rs ?
c o n se c u t
iv e
e
ven
Q UA D RATI C
6
.
7
.
t h e pr i c e
Fi n d
1 39
20 t im e s t h e
l a rger
Th e
su m
of
.
Th e
duct
is
d o z en w h en
u m bers
s um o f
t he
n umb e r
l ess fo r 50 6
5
b y 94
n um b e r s
tw o
a nd
r
p o
is 24 ,
v e yo u r
and
t h e i r pr o d u c t is
a nswe r
.
di ff ere n c e b e t wee n t w o n u m b e r s
1 380
Fi n d t h e n um b e r s
is 1 6,
.
13
.
t hei r
14
sq
.
3 98 i
15
Th e
.
.
t i m es
16
.
17
.
th an
t h r ee
Th e
Fi n d
T he
as
t he
Of
n
s um o f
m uc h
tw o
.
At 1 5 ¢
a
a
2
of
= b2 +
r
a
l o ng e r
t r i a n gl e
sq
is 40 ,
is 96 ,
c
2
igh t
ds
,
and
and
t he
s um o f
.
t h e i r pro du c t is
a n sw e r
a nd
.
and
t h e i r p ro du c t is 1 8
.
fo r b
a nd
c
.
t r i a n gl e is 9
t h a n t he
ot h e r
l eg
.
f e et l o ng e r
Fi n d t h e
.
u a re f o o t it
fl o o r in a r o o m wh ose l e n gt h is
Fi n d t h e di m e n sio n s of t h e fl o o r
18
n um b e r s
n u m b e rs
t he
fo rm u l a
the
ro
.
is 4 8 ,
P r o v e yo u r
.
n u m b e rs
T h e h y po t e n use
o n e l e g a n d 2 fee t
Sid es of
th e
n um b e r s
u m b e rs
t wo
n u m b e rs
e
.
Fi n d
.
Solv e t h e
é
— 2
=
(t
t
f
n
o
r
t
a
d g
g
ven
Fi n d
two
is 1 2 24
s um
t h ei r
.
f o rm u l a
sum o f
u a res
and
.
.
So lv e t h e
the
.
T h e pe r i m e t e r o f a r e c t a n g ul a r fi e ld is 1 1 4
Fi n d t h e di m e n s io n s
a r e a is 5 a c r e s
.
and
,
’
.
12
w h os e
.
gu l ar fi e ld is 84 r od s
Fi n d t h e di m e n s i o n s
.
.
30 3
a rec t a n
Fi n d t h e n u m b e r s
11
th e
ds
T h e p e r i m e t er o f
is 4 3 2 s q u a r e r o d s
.
10
pr o
odd n
c on s e c
.
a re a
9
ut iv e
two
s q u a r es e x c e e
8
prIce o f e gg s pe r
?
n
ze
6¢ a d o
Wh at is t h e
i n c r ea s e s
Q UATIO N S
E
,
$ 99 t o l a y a p a r q u e t
8 f ee t m o re t h a n it s w id t h
c ost
.
.
di m e n s i o n s of a c e rt a i n r ec t a ngl e a n d it s di a g o n a l
re pr e se n t e d b y t h re e c o n se c ut iv e e v e n n u m b ers
W h at
?
f
n
h
m
n
n
t
h
re
t
e
c a gl
e
t e di e s i o s o
19
a re
a re
.
The
.
LE M E N TARY AL GE BRA
E
3 04
A
d ays m o r e t h a n h e r e c e iv e d
do ll a rs pe r d a y for his l a b o r a n d e a r n e d $ 1 7 5 H ow m a ny
d ays did h e w o rk a n d h o w m u c h did h e r ec e iv e p e r d ay ?
20
.
w o rke d 3 0
c a r pe n t e r
.
21
e x c ee
22
ds
di ff e r b y 1 T h e
s u m o f t h e i r s q u a r es b y 2 2 0
n um b e rs
Two
.
t he
T h ere a r e
t h e w id t h
.
t im es
'
32
sq
y d in
.
the
the
Fi n d
.
s um
n u m b e rs
gl e w h o s e l e n gt h is
f eet
l e ng t h
in
.
18
.
Fi n d t w o n um b e rs w h o s e di ff e r e n c e is
s um m ul t i pli e d b y t h e s m a ll e r n u m b e r is 7 5 6
6,
Fi n d t h e Sid e o f a
i n c r e a s i n g it s l e n gt h 9 y d
is
23
t heir
a re ct an
.
Fi n d
.
of
s q ua re
.
.
and
wh ose
.
24
s q ua re
.
25
On e
.
a r ea o f
26
an d
b o t h is 1 1 0 8 s q
Fi n d
.
Th e
.
exc e e
ds
Fi n d
the
28
.
the
s um o f
t he
27
fi e ld is
s q ua re
t he
n
rd
.
u m b e rs
10
.
a r ea
it s w id t h 6 yd
and
.
whose
.
t he
s um
Of
w h o se two
is 8 9
digi t s
n u m b e r O f s q ua re
i nc hes
in t h e
i n c h es
vo lum e of t h e
A m an bo ug h t
a
the
c ub e
in t h e
u rfa c e o f a c u b e
o f it s e dg e s b y 1 1 7 0
s um
s
.
.
l a n d fo r $4 050 H e
pr o fi t e q u a l t o t h e c o s t o f
pi e c e
is 1 3
digi t s
.
of
.
35 3 a n a c re , m a ki n g a
Ho w m an y a c r e s did h e b uy ?
A m er c ha n t
,
an d
.
whose
n um b er o f
by
.
l o n g e r t h a n a n ot h e r
Fi n d t h e l e n gt h o f ea c h
rd
s q ua r e s o f
t he
d o ub l e d
so
ld
16
it
at
a c r es
.
ld s o m e d a m a g e d g o od s fo r $ 24 a n d l o st
a pe r c e n t e qu a l t o t h e n u m b e r O f d o ll a r s h e pa id fo r t h e g oo d s
Fi n d t h e c o st of t h e g o o d s
29
.
so
.
.
30
If
5
.
Th e
the
.
.
.
a c re s
31
l e n gt h of a r e c t a n gl e e x c ee d s it s w id t h b y 7 r d
di m e n s i o n s w e re i n c r e a se d 5 r d it would c o nt a i n
Fi n d t h e di m en s i o n s o f t h e re c t a n gl e
.
A
,
.
m e r c h a nt b o ug ht
lace
fo r 3 1 0 0
.
H e k e pt 3 0 ya r d s
old t h e r e m a i n d er fo r as m u c h as it a l l
H o w m an y ya r d s did h e b uy ?
a ya r d
a nd s
.
c os t ,
g ai n i n g
75 ¢
E
3 06
L E M E N TARY ALG E BRA
On t h e
s a m e r e e r e n ce
f
li n es g ra ph i n g
,
2
f
r
o
)
(
x
+ 2,
2
y
+ 3,
5
gi ve s t h e s t r a igh t li n e m a rke d y 2x 1
in t h e fig u re
T h e s o l u t i o n s s o ug h t a r e t h e x a n d
n
f
n
n
di
s
a
c
s
h
e
c
r
ss
i
g
i
e
t
t
o
o
ts o f
o
y
p
t h e g ra ph s o f ( 1 ) a n d
.
-
The
Sc ab
h o r iz o nt a '
1
"
-
“”
mm
spa ce
a
a nd
x
tha t both
'
“
to
the
sa
y
-
va l u es
n u m ber s
f
o
m u s t be
each
so
i
a
p r
pa ired
bel on g
me cr ossin g p oin t
-
.
lut i o n s a re : x = 0 y
1 and x = + 2 y = + 3
41 4 T h e g r a ph o f ( 1 ) is a pa r ab o l a a n d a ny t wo-l e t t e r
e q u a t i o n o f t h e s e c o n d d e g ree w i t h o nl y one va ri a bl e ra ised to
the s econ d power a n d w i t h o u t a n x y t e rm giv es a par ab ol a
fo r it s g ra ph
The
so
,
,
.
,
.
-
,
.
li n e to st a rt f ro m t h e po s i t i o n m a rke d
i
n
1
m
n
2
r
r
r
x
o
v
g
a
c
o
s
t
h
a
a
b
l
a
a
a
l
l
e
l
h
r
i
g
s
s
e
o
t
o
t
e
t
a
t
y
p
p
n
n
h
e
2
x
n
r
n
h
i
i
li
v
i
i
s
t
t
o
t
e
I
e
e
s
er e w o uld b e
o
o
o
t
o
t
p
y
y p
two c r o ss i n g po m t s u nt il t h e po s i t io n y = 2 x is r e a c h e d
At
t h i s po s it io n t h e t w o c r o ss i n g po i n t s b l e n d i n t o on e t h e li n e
b e c o m i n g ta n gen t t o t h e pa r ab ol a
B eyo n d t h e p o s i t io n y = 2 x t he re would be n o c r o s s i n g
f
h
n
n
h
i
n
t
t
e
li
e
a
t
e
r
b
l
d
a
a
O
o
a
o
p
p
—
=
St a rt i n g f ro m t h e li n e y 2 x 1 a n d m o vi n g pa r a ll e l t o
i t se lf t o w a r d t h e righ t t h e re w o u ld a l ways b e two c r o ss i n g
n
f
n
a
h
v
n
n
r
i
t
s
R
e
c
lli
g
t
a
t
er
c
i
g
i
giv
es
a
v
a
lu
e
o
t
o
e
oss
o
y
p
p
x an d o f y w e o b ser v e t h at :
I Ther e a r e in gen era l two s ol utio ns of a s ys tem ma d e u p
of a pa r a bo l ic a n d a l i n ea r equ a ti on
41 5
S u ppo se
.
a
,
.
-
.
-
,
.
.
,
-
.
,
.
.
II
.
Whe n
s o l u tio n, o r
two
the l i n e i s ta ng en t to the pa ra bol a there is bu t o n e
s in ce the two c ross in g p oin ts coa l e s ce , we ma y s a y
e qua l sol u ti on s
-
.
.
S I M ULTA N E O US S Y S TE M S
III
F or
.
o
s
i
t
i
n
o
p
two
of
th e
re al
there i s no
l
Th e
.
l in e b eyo nd the ta n gen t
A l g ebr a s hows tha t there a re
a n equa ti on r epr es en tin g a
s o l u tio n
.
her e, bu t tha t they
so l u tion s even
416
s ys t e m
.
.
j u s t giv e n is t h e g raph i c a l s o l ut i o n
We n o w g i v e t h e al g e b r ai c s o l ut i o n O f t h e
i
.
Wr i t i n g t h e
i
e q u at o n s
x
thus
t he
(1)
( 2)
2
2x
y
i t ut e
imag ina ry
a re
s o ut o n
s a m e s ys t e m
s ub s t
30 7
l
v a ue Of
y
f rom ( 2 )
1
in
s
i m plify
a nd
,
find :
—
2x = 0
x
2
Wh e n c e
x
,
Su b s t i t u t i n g t h e s e
v a l ue s
T he
so
lu t i o n s
a re
of x
1,
y
the
n
um ber
an d
w i t h t ho s e
+2
we fi nd
+3
pa i r s :
=0
y
a nd
,
in
and
x
T h e s e v a l ue s a g r e e
=0
x
,
—
of
l
,
= +2
,
y
t he
=
+3
g r a ph i c a l
l
i
s o ut o n
.
E xe r c i s e 1 8 1
fo ll o win g
So lv e t h e
x
2
+ 3x
-
= 18
y
— 2 = 2x
y
—2
=
x
5
y+
y
—
x
2 =3
2
s ys t e m s a
l g e b r a i c a ll y
[
l
2
2x
'
v
Q—
f
—
x
3
y
l
5 y+ 3 x = 6
2y
y
-
— 6x
=
+y 8
— 4
— 4x
y
f
l
—
3x
=5
y
-
5
2
—
3x = 4
—
9x
=2
y
—
3x
=2
y
E
30 8
41 7
So lv e
.
ne xt
L E M E N TARY ALG E BRA
the
s ys t e m
x
2
2
25
+y
x/ fl
d:
y
fl)
—
—
t
'
G ra ph i n g
( 1)
y
=
+ 6, + 5 ,
e t c , a n d c a l c u l a ti n g y
= ima
+ 4
i
0
3
y
g
,
,
,
33
.
x
-
—x 2 u s i n
i
+ 4, + 3 )
:
l
x
+ 2,
fr o m
+ 1;
=
l=
V
y
l
g
1’
_
_
2;
_
4)
_
—6
5)
y
25 — x 2 fi n d
,
=5
,
zi
.
etc
07
(1)
(2)
2
4 0,
t
d
+
:
.
4,
+
0 , imag
3,
,
.
G ra p h i n g t h ese pa i r s , l ayi ng
o ff t h e v a l ue s w i t h d o u b l e s ig n
a nd
upwa rd
bo th
d ow n wa rd ,
Ob t a i n t h e c i r c l e O f t h e fig u re
G r a ph i n g
Ob t a i n
of
figu re
th e
the
li n e y
s t r a ig h t li n e
n ow
—
x
1,
.
th e
.
i g po i nt s giv e t h e
fo ll o w i n g s o l ut i o n s
The
c ro ss n
x=
y
:
-
—3
+ 4, x
+ 3) y
T h i s is t h e
_
,
4
i
ca
l
h
r
a
g p
s ol u tion
.
—
Su pp o se a li n e s h o uld st a r t fro m t h e p o s i t i o n x y = 1 a n d
—
m o v e u pw a r d a c r os s t h e c i r c l e , ke e pi n g p a r a ll e l t o x y = 1 ,
t h r o ug h
—
t h e po s it i o ns x y =
0,
= — 3 to x — =
y
y
,
—
x
—
x
= 4 to x
or d o wn wa r d t h r o u gh t h e po s it io n
y
In e v e ry p os i t i o nt h e li n e giv e s t wo c r o ss i n g p o i n t s w i t h
c i rc l e
u n t il t h e ta ngen t po s it i o n s a re re a c h e d w h e re
t wo c r oss i n g p oi nt s b e c o m e one po in t O f c o nt a c t
,
’
-
,
,
th e
th e
-
t a n g e n t pos i t i o n s t h e sys t e m w ould
giv e t w o ima gin a r y so l ut i o n s F o r t h e t a n g e n t po s i t i o n s
O f t h e li n e w e m ig h t a g a i n s a y t h e r e a r e two equ a l so lu t io n s
Fo r
a
li ne beyon d t h e
.
.
Fo r t h e
fo r t h e
u pp e r
l o we r
t a n g e n t p oi n t
-
t a nge n t po i n t ,
-
x
fl
3—
x
g
-
+1
2, y
,
y
=
and
E
10
4 20
the
The
.
v a lu e
s ol u ti on
a l gebr a i c
of
R e du c i n g
LE M E N TARY ALG E BRA
n
f
r
m
1
i
o
( )
y
— 1 60 x
2
3 2x
,
Or ,
W
+ 4 00
hen ce , x
—
g
4
g et h er
=0
,
and
5x = 0
an d
f om ( 1 )
Of
v a lu e
r
and
y,
ub st i t u t i n g
—4
y
,
and
t h at t h e
Of x
g ra ph for t h e
M o vi n g
e lli ps e s h o w s t h er e
eq
1 6x + 2 5 y = 4 00 , is
u at i o n
2
2
—
t h i s li n e 4 x 5 y = 2 0 p a r a ll e l t o
two p a ir s
f
o
a re a
va l u es o
f
two equ a l pa i rs
a
y,
two
i t self
c r o ss i n g po i n t s
an e
a c r oss
and
-
,
,
.
t he
h en c e
v e fo r t he ta ngen t posi tion s
pa i r o r as we pre f e r t o s a y
sa
on l y one
ll ips e
,
,
.
lg e b r a i c
t ou c h t h e
of x and
l w ays
x a nd
w h er e t h e re w o uld b e
n ot
.
.
Th e
An
0
m u s t b e pa i re d wi t h
+ 5 an d 0 a l s o b e l o n g t o
t h at t h e 0- v a lu e
r a p h sh o w s
-
+ 5,
s
4 00
—
x
2
Th e
t he
is O b t a i n e d b y
O b t a in i n g
l io n w ould Sh o w t h a t wh en t h e li n e do e s
e lli ps e t h e re w o uld b e t w o im a g in a r y v a lu es
s o ut
y
.
A
dr a t i c e q u at i o n w i t h n o xy t erm b ut c o n t ai n i ng
t h e s q u a re t e rm s of b o t h v a r i ab l e s t h e c o e ffi c i e n t s o f t h e se
t e r m s b e i n g of t h e sa m e Sig n giv es a g r aph t h a t iS a n el l i pse
4 21
.
q ua
-
-
,
'
.
,
E xe rc i s e 1 83
So lv e t h e
fo llo w i n g
4 x + 9y
it
—
7x
x
‘
422
.
2
2
s ys t e m s a
36
4y = 1 0
=
1
16
6
+ y
2
lge b r a i ca ll y
2
4 23 + 9y
36
2
—
5x
.
:
3y = 3
9x + y = 9
2
2
So lv e t h e sys t em :
—
2
-
y
= 16
y
=2
or
or
—
l
x
6
/
x
y
=x— 2
=
y
i
2
S I M ULTA N E O US S Y S T E M S
i
In e q ua t o n
v a l ue s
va l ues
of
,
1)
al l
v a l ues
i m a gi n a ry
a re
y
fo r
of
311
—
x b e t w ee n
4
C a l c u l a t e y fo r t h e
.
+ 4 the
and
gi v e n
x
find
x=
+ 10,
y
+
92
+ 8, + 5, + 4,
69
+
,
0,
3,
+
,
4,
+
Pl o tt i n g
t h e s e po i n t s ,
=0
2
f
r
o
x
( )
, y
5,
0
3,
+
dr aw i n g
s
the
+
69
g r a ph
—
and
,
—
2 and x
4 y
i
t h e p i c t u r e o f t h e figu re s h o w n
T h e g ra ph o f e q u a t i o n ( 1 ) is a h ype rb ol a
di s c o n ne c te d pa r t s o r br a n c hes T h e r e IS b ut
e qua t o n
,
—
,
,
g r a ph i n g
6 Ob t a i n
,
.
.
.
,
S
1 h
2
po i n t
of
g r a ph
s h o ws
x
=
+ 5,
423
the
x
e
or o n a sp ce
e r ca spa ce
iz
t l
t
l
i
a
c
.
.
-
= +3
y
a nd
-
a
,
,
,
s u b st
+5
it u t e d in
y
T h e s e v a l ues o f
to be
the
v a lu e
,
(I)
T h i s v a lu e o f x
Th e
.
lg e b r ai c so l ut i o n giv es b y s u b s t i t ut i n g
2
o f y f r o m ( 2 ) in ( 1 )
x
16
(x
R e d uc i ng w e fi nd
4 x = + 20
.
Th e
the
v
i g
one c r oss n
u r v e T he fig ure S h o w s w h y
a n d y v a l ue s fo r t h i s c ro s s i n g p o i n t
and
li n e
1
ca l
It h as t w o
x an d
y
a
i
e q ua t o n
+3
g ree
.
giv es
.
wit h t h e
g r a ph i c a l
so
lu t i o n
.
E
3 12
42 4
A
.
wi t h
q u a d r at i c
a
,
st r a
igh t li n e b e
of
t ur n e d
f o ll o wi n g
i
r
n
e
se
t
p
,
a
l w a ys
und
t h a t it w o uld
so
H ow m a ny v alu es
z—
lg e b ra i c a ll y
2 =
—
x
7
y
—
=1
x
y
= 13
x
y
—
=1
x
y
2=
—
27
x
y
—
=3
x
Of x
2=
—
x
5
5
y
'
2=
—
45
x
y
—
=5
x
2
2
y
f y
—
=6
x
3
y
l
z—
2
—
x
y
y
x
=5
2
= 80
So lv e t h e s ys t em
xy =
1 2,
eq
u at io n ( 1 )
v a lu e s o f x
l ul at e
ca c
( 1)
y
or
—x= 1
y
,
y
y t erm s
1 84
xe rc s e
s ys t e m s a
2
x=
aro
t h e h yp e r b o l a ?
2
In
a nd
2-
h
e
s
e
?
ould
t
e
b
w
y
So lv e th e
.
x
2
.
E
42 5
b oth
b e i ng
-
both br a n ches
of
h a vi n g
u at i o n
i t e Sig n s n o x y t e r m
h ype rb ol a fo r it s g r a ph
C o u ld t h e
an d
eq
o ppos
giv es
ou t
L E M E N TARY ALG E BRA
or
y
= x+ 1
y fo r t h e
( 2)
follo wi n g
as s
umed
,
+ 3, + 2, + 1 ,
+ 6, + 1 2 ,
+ l 2 , + 6,
=
d r aw
1,
—
1 2,
— 2
,
—
6,
-
3,
-
— 4 —
,
4,
—
3,
—
6,
— 12
2,
— 1
g r a ph O b t a i n i n g a c u r v e
for x y = 1 2
Sh o w b o t h b r a n c h e s o f t h e c u r v e
B ot h b r a n c h e s t o g e t h e r a re spo ke n o f a s a s i n gl e c u r v e
P l o t t h e se
n
n
o
t
s
a
d
p
,
i
the
,
.
.
,
t h e hyp er bol a
G r a ph i n g
.
eq
u at io n ( 2 )
on
the
sa m e axe s ,
u s i n g t h e f o ll o w
in g p oi nt s ,
x=
0,
+ 3,
= +4 + 1
y
)
)
th e
s t r a ig h t li n e g ra p h
-
1,
0)
-
_
4
3)
fo r y = x + 1 is O b t a i n e d
.
LE M E N TARY ALG E BRA
E
3 14
E x erc i s e 1 8 5
—5
3 xy = 2 1
—
—
x
8y
1
42 9
Th e m ai n
7 xy = 98
5 x y = l 50
x
—
y
—
—
x
1
—
5y
3
g r a ph i c a l s olu ti on O f e q u a t i o n s
t o pu pil s is t o e n a b l e t h e m t o s ee t h e m ea n i n g o f s o lu t io n s
a n d t o u n d e r s t a n d w h y r oo t s a r e p a i r e d in a c er t a i n w a y
.
u se o f
the
,
.
F o r pr a c t i c a l w o r k
so lu t i o n , a s giv e n in
of
so
lvi n g
.
t he
lg eb r a i c
4 2 3 a n d 4 28 s h o uld
foll o w t h e a lg eb ra i c
u a t i o ns
4 1 6, 4 1 8 , 4 2 0 ,
l w a ys b e u s e d In t h e e xerc i s es
m et h od is t o b e e m pl oye d
a
eq
t h at
a
,
.
E x e rc i s e 1 8 6
So lv e t h e
x
2
fo ll o w i n g sys t e m s a n d pa i r t h e r oot s pro per l y
27
+y
x
3
y
y
x
2
+
3x
2
x
x
1
x=
— 4
y
2
—
xH y
—
x
—
x
26
1 47
+ 8y
=2
y
5x + y
x + 2y
2
2
45
12
18
xy
x+y= 9
x
=
2
10
+ y
2
—
x
y
=
+ y 10
=
20
+y
=6
x+
2
y
2
2
+y
= 73
— 2x = 1 3
y
2
x
y
2
x
=6
y
—
x
=4
2
=7
2
x
=
2
+ y 89
x+ y= 1 1
2
xy =
— x
y
=3
xy =
—
x
lo
24
=2
y
S I M ULTA N E O US S Y ST E M S
3 15
= 12 — y
2
x
— 14
=0
y
—
x
—
m n=3
x
2
2
+ xy + y
=
9
x+y
mn = 1 8
ac
+
2
76
2
0
x
z=
+y
m
—
mn + n
m
= 19
2
— n=
3
=
m
n
39
m +n +
2
2
—
m n =3
—
2a
~
x y+ y
+
a
-
—4
3x
-
m + 2x = 2 7
mx = 8 5
b= 18
= 40
2
—4
—
b 3a
Z
= 40
+b
a
c
2
y
=0
— a
+b
ab
—
=
x
4x
3 y 91
— 2x = 1
3y
2
2
2
x
2
=
13
3y
—
a
2
a
5y
65
= 11 —
y
a + c = 14
2
= 61
76
5d
2
—
4 c 5d = 2 9
3x y + x
2
—2 2 = 5 2
y
2x + 3y
2c
-
36
3m
m
= 57
xy = 4
z=
—
16
n
m
—
x
=
13
m
2
u+
—
9n
z
= 19
=
19
m
2
3n +
m
2
2 mn = 4 5
2
2
4m
2n = 23
d = 35
cd
2
—
—
+n
2
—
m n = 50
mn = 30
y
=0
=
7
4
6
+
q
p
—
3p q = 2 1
2
2 =
—
275
3x
y
—
—
5
3
x
2
y
E
3 16
43 0
.
ie n t l y
LE M E N TARY ALG E BRA
S pe c i a l M e th o d s
so
lv e d b y
So m e
.
i l m et h od s
s p ec a
sys t e m s
w e ll
as
m ay b e
as
by
s
c o nv en
u b s t i t ut i o n
.
i l m et h od s is t o divid e t h e giv e n
e q u a t io n s m em b er b y m e m b e r o b t a i n i n g a d e r iv e d e q u a t i o n
w h i c h w i t h o n e o f t h e giv e n e q u at io n s fu rn i sh e s a s ys t e m O f
e q u at io n s e q uiv a l e n t t o t h e giv e n sys t e m a n d t h e n t o s o lv e
t h e d e r iv e d syst e m
r
r
h
n
a
O
b
e
v
e
c
a
e
full
t
e
f
ll
i
g s o lu t i o n O f t h e syst e m
s
o
o
w
( )
y
43 1
.
One
t h ese
of
spe c a
,
,
,
,
,
.
2
x
—
2
y
= 33
x+ y=
—
x
Dividi n g ( 1 ) by
y
(1)
( 2)
(3)
11
=3
i i g o f ( 2 ) a n d ( 3 ) is s i m pl er t h a n t h e
giv e n s ys t em a n d t h e s i m pl er sys t e m giv es x = 7 a n d y = 4
T h es e are a l l t h e roo t s fo r ( 1 ) r e pres en t s a h yper bol a a n d
n
n
n
n
2
h
n
h
r
i
n
ig
li
o
l
i
a
r
a
t
e
a
d
t
e
c
oss
e
o
t
s
t
o
( )
p
y
y
Th e
sy s t e m
c o n s st n
‘
.
,
.
,
( b) Solv e
the
syst e m :
—
2
3 6m
—
6m
2
p
p
= 8 19
—
39
21
Dividi n g ( 1 ) by
6m + p
Th e sys t e m ( 2 ) a n d ( 3 ) is e q uiv a l e nt
a n d it s ro o t s a r e :
-
—
=
m
5 an d p = + 9
E x e r c i s e 1 87
f o ll o w i n g s ys t em s
a n d p a i ri n g res ul t s p r o pe r l y :
So lv e t h e
2
9x
3x
2
4y
2y
30 8
14
,
'
dividi n g
w h e n po ss ibl e
LE M E N TARY ALG E BRA
E
3 18
The
giv e n
_
sys t e m
z
5
_
y
x+ y=
is t h en
Dividi n g t h e fi rst
i
e q ua t o n s
—
—
y
1
w e h ave :
x = 2 an d
( c) So lv e
=3
,
y
113
M ul t i pl yi n g ( 2 ) by 2
2
x
by t h e
sec o n
d
2
40
xy
12
+y
and a
ddi n g
—
x
N ow from ( 3 )
(4 )
and
bt ain
+1
i
the
+y
y
=
=
x+ y
—
x
y
giv e s
6,
=
( I)
d e riv e d
b t ai n
1 6,
4
i
y
fo rm t h e fou r sys t e m s
giv e n sys t em v iz :
wh i c h
=
+8
+8
x
+4
—
x
—
x+ y
-
—
x
—4
8
+4
and
lu t i o n s of t h e giv e n
x=
=
I V giv e s
—
x
x
are
.
,
=6
= 2 II
iv e s
, g
, y
,
y
o
f ro m ( 1 )
’
x
,
we
to the
iv a l e nt
—
x
so
,
5
e q uat o ns o f
to
24
2 xy + y
Or ,
x
o
—
8
2
x
—
y
=
d
5
=
2
x
+ y+ y 64
2
— 2
, y
se c o n
_
2
2 xy
Sys t em I
y
=
s
— 2 a nd
y
x+ y
x
2
_
s ys t e m
sys t e m
th e
e qu
2
x+y
and x
2
t o g et h e r
t o the two
x
—
n
a d x
C o m b i n i n g t h ese w i t h t h e
s ys t e m ,
uiv a l e n t
a nd
+5
x
eq
=2
,
6, y
+y
—4
y
y
=6
,
y
2
-
8
I I I giv es
H en ce , the
.
,
s ys t e m a r e
+ 6, + 2 ,
+ 2 , + 6,
—2
-
,
6,
an d
—
and
— 2
6,
.
b o th qu a d ra ti c e q u ati o n s s o t h a t t h i s
h
k
B
h
m
h
o
d
s
l
li
littl
n
h
l
i
m
it
s
s
e
t
f
or
t
i
b
oo
u
t
t
e
e
t
ro
b
e
m
es
a
e
b
e
o
d
t
e
p
y
in m os t o f it s p a r t s is so li ke t h a t for s ys t e m s m a d e up of one q uad r a ti c
Th e
t
s ys e m
( 1)
a nd
( 2)
ar e
,
.
S I MU LTA N E O US S Y S TE M S
3 19
it w ith i n t h e pupil s c o m pr eh en s i o n Th e
r e as o n t h e r e a re so m an y so l u ti o n s li es in t h e f a c t t h a t t h e g r a p h o f
( 1 ) is a c i r c l e a n d O f ( 2 ) a h yp e rb o l a s i n c e a c i r c l e a n d a h yp erb o l a
in g e n e r a l c ross e a c h o t h e r in fo ur po i n t s
an d on e
li n e ar
as
t o b r in g
’
.
,
,
.
.
,
fo ll o wi n g li st o f e xe rc i s e s w e Sh a ll i n c l u d e
s ys t e m s in t w o q u a d r a t i c s o f t h e t yp e O f t h e l as t
433
fe w
In t h e
.
a
.
E xe r c i s e 1 8 8
S o lv e t h e
fo ll o w i n g
sy s t e m s O f e q u a t o n s :
i
4r
2
2
13
=
n
4
m
3 56
6
+
2
8 mn = 1 60
2
x
xy
—
xy
73
rs
= 12
6 1 s + 93
2
2
2
’
= 49
3
25
22
x
2
+ 4 xy+ 36y
2
224
=
1 2xy 96
= 18
y
2
E xe r c i s e 1 8 9
1
.
Th e
s um O f
t h e i r s q u a re s is 2 1
tw o
n um b e r
( o r b)
.
sis
Wh at
7
a re
( or
a
)
n um b e rs
t he
Fi n d t w o n um b e r s t h e di ffe re n c e O f
( o r m) a n d t h e p ro d uc t o f wh o se s q ua r e s
2
33
,
,
?
whose
.
s um o f
the
and
is 784
s q ua re s
(or n )
1
c om b n e
d a r e a o f t w o s q u a re fi e ld s is 8 3;
a n d t h e s u m o f t h e i r p e r i m e t e r s is 2 00 r o d s
Wh a t
a r e a O f ea c h fi e ld ?
3
.
Th e
i
.
4
t he
5
.
Th e
di ffe r e n c e
.
of
t he
the
Th e
pr o d u c t
a re
s um o f
t he
s q u a r es o f
n um b e r s
di ff e r e n ce o f
is e q ua l t o t h e
n u m b e rs
?
t wo
t wo
is 5
n um b e rs
( o r q)
n um b e r s
c ub e O f
t he
.
is 9 1
Fi n d
is 2 8 ,
sm a
ll e r
.
a c r es ,
is t h e
(or
t he
a nd
is
n
um be r s
.
h a lf t h e i r
numbe r
.
Wh a t
E
3 20
6
Th e
.
L E M E N TARY ALG E BRA
of
a re a
t he
ili n g of a h a ll is 700 s q u a r e fe e t
fee t l e s s t h a n fo ur t i m es t h e w id t h
l e n gt h is s ix
Fi n d t h e di m e n s i o n s
it s
a nd
ce
,
.
.
7
The
.
is 2 1 0
8
(
sum O f
p)
or
I f t h e di m e n s i o n s
.
is 1 3
n um b e rs
the
Fi n d
.
n u m b e rs
t wo
(
or s
)
,
and
t h e i r pro du c t
.
an gl e w e re
o f a r ec t
ea c h
i n c re as e d
fo ot t h e a r e a w o uld b e 99 s q ua r e f ee t ; if t h ey w e re e a c h
di m i n i sh e d 1 f o o t t h e ar e a w o uld b e 63 s q u ar e fe et W h a t
a r e t h e di m e n s i o n s ?
1
‘
,
,
9
A
.
n um b er
w h i c h is 1 4 ,
the
n
u m b er
The
an d
.
is
e x pr es s e
su m o f
the
by 1 1
Fi n d
.
c om b n e
d by
t he
t he
of
two
fig ure s
s q u a r es o f
n u m b er
t he
the
digit s
s um
Of
e x c ee
ds
.
dj o i n i n g s q u a re fi e ld s is
90 0 s q u a r e r o d s a n d it r e q u i r es 1 50 r o d s o f f e n ce t o i n c l o s e
I f t h e y a r e s o Si t u at e d a s t o r e qu i r e t h e l e a st a m o un t
them
o f f e n c e w h at is t h e di m e n s io n o f e ac h ?
10
.
i
d
a rea
tw o
a
,
.
,
11
Th e
.
a re a O f a r e c t a n
di a g o n a l is 20 i n c h es
12
.
A
sq
13
th e
ca
.
.
.
A f a rm e r
i
.
.
b o ug ht 1 2
sh e e p a n d
4
ould b uy 3 m o r e
Fi n d t h e pr i c e O f e a c h
r
c
es
p
,
lv e s fo r $3 0
he
c
ca
lv e s
sh e ep
.
.
fi e ld
15
.
it s
two
s
At
fo r 324 t h a n
of
gr ou n d is
di m e n s io n s o f
.
Th e
and
fo r
.
T h e pe r i m e t er of a r e c t a n gul ar p i e c e
2 0 0 r o d s , a n d it s area is 1 5 a cres
Fi n d t h e
the
it s
,
sa m e
14
,
an d
g u l ar fi e ld co nt a i n s 2 7 0 s q u a re r o d s I f it
r o d s lo n g er a n d o n e r o d w id e r
it w ould c o n t a i n
ro ds m o re
Fi n d t h e di m e n s i o n s o f t h e fi e ld
u a re
.
.
rectan
w er e t w o
50
gl e is 1 92 s q u a r e i n c h e s
Fi n d t h e di m e n s io n s
a re a
id e s
.
h yp o t e n us e
is 2 1 6
of a
s q uare fe e t
.
ig h t t r i a n gl e is 30 fe e t
Fi n d t h e l e n g t h O f th e o t h e r
r
,
S UM M ARY O F DE FIN ITIO N S FOR
( D e fin iti ons w it h o u t pa g e n um b ers
RE FE RE N CE A ND RE VIE
ar e
on
W
l as t i n d i c a t ed )
pa ge
.
C H A PT E R I
m
factors of a n um b e r ar e it s ak er s b y m ulti p li c a ti o n ( P a g e
e qua ti on is a n e xpre s s i on of e qu ality b e t w e e n t w o e q ua l n um b ers
Th e
An
.
.
( Pa ge
The
n um
b er s
An
l
it
v a ue o f an y
t
r e pr esen s
l e tt er
.
a n um
in
b er
i
is t h e
e x pr e s s o n
n um
b er
or
( P a ge
unkn o wn n um b e r
is
a
l e tt er
wh os e
l
v a ue
in
an
e
qu a ti on
is t o
foun d ( P a ge
S ol v in g an e qua tion is fin d i n g t h e v a l ue o f th e unkn o wn n um b er o r
n umb ers in it
A n axiom is a s t a t em en t so ev i d en tl y t r ue t h a t it m a y b e a cc ep t e d
wit h o u t pr oo f
In pro bl em s o l v i n g th e n o ta tion is t h e r e pr e s en t a ti o n in a lgeb r a i c
s ym b o l s o f t h e un kn o wn n u m b e rs o f t h e pr o bl e m
( P a ge
T h e s ta t e m e nt is th e exp r ess i o n of t h e c o n d iti on s o f t h e pro bl e m
in o n e o r m or e e qu a ti o n s
be
.
,
.
.
-
.
.
C H A P T E R II
D ir e c t e d
po s itiv e
ign e d
( P a ge
numb er s or s
n um b er s a r e n um
b er s
w h o s e un it s
are
gativ e
T h e ab s ol ut e v al u e o f a n um b e r is t h e n um b e r o f un it s in it r e g a rd
l ess o f s ign ( P a g e
The
s ig n s m a y d e n o t e e ith er ope ra ti on s o r oppos ing qua l
an d
iti e s of n um b ers ( P a g e
Alg e b ra i c n ota ti on is a m e t h o d o f e xpr ess in g _n um b e rs b y fi gur e s
a n d l e tt e r s
( P age
An a lg eb r a i c e xpre s s i on is t h e r epr e s en t a ti o n o f an y n um b er in
a lg e b r a i c n o t a ti o n
A te rm is a n um b er expr ess i o n w h o s e par ts ar e no t separa ted b y
th e
or
s ign
A mon omia l is an expres s i on of one t er m ( P a g e
A po l yn om i al is a n expr es s i o n o f two o r mor e t e rm s
or n e
.
,
.
.
.
.
.
.
.
322
S UM M ARY O F
DE
F I N ITIO NS
3 23
A b inomia l is a po lyn o m ial of two t erms
A trinom ial is a po lyn om ia l of three t er m s
A coe ffi c i en t of a t er m is an y f a c to r o f t h e t e rm which sh o ws h o w
m an y tim es t h e o t h er f a c t or is t a k e n as an a d d e n d
S imil ar t e rm s ar e t e r m s w h i c h d o n o t d i ff er o r wh i c h d iff e r o n l y in
t h e ir n um er i c a l f a c to r s
.
.
.
,
.
D i ssim il a r t erm s
t e r ms tha t ar e n o t s im ilar
P ar tl y s imil ar t e rm s ar e t e rm s t h a t h av e a c o mm on f a c t o r
Th e v al ue o f an al g e b ra i c e xpr e s s io n is t h e n um b er it repr e s e n t s
wh e n so m e p ar ti c ul ar v al ue is a ss ign ed t o ea c h l e tt e r in t h e e xpr e s s i o n
( P ag e
ar e
.
.
.
C H A P TE R III
A d d iti on is t h e pr o c ess
n um b e r
( P age
o f u ni
tin g
tw o
or
m ore
n um b e r s
in to
one
.
be rs to b e ad d ed
Th e sum is th e n um b er o bt a i n ed b y ad d iti o n
Th e fun dam e n ta l l aw s o f a d dition ar e t h e l a w of or der ( t h e c o m
mu t a ti v e l a w ) a n d t h e l a w of grou ping ( t h e as so c i a ti v e l a w )
( P age
Th e l a w of or d er s t a t es t h a t n um b e r s m a y b e a d d e d in a n y o rd e r
T h e l a w of gr oupin g s t a t es t h a t a d d e n d s m a y b e g ro up e d in a n y w a y
Th e
a d d e n d s are
the
n um
.
.
,
.
,
‘
.
.
C H A P T E R IV
S ub tr a c tio n is t h e pr o c es s of fi n d i n g o n e o f t w o n um b ers wh e n t h e i r
s u m a n d t h e o t h er n um be r ar e k n o w n
( P ag e
T h e mi n u e n d is t h e n um b er th a t re pres e n ts t h e sum
T h e s ub tr ah e n d is t h e gi v e n a d d e n d
T h e difi e r e nc e o r r e m ain d er is t h e n um b er wh i c h a d d ed t o t h e s ub
t r a b e nd gi v es t h e m i n ue n d
T h e sym b o l s o f a ggr e g a ti o n a r e th e paren thesi s
th e br a ce
br a c ket
a n d t h e vi nc ul u m
( P age
.
.
.
.
C H AP TE R
Alg e b raic num b e r an d fun c ti on
Th e in d e pe n d e n t
d epe n d s
.
n um b e r
V
h av e th e
is t h e
s am e
n um b e r
on
m eani n g
.
( P a ge
wh i c h t h e
f un c ti on
E
3 24
A fun c ti o n
l
v a ue
is
LE M E N TARY ALG E BRA
a n um
b er t h a t
d e p en d s
o n so m e o
n um b e r
fo r it s
( P ag e
.
lg e b ra i c fun c ti on is a n u m b er w h o s e
n u m b er is e xpr e sse d in a lg e b r a i c s ym b o l s
An
th er
a
d e pe nd e n c e
on
an o
t he r
.
C H A P T E R VI
E qu a ti on s
are
of
t w o k i n d s , i de n ti ti es
an d
co nd i ti o n a l
equa tio ns
.
( P a ge
An
i d e n tity
m ay be
c
r ed u ed
is
an
e
t o th e
qua ti on
sam e
w it h l i ke m em b er s ,
fo r m
or
m e m b ers w h i c h
.
S ub s tituti on is pu tti n g a n u m b er sym b o l i n t o a n u m b er e xpr e ss i o n
in pl a c e o f a n o th e r w h i c h h a s t h e s a m e v a l u e
A n e qua ti on is s a ti s fi e d b y a n y n u m b er w h i c h w h en s ub s tit u t e d
fo r t h e u n k n o wn n u m b e r r e d u c es t h e e qu a ti o n to a n i d e n tity
A c on d iti on a l e qu a ti on is a n e qu a ti o n t h a t c an b e s a ti s fi e d b y o n ly
o n e o r b y a d efi n i te n u m ber o f v a l u e s o f t h e l e tt e r s in it
( P a ge
A r oot o f a n e q ua ti o n is a n y v a l ue of t h e un k n own n um b er th a t
s a ti s fi e s t h e e q u a tion f
Tran s pos iti on is t h e pr o c ess o f ch a n gi n g a t erm fro m o n e m e m b e r o f;
b y a d d i n g o r s u btr a c ti n g t h e s a m e n um b er
a n e q u a ti o n t o t h e o t h e r
in b o t h m e m b er s
( P a ge
.
,
,
.
.
,
.
C H A P T E R VII
G r aph in g m ea n s
b er p a irs r e l a t e d s e t s o f n um b e rs
a n d n um b e r l a w s b y p i c t ur e s a n d d i a gr a m s
( P a ge
A lin e ar e qu atio n is a n e qua ti o n in t w o un k n o wn s b o th w ith e xpo n en t
1
( P a ge
T h e graphi c al s ol u ti o n o f t w o lin e ar e qu a ti o n s is t h e p o i n t o f i n t e r
s e c ti o n o f th e gr a p h s o f th e e q u a ti o n s
( P a ge
S im u lta n e ou s e qu a ti on s ar e e qua ti o n s th a t c a n b e s a ti s fi ed b y th e
s a m e v a l ues o f x a n d y
A s ys te m of e qua ti o n s is t w o or m o r e e qu a ti o nsc o n s i d e r e d t o ge th e r
( P a g es 8 2 a n d
No n s im u ltan e o u s o r in c on s i s t e n t e qu a ti o n s a r e e q ua ti o n s w h i c h
c an n o t b e s a ti s fi e d b y a ny v a l ue s o f th e unk n o wn s ( P a g e
D e pe n d e n t e q u a ti o n s a r e e qu a ti o n s in wh i c h o n e o r m o r e c an b e
d e r i v ed f ro m a n o t h e r o r o t h e r s b y so m e s i m pl e a r it h m e t ic a l o pe r a tio n
( P a ge
r e p re se n
ti n g
n um
-
,
,
.
.
.
.
.
-
.
'
.
or
LE M E N TARY ALG E BRA
E
3 26
A polyn omia l is arran g e d wh e n t h e e xpo n en ts
d e c r e as e w it h ea c h s ucc ee d i n g t e r m
( P a ge
o f so m e
letter in c r ease
.
C H AP TE R X
T h e d e gr e e
lit er a l f a c t or s
T h e d e gr e e of
the
.
hi ghes t power
A
t erm is in d i c a t ed b y t h e s um of t h e e xpo n en t s o f
( P age
an e quati o n in o n e unk n o wn is t h e d e gree o f t h e
of a
of
n um b er
t h e un kn o wn
lin e ar e quati on is a n e qu a tio n wh i c h w h en
c l ear ed a n d s i m plifi ed is of t h e fi r s t d e gr ee
Ch e c kin g o r v e rifyin g a r oo t o f a n e q ua ti o n is t h e pr o c ess o f pro v i n g
t h a t t h e r oo t s a ti s fi e s t h e e qua ti o n
l
s im p e
e
quati on
.
or
,
,
,
,
.
.
C H A P T E R XI
D iv i s i on is t h e pr o c es s o f fi n d i n g on e
prod u ct a n d t h e other n u m b e r a r e kn o w n
.
Th e d iv i d e n d is t h e
of
t h e two
n um
b er s
the d
n um
b er s
t h e ir
wh e n
( P a ge
t o b e d i v i d ed
t
a n d r e pr e s e n s
n um
b y wh i c h we d iv i d e
b er
i v i d en d
Th e qu oti e n t is t h e r es u lt o f d i v i s i o n
A n y n um b er w it h a z er o e xpo n e n t e qu a l s
of
two
th e pr o d u
ct
.
Th e d iv i s or is t h e
fa c t o r
n u m b er
of
t
an d re pr e s e n s o n e
.
.
-
1
( P age
.
C H A P T E R XIII
A g en e ral numb er is a l e tt er o r o th er n um b er sym b o l th a t m a y r epre
sen
t
a n y n um
A f orm ul a
b er
.
( P a ge
i
gen er a l pr i n ci pl e o r r u l e in gen er a l
n um b e r s ym b o l s a n d in t h e f o r m o f a n e q ua lit y
( P a ge
T o s o l v e a f ormul a c om pl e t e l y is t o fin d t h e v a l ue o f e a c h g en e r a l
n um b er in t e r m s o f t h e o t h e r s
( P a ge
is
a n e x pr e ss o n o f a
,
,
.
.
C H A P TE R XI V
A
b er is o ne o f it s e q ua l fa c to r s ( P a g e
T h e s qu ar e roo t o f a n um b er is o n e o f t h e t wo e qu a l f a c t o rs
( P a ge
pr o d u c t is t h e n u m b e r
T h e c ub e roo t o f a n u m b e r is o n e o f t h e t hr ee e q u al fa c t ors
pr o d u c t is t h e n um b e r
t
roo
o f a n um
.
w h os e
.
.
w h o se
S UM M ARY O F
DE
F I N ITIO NS
3 27
C H AP TE R X V I
A
d iv i s or ,
comm on
d i v i s or
ch
or
f a c t or
( P a ge
c o mm on
,
of
tw o
or
m or e
num b er s
is
th em
o f t w o o r m o r e n u m b e r s is t h e
Th e h igh e s t c omm on f a c tor
pr o d uc t o f a l l t h e i r c o mm o n f a c t or s
A m ulti pl e o f a n um b e r is a n um b er t h a t is exa ctl y divi si bl e b y it
( P a ge
A comm on m ulti pl e o f t w o or m or e n um b ers i s a n um b e r th a t is
e xa c tl y d i v i s ibl e b y e a c h o f t h e m
T h e l ow e s t c ommon m u ltipl e
o f t w o or m or e n u m b e rs is
t h e pr od u c t o f a l l t h e ir difi er en t f a c t o r s
a n ex a c t
of e a
of
.
.
.
.
.
C
H AP TE R XVII
lg eb ra i c fra c ti on is t h e i n d i c a t e d d i v i s i o n in f r a c ti o na l form
of o n e n um b e r b y a n o t h er
( P a ge
T h e n um e rator is t h e n um b e r a b o v e th e li n e
T h e d e n om in a tor is t h e n u m b er b e l o w t h e li n e
T h e t erm s o f a fr a c ti o n a r e t h e n um era t or a n d d e n o m i n a t o r t o g e t h e r
A n i n t e g e r o r in t e g ra l n um b e r is a n u m b e r n o p a r t o f w h i c h is a
f ra c ti o n
Th e Sign o f a fra c tion is t h e s ign w r itt e n b e f o r e t h e li n e t h a t s e pa r a te s
t h e t erm s
( P a ge
R e d uc ti on of fr a c ti on s is th e pr o c e s s o f c h a n gi n g t h e i r for m w it h o u t
c h an gi ng t h e i r va l u es ( P a ge
A mix e d num b e r is a n um b e r o ne par t o f w h i c h is i n t e gr a l a n d t h e
o t h e r p ar t f r a c ti o n a l
( P a ge
A proper fr a c ti o n is a fr a c ti o n w h i c h c a nn o t b e r e d uc e d t o a w h o l e
o r a m i x e d n um b e r
A n im pr ope r fr a c ti o n is a f r a c ti o n w h i c h c a n b e r e d u c e d t o a wh o l e
n um b e r
o r a m i xe
d
T h e l ow e s t c omm on d e n om in a to r
o f t w o o r m o re f r a c ti o n s
is t h e l c m o f t h e i r d e n o m i n a t o r s
( P a ge
Th e r e c ipr ocal o f a fr a c ti o n is t h e fr a c ti o n i n v er ted
( P a ge
An
a
.
.
.
.
,
,
.
.
.
.
.
.
.
.
.
.
.
C
A litera l
ge n era l
i
equ a t on
n u m b er s
.
is
an
( P a ge
H A P TE R XVIII
e
qua ti o n
in wh i c h
t h er e
ar e
t wo
or
m o re
ELE M E N TARY ALG E BRA
3 28
A g e n eral
prob l e m is
n um b e rs
(P a ge
g e n er a l
a
b l em
pr o
al l
of
the
n u m b ers
in wh i c h
ar e
.
C H A P T E R XX
ti o o f o n e n u m b er t o a n o t h er
n u m b e r d i v i d ed b y t h e s e c o n d
( P a ge
Th e an t e c e d e n t is t h e fi r s t n u m b e r o f
Th e
is t h e
ra
quo ti e n t
of
fi rs t
th e
.
is t h e
c
s e o n d n u m b er
a ra
ti o
a nd
,
the
c o ns e
qu ent
.
t erm s o f a r a ti o a r e t h e a n t e c ed e n t a n d c o n s e q ue n t
Th e v a l u e of a ra tio is t h e quo ti e n t exp r e ss ed in it s l o w es t t e rm s
A ra ti o of gr e at er in e qua lity is a r a ti o in wh ic h th e a n t e c ed e n t is
gr ea t er t h an t h e c o n se qu en t ( P a g e
A rati o of l e s s in e qua lity is a r a ti o in w h i c h t h e an t e c ed e n t is l e ss
t h a n t h e c o n se q ue n t
A proport i on is an e qu a ti o n o f r a ti os ( P a g e
Th e t e rm s o f a p ro p or ti o n a re t h e t er m s o f t h e r a ti o s
Th e e xtr e m e s o f a pr o p o r ti o n a r e t h e fi rs t a n d fo ur t h t e r m s ; t h e
m e an s a r e t h e s e c o n d a n d t h i r d t e r m s
A m e an pr opor ti o na l is th e se c on d o f thr e e num b e rs wh i c h f o rm a
x zb
c o n ti n ue d pr o po r ti on a s a: in a zr
( P a ge
A th ir d proport i on al is t h e t h ir d o f thr ee n um b ers t h a t f or m a c o n
ti nn ed pro p o r ti on
A fourth pr oport i on al is t h e f our t h o f four n um b ers t h a t fo r m a
pr o p or ti o n
A v ar i ab l e num b e r o r a v ar i ab l e is a n um b er wh i c h in a gi v e n
f
f
r
d
o
n
m
h
d
l
i
c
s
i
i
r
o
b
m
o
s
u
s
a
a
v
e
e r e n t v a l ue s
e
( P a ge
y
p
A c on s tan t num b er o r a c on s ta nt is a n um b er t h a t is n o t a v a riab l z
O n e v a r i a bl e v ar i e s a s a n o t h e r if a s t h e y v a ry t h e ir r a tio r e m a i n s
Th e
.
.
’
.
.
.
.
.
.
,
.
.
,
,
.
,
,
,
.
,
,
,
c on sta n t
.
C H A P T E R XXI
Inv ol uti on is t h e pr o c e s s o f r a i s i n g
e xpo n e n t is a po s iti v e in t e g e r
( P age
n um b e r
a
to
po wer
a
wh o s e
.
Th e
ti m es
expo n e n
th e
t
n u m b er
The b a s e
of a
c i nd i c a tes
fa c t o r
t h e p o w er is th e n u m b e r w h i h
( th e ro o t o r b a s e ) is t ak e n a s a
of
p o w er
is t h e
E v ol uti on is t h e pr o c e s s
of
num b e r
find in g
w h i c h is
a ro o
t
h o w m a ny
.
i
r a se d
o f a n um
to
b er
.
a
po w e r
( P a ge
.
E
33 0
A
L E M E N TARY ALG E BRA
qua d ra ti c s ur d is a b i no m i a l s urd w h os e s ur d t e rm or
t er m s a r e o f t h e s e c o n d o r d e r ( P a g e
Con j u g a t e s ur d s a r e t w o bi n o m i a l q u a d r a ti c s ur d s t h a t d iff e r o n l y
b in omial
.
in t h e
s
An
ig n o f o n e o f th e t er m s
irra ti ona l o r ra d i ca l e q ua ti o n
.
i rra tiona l
r oot o f
is
,
,
t h e un k n ow n
n um b e r
’
.
an
e
( P a ge
q ua tio n c on t a in i n
g
an
27
C H A P T E R XXIII
A qua d ra ti c e qua ti on
is
an
e
q ua ti o n
c
the
of
s e on d
d e gr e e in t h e
b er ( P a g e
T h e c on s ta n t t e rm in a q ua d r a ti c e q ua ti o n is t h e t e r m t h a t d oes
n o t c o n t a i n t h e un kn o w n n um b er
A pur e qua d r a ti c e qua tion is a n e q ua ti on th a t d oes n o t c o n t ai n t h e
firs t po w er of t h e un kn o w n n um b e r
An a fi e c t e d qua d r ati c e qu a ti on is a n e q ua ti o n t h a t c o n t a i n s b o t h
th e fir s t a n d se c on d p o w e r s o f t h e u nkn o wn n um b e r
P ur e q u a d r a ti c s ar e o ft e n c a ll ed in c om pl e t e qu a dr a ti c s a n d a ffe c t e d
quad ra ti cs a r e a ls o o ft en c a ll e d c om pl e t e qua d rati c s
2
2
=
4ac
Th e d i s cr i m in an t o f t h e r oo t s o f a x + bzc + c 0 is b
( P a ge
u n k n o wn n u m
.
.
0
.
.
,
.
.
A
l
comp e x n u m b er
d e n o ti n g
in t h e
s
ign
of
a
b
er
f
o f th e
o rm
b er s
c om pl e x n umb e rs a r e c o m pl e x
t h e i m a gi n a r y t e r m
rea l n u m
C on j u g at e
is
n um
a
— 1
/
\
b
+
,
a
and
b
.
n um
b ers
wh i c h
d i ff er
.
C H A P T E R XXIV
A qua dr ati c e quati on in two v ar i ab l e s is a n e q u a ti o n in t w o v a r i abl es
o n e o r b o t h o f w h i c h ar e o f t h e se c o n d d e g r e e
( P a ge
A s ys te m of qua d rati c e qua ti on s is t w o o r m o r e q ua d ra ti c e q ua ti ons
c o ns i d er ed t o g e t h er
A s im ultan e ous s ys t em is a syst em in w h i c h a l l th e e qua ti o ns c a n
,
.
.
be
sa
ti s fi ed
b y th e
l
s a m e v a ue s o f
th e
i
l
v a r a b es
.
IN DE X
P A GE
Ab so lut e v a l ue o f a n um b e r
Ad d end s
Addin g in d i c a ted pr o d u c ts
s e v er a
l
pos iti v e
an d n e
22
27
27
ga
ti v e t er m s
s im il ar t e rm s
f
Ad d iti o n a n d subt r a c ti o no
fra c ti o ns
A dd iti o n
d e fin ed
l aw
of
of ord er
d iss im il ar
ana
l ys i s
29
29
t e rm s
30
27
32
s
i m il ar
48
237
of
,
A lgeb r a i c
for
t
i g
s ud y
n
i
e xpr es s o n
l
v a ue of an
fr a c ti o n
f un c ti on d e fin ed
fun c ti o ns
l a n gua g e
n o t a ti o n
n um b ers
1 6
-
24
26
1 79
51
50
8
24
2 1 , 50
9
Alt ern a ti o n pr o por ti on b y
An t e c ed en t
A ppro xim a t e v a lues o f s urd s
Arrang ed polynom i al s
,
t
B a l a n c e o f v a l ue s
B as e of a pow er
B in o m i a l d e fi n ed
q uad r a ti c s ur d
s ur d
t h eo re m
B i n o m i a ls
,
po w e r s
B ra c e
B ra c ke t
238
Affe c ted qua d ra ti c e qu a ti o n 2 8 2
28 5
so l v ed b y f a c t o r i n g
41 43
A gg r e ga ti o n sym b ol s o f
A lg eb r a d efi n ed
7
r easo n s
As s ump ti o n fo r i rra tio n a l
e q u a ti o n s
Axi o m po w er
188
29
m o n o m i a ls
o f po lyn o m i a l s
o f term s par tl y
pro po r ti o n b y
iti o n
A xi o m s
for
of
of a d d
m u lti p li c a ti o n
roo
27
fun d a m e n t a l l a w s o f
l a w o f gr o up i n g fo r
of
l aw
,
2 70
o f s ur d s
Ass o c i a ti v e
2 37
2 29
2 78
97
C he ck
Check
t es t
o n a lg e b r a i c w or k
d e fi n ed
C he c k in g
a d d iti o n b y s ub s tit u ti o n
a pr o bl e m
o r v er if y i n g a ro o t
Clear in g e q u a ti on s o f
fr a c tio n s
pr in c i p l e o f
a ppl i c a ti o n o f
Clo c k prob l em s
C oe ffi c i en t
o f a r ad i c a l
C omm on c om po und fa c t or s
C om m on d i v i sor
fr a c ti o n s quar e r oo t of
m ulti pl e
or
,
INDE X
33 2
PA G
C o m pa r i so n ,
e
li m i n a ti o n
C o m p l e t e d i v i so r
q uad r a ti c e q ua ti o n
a p pr o x i m
a t e v a l ue s
roo t s
n or m a l f o r m
roo t s o f t h e
q uad r a ti c s
C o m p l e ti n g th e s qu a re
by
2 13
D i s cr i m i n an t
2 56
D i s s i m il a r
28 7
D i s tr i b uti v e l a w
29 2
1
C o m pl e x n u m b e r
C o m po s iti o n pr o p o r ti o n b y
C o m p oun d e x pr e ss i o n s o per
,
2 92
D e fin iti o n
of a
of n
o
?
”
D e fi n iti o n s ,
D e gr ee
sum m ar y
o f an e
of
qu a ti o n
D e n o m in a t o r d e fi n ed
D e pe n d e n c e
f un c ti o n
D epe nd e n t e q ua ti o n s
D er i v i n g f o r m u l a s
D e t er m i n a t e e q ua ti o n s
D i ff e r en c e d e fin ed
of a
o f sa m e o d d
po w er s
q uar e s
D igit s A r ab i c
D i r e c t e d n u m b e rs
D i re c ti o n s for s o l v i n g e q u a
of
tw o
s
,
ti o n s
.
a
p o l yn o m i a l b y
a
po l yn o m i a l b y
2 98
i n d i c a te d
o f fr a c ti o n s
pro p o rti o n b y
s ig n l a w o f
D i v i so r c o m m o n
c o m pl e t e
d efined
p a r ti a l
D o ubl e m e a n i n g o f
27 6
2 98
2 29
24 1
mo no
a
po l y
i l
D iv i s i o n d e fi n ed
61
a
nom a
2 89
2 38
a
m ial
2 88
43
C o n d iti o n a l e q ua ti o n
C o n j u g a t e s ur d s
C o n j u g a t e c o m p l e x n um b e r s
C o n se quen t
C on s t an t
term o f a qu a d r a ti c
C ub e d e fi n e d
roo t
‘
a
28 2
,
a t io n s o n
t er m s
m o n o m 1 al b y
m on om i a l
2 87
'
of a
D iv id in g
288
a not
t
o f ro o s
D iv id end d e fined
of
=1
a
E
,
and
28 2
95
1 40
li m i n a ti o n d e fi n ed
86
b y a d d iti o n or s ub tr a c ti o n
87
b y c o m pa r i so n
2 13
b y su b s tit u ti o n
1 20
E lli ps e
3 10
E q u a ti o n d e fi n e d
11
d e gr e e o f
1 00
d e term i n a t e
85
h i s t o ry o f
5 9 60
i n d e t er m i n a t e
85
li n ea r
1 00
lit er a l a n d fr a c ti o n a l
1 98
qua d r a ti c
28 2
1 00
s i m p l e o r lin ea r
in q ua d r a ti c fo r m
2 93
11
m e m b er s o f a n
ro o t of
61
13
s o lv i n g a n
E q u a ti o n s d e pe n d e n t
83
i n c o n si s t e n t
83
E
,
,
,
INDE X
3 34
PAG
I n c o m pl e t e qua dr a ti c
e q ua ti o n s
t r i no m ia l s qu ar es
I n c ons i s t en t e qua ti o n s
I n d ep en d en t e qua ti o n s
I n d e pen d e n t n u m b e r
I n d e term i n a t e e qu a ti o n
I nd ex o f t h e r oo t
I n d i c a ti n g d i v i s i o n
m u lti p li c a ti o n
I n e q ua li ty r a ti o o f gr ea t er
E
Mi
28 2
15 1
8 4, 8 6
9
23 1
27 2
2 37
244
,
2 80
27 8
8
,
17
t for d i v i s i o n
for m ulti p li c a ti on
La w o f gr o upi n g for a d d iti o n
fo r m u lti p li c a ti o n
L aw o f o r d e r fo r a d d iti o n
fo r m ulti p li c a ti o n
L i n ear e qu a ti o n s
81
Lit er a l a n d fr a c ti on a l
e q u a ti o n s
L o w est c o m m o n d e n o m i n a t or
L o w es t c o m m on m ulti p l e
o f m on o m i a l s
o f p o l yn o m i a l s
o f exp o n e n s
.
,
i g o f e xp o n e n t 0 1 0 8
t ype for m s
M e a n pr o p or ti o n a l
M e a n s o f a pro por ti o n
M eas ur in g is r a tio in g
M em b ers fi r s t a nd se c o n d
,
an e n
tir e
s ur d
M o n o m i a l d e fi n ed
e,
o mm o n
o f s ur d s
ign l aw o f
M u lti p li er d e fi n e d
n e g a ti v e
M ultip lyin g m on om i a l
a po l yn o m i a l b y a m on o
s
2 65
L a n gu a g e usi n g a lg eb r a i c
L e tt er s r epr es en ti n g n um b e rs
-
to
l ow es t c o m m on
M ulti pl i c a n d d e fin ed
M ulti p li c a ti o n d e fi n ed
i n d i c a t ed
l a w o f e xpo n e n t s fo r
o f fr a c ti o n s
23 1
n u m b er
s ur d
85
9
,
d
xe
M ultipl c
~
ti o o f l e s s
s ign s o f
I nv er s i o n pro po r ti o n b y
I nv o luti o n
I rr a ti on a l e qua ti o n s a ssum p
ti on for
e qua ti o n s in o n e un k n o w n
ea n n
Mi
‘
50
2 50
ra
M
s ur d
83
,
La w
m b er
x ed n u
,
10 8
93
29
a
po l yn o m i a l b y
a
po ly
i l
n om a
t
qua d r a ti c 2 98
N e ga ti v e m ulti pli er
91
N o n s i m u lt a n eo us e qu a ti o n s
83
N o t a ti on
7
a lg eb r a i c
24
in pr o bl em so lv i n g
15 16
s ys t e m o f
24
N um b er
13
i m a gi n ar y
253
i n d e pe n d e n t
50
irr a ti on al
264
m i xe d
1 84
o f r oo t s
252
r a ti o n a l
2 64
re al
253
N a t ur e
o f ro o s o f
-
94
29
94
-
1 00
1 98
187
175
175
1 76
2 63
1 30
234
232
2 30
11
,
'
u n kn o wn
N um b ers d ir e c t ed
g e n er a l
o f ar it hm e ti c
,
13
21
1 23
20
INDE X
PAG
N u m b ers , pos iti v e
and
ne
ga
ti v e
t
r epr es e n e d
335
E
P r i n c i pl e
l ti on
P r i n c i p l es o f pr o po rti o n
P r o b l e m g e n er a l
so l v i n g a
qua d r a ti c s
P ro bl e m s in s um ul t a n e o us
e q ua ti o n s
t hr ee o r m or e unkn o wn s
,
N um er a t o r d e fi n ed
O d d p o w ers
t
O per a ti o n s o n c o m p o und e x
pr e s s i on s
O ppos it e q u a liti e s o f a lg e
b ra i e n um b ers
O r d er o f a r a d i c a l
s e c o n d a n d t h ir d
ro o
t w o un kn o w n s
P r o bl e m - s o lv
P ara b o l a
P a r e n t h es i s
d e fin e d
P ar ti a l d iv 1 s o r
i m ilar term s
P a s c a l s t r i a n gl e
P i c t ur i n g f un c ti o n s
P o l yn o m i a l a r r a n g e d
d e fin ed
s q u ar e ro o t o f a
P o l yn o m i a l s f a c t o r ed b y
gr o upi n g
P o s iti v e a n d n e g a ti v e n u m
’
,
b e rs
pr o b l e m s in
P o w er d e fi n ed
of a
of a
of a
of
th e
fr a c ti o n
m o n om ial
pr o d u c t
s urd
i
gg es ti o n s
P r od u c t d e fi n e d
o f su m a n d
d i ffe r e n c e
of
b er s
o f t w o bi n o m i a l s w it h a
c o mm o n t e r m
o f t w o n um b ers e q u a l t o
Pr od u c t s ign o f t h e
P r o d uc t s h o w wr itt en
Pr o p e r fr a c ti o n
P r o po r ti o n d e fin e d
b y a d d iti o n
b y a d d iti o n a n d s ub
t ra c ti o n
b y a lt er n a ti o n
b y c o m po s iti o n
by d i v i s io n
b y i nv er s i o n
b y s ub t r a c ti o n
e x tr e m e s a n d m e a n s o f
pr i n c i p l es o f
P r o p or ti o n a l m ea n
f o ur t h
t h ird
P r o po rti o n a lit y t e s t o f
P ur e qu ad r a ti c e qu a ti o n
n o rm a l f o r m o f
so l v ed b y f a c t o r i n g
n um
0
,
,
c
s e on d
t hir d
of
su
,
a x om
P o w er s
,
,
s
b as e
in g
on
t wo
P a r tl y
o f ev o u
t
a n d r oo s
bi n o m i a ls
P r i m es
c i pt s
a n d sub s r
Pr i n c i p a l
r oo
t
Q ua d r a ti c
t
e
qua ti on
t
,
n a ur e o f r oo s o f
f c t ed
af e
I NDE X
3 36
Ro o t pr i n c i pa l
s q uare o f a d e c i m a l
s q uar e o f n um b er s
R o o ts im a gi n a ry
o f co m p l e t e q ua d r a ti c
Q ua d r a ti c s pur e
Q ua d ra ti c e q ua ti o n s
,
,
l
s o v ed
by
Q uad r a ti c
,
for m ul a
,
s ur d
,
bi n o m i a l
t r i nom ia l
Q ua li t y o f n um b er
Q uo ti en t d e fi n e d
Ra d i c a l c o e ffi c i e n t
,
se ts o f
S a ti s fyi n g a n e qua ti o n
S e c o n d n um b e r
of
d e fin ed
d e gr e e
r ed u
p o w er
Se t s o f ro o t s
o r o r d er o f
c ti o n
of
ign
R a d i c a nd
R a ti o a n t e c e d a n t o f
c on se que n t o f
d e fin ed
o f gr ea t er a n d l es s in e q u a l i
s
,
23 1
ty
Ra ti o n a l n um b e r
R ea l n um b er
2 64
25 3
t
2 53
r oo s
Re aso n s fo r s t ud yi n g a lg eb ra
R e c i pr o c a l o f a n u m b e r
R e d uc ti on o f fr a c ti o ns
o f i m pr o pe r f r a c ti o n s
of
m i xed
o f r ad
i
e xpr ess o n s
i c als
o f s ur d s
to
s a m e o r d er
g a ti e n
R e v i ew o f f a c t or i n g
Roo t o f a fr a c ti o n
a n e q ua ti o n
b er
a
po w er
a
pr o d u c t
c ub e a n d s q u a r e
i nd ex
of
1 93
181
184
the
27 1
35
to
a
fa c
t or
Si m u lta n eo us e qu a ti on s
fin
1 86
2 67
Re m a in d er in s ubtr a c ti o n
Re m ov i n g s ym b o ls of a g g r e
a n um
1 —6
Sig n ed n um b e rs
Sign l a w o f d i v i s i o n
o f m u lti pl i c a ti o n
Sig n o f a fr a c ti on
c on ti nua ti o n
n e g a ti v e n um b e r s
po s iti v e n u m b e r s
pr od uc t
quo ti e n t
r ea l r oo t
Sign s o f i n e qu a lit y
Si m il ar t er ms
Si m il ar with r es pe c t
de
ed
82
t
Si m ult aneo us Sim ple e qua ti o n s
S o luti o n o f e qua ti on s b y fa c
tor i n g
f orm ul as
So lv i n g an e qua ti on
a pr obl e m
s ys e m o f
e
qua ti on s li n e ar
1
in
:v
and
-
85
1
2 15
y
pr obl em s
t h e e q ua ti o n
So lv i n g o n e l e tter
gr aph i call y
82
88
15
e
qua ti on s
INDE X
33 8
PAG
Typ e form s m ea n i n g o f
Type form s i n t er pr e t ed
-
,
V a l ues
of
-
,
n um
i m at e
qu a d r a ti c s
a pprox
,
t
o f ro o s
V a l ues o f s ur d s
V ar i ab l es
V a r i a ti o n
dir e c t
l
e x a m p es o f
Un k n o w n
,
b er
ign o f
V ar i e s a s
V i n c u l um
s
V a l ue o f
an a
lg eb r a i c
pr ess i o n
l e tt er
qua d r a ti c
ex
,
or
d i r e c tl y
as
o f any
of a
s ur d
Z ero - e xp o n e n t , m ea n i n g
E