1.4 'til,,
in TwoVariables
LinearEquations
39
I
I
f)
Replace W and A with your actual weight and height. Now draw a graph showing how B dependson height for suitablevalues of H.
s)
For a fixed weight and age,doesthe basic energyrequirementincreaseor decrease
for taller persons?Increases
h)
For the three graphs that you have drawn, explain how to determine whether the
graph is increasingor decreasingfrom the formula for the graph.
\
*ir
I
In Section 1.3 we graphedlines, including horizontaland vertical lines.We learned
that every line has an equation in standardform Ax * By : C. In this section we
will continueto study lines.
1x2, !t)
Slope of a line
A road that has a 5o/ograde rises 5 feet for every horizontal run of 100 feet. A roof
that has a 5-12 pitch rises 5 feet for every horizontal run of 12 feet.The grade of a
The steepnessor slope of
road and the pitch of a roof are measurementsof steepness.
a line in the xy-coordinatesystemis the ratio of the rise (the changein y-coordinates)
to the run (the changein x-coordinates)betweentwo points on the line:
_
slooe
'
Figure 1.27
Definition:Slope
changeinl-coordinates_ rise
changein r-coordinates
mn
If (x1,y1) and (*2,y) are the coordinatesof the two points in Fig. l.27,then the rise
rs lz - y1 andthe run is x2 - x1:
Theslopeofthe1inethrough(xr,,1J'1"z,yz)withx1*x2is
lz-lr
xz-xt
i
Note that if (x1, y) and (*t, yr) are two points for which xl : x2 then the line through
them is a vertical line. Sincethis caseis not includedin the definition of slope,a vertical line doesnot havea slope.We also saythat the slopeof a vertical line is undefined.
If we choosetwo points on a horizontal line then y1 : 12dnd lz - lt : 0. For any
horizontalline the rise betweentwo points is 0 and the slopeis 0.
40
Chapter 1 ruffiffi
Equations,
Inequalities,
and Modeling
e**rr./t
E Finding the stope
In eachcasefind the slopeof the line that containsthe two givenpoints.
a. (-3,4), (-1, -2)
b . ( - 3 , 7 ) , ( 5 , 7 ) c . ( - 3 , 5 ) ,( - 3 , 8 )
$olution
a. Use(*vyr) : (-3, 4) and(*z,yz) : (- l, -2) in theformula:
-2-4
.
rl-Vr
s l o p e: ' r ; :
lxq, y q)
_l _ (_3)
The slopeof the line is -3.
b. Use (*v y) : ( -3, 7) and (rr, yr) :
.
vt-lr
-6
- -3
2
(5, 7) in the formula:
7-7
slope:-;=:5_(_"r:8:o
(xs,y)
:
0
The slope of this horizontal line is 0.
c. The line through (-3, 5) and (-3, 8) is a vertical line and so it does not have a
slope.
I
(xvy)
Figure1.28
The slope of a line is the same number regardless of which two points on the
line are used in the calculation of the slope. To understandwhy, consider the two triangles shown in Fig. 1.28. These triangles have the same shapeand are called similar triangles.Becausethe ratios of correspondingsidesof similar trianglesare equal,
the ratio of rise to run is the same for either triangle.
Point-StopeForm
Supposethata line through(xr, y1) hasslopem. Everyotherpoint (r, y) on the line
mustsatisff the equation
=:*
becauseanytwo pointscanbe usedto find the slope.Multiply both sidesby x - x1
to get! - lr : m(x - xr),whichis thepoint-slopeform of the equationof a line.
Theorem:
Point-SlopeForm
In Section 1.3 we started with the equation of a line and graphed the line. Using
the point-slope form, we can start with a graph of a line or a description of the line
and write the equation for the line.
1.4 wmtur LinearEquations
in TwoVariables
e-*r"^fb
41
A The equation of a tine given two points
In each case graph the line through the given pair of points. Then find the equation
of the line and solve itfor y if possible.
c . ( 3 ,- 1 ) , ( 3 , 9 )
^ . ( - 1 , 4 ) , ( 2 , 3 ) b . ( 2 , 5 ) ,( - 6 , 5 )
$olutisn
246810
a. Find the slope of the line shown in Fig. t.29 as follows:
!z-!t
"'-*r-x1
Figure1.29
3-4
-l
1
2-(-l)
3
3
Now use a point, say(2,3), and m - -{
!-lr:m(x-xr)
- -; L
y
3
(x -
;2
y - 3 : -r,
+
I
-:7 4
-5 -4 -3 -2
*
i*
2)
in ttt. point-slope form:
The equation in point-slope form
i
ll
The equation solved fory
T
b. The slope of the line through (2,5) and (-6, 5) shown in Fig. 1.30 is 0. The
equationof this horizontal line is y : 5.
c. The line through (3, - 1) and (3, 9) shown in Fig. 1.31is vertical and it doesnot
I
have slope. Its equationisx : 3.
Stope-InterceptForm
Theline y : mx * bgoesthrough(0,b) and (7,m t b). Betweenthesetwopoints
the rise is m andthe run is 1. So the slope is ru. Since (0, b) is the y-intercept and m
is the slope, y : mx * b is called slope-intercept form. Any equation in standard
form Ax * By : C canbe rewritten in slope-interceptform by solving the equation
fory provided B + 0.lf B : 0, then the line is vertical and has no slope.
F i g u r e1 . 3 1
I
I
i
Theorem:
Slope-lnterceptForm
i*m;;ry.*ffffi;;ry'i
Every nonvertical line has an equation in slope-intercept form.
i:'
!l
r1..",,,
-
If you know the slope and y-intercept for a line, then you can use slope-intercept
form to write its equation. For example, the equation of the line through (0, 9) with
Chapter1 ffiwii# Equations,
Inequalities,
and Modeling
slope4 is y - 4x * 9. In the next examplewe use the slope-interceptform to determine the slope andy-intercept for a line.
E Find the stopeand y-intercept
Ar*r""N,
Identi$r the slope andy-intercept for the line 2x - 3y : 6.
$otution
First solve the equationfory to get it in slope-interceptform:
2x-3y:6
-31,y
: -2x -t 6
2
!:1x-2
Figure1.32
So the slope is ] and they-intercept is (0, -2).
I
Using Stopeto Graph a line
Slope is the ratio ffi that results from moving from one point to another on a line. A
positive rise indicatesa motion upward and a negativerise indicatesa motion downward. A positive run indicates a motion to the right and a negative run indicates
a motion to the left. If you start at anypoint on a line with slope thenmoving up
),
1 unit and2 units to the right will bring you back to the line. On a line with slope -3
F i g u r e1 . 3 3
ot t',moving
down 3 units and 1 unit to the right will bring you back to the line.
e**r"./t
A Graphinga tine using its stope
and y-intercept
Graph eachline.
a.y:3x-
I
b.
2
1x+4
$olution
- t hasy-intercept
(0, -1)andslope: or
(0, -1)
J. Startingat
we obtain a secondpoint on the line by moving up 3 units and I unit to the right.
So the line goesthrough (0, - 1) and (1,2) as shown inFig. 1.32.
-!, * 4 hasy-i ntercept(0,4) and sl ope -?otf. . S tart ingat
b . T h e \i n ey :
(0, 4) we obtain a secondpoint on the line by moving down 2 units and then 3 units
to the right. So the line goesthrough (0, 4) and (3, 2) as shownin Fig. 1.33.
I
a. The liney:3x
Graphsof y=vv1;s
, Figure1.34
As the .r-coordinateincreaseson a line with positive slope, the y-coordinate
increasesalso. As the x-coordinate increaseson a line with negative slope, the
y-coordinatedecreases.Figure 1.34 showssome lines of the form y : mx with positive slopesand negativeslopes.Observethe effect that the slope has on the position
of the line.
t;
L-
1.4 ffiffii4rniLinearEquationsin Two Variables
l,
The Three Formsfor the Equationof a Line
There are three forms for the equation of a line. The following strategywill help you
decidewhen and how to use theseforms.
STRATEGY
Findingthe Equationof a Line
1.
2.
3.
4.
5.
A-*rr"/t
tr $tandardform using integers
Find the equationof the line throurn (O,J) *i.n slope]. Wrlt. the equationin
standardform usingonly integers.
Sotution
Sincewe know the slopeandy-rntercept,
startwith slope-intercept
form:
11
-
V : -x I
"2J
Slope-interceptform
- - xl 1 * v : -/a
LJ
- 6 /( - t 1* * y\) :
-6
'
I
:
.
)
L
J
Multiply by -6 to getintegers.
J
3x- 6y:
-2
Standardform with integers
Any integral multiple of 3x - 6y : -2 would also be standard form, but we usually use the smallestpossiblepositive coefficient for x.
T
ParattetLines
Two lines in a plane are said to be parallel if they have no points in common. Any
two vertical lines are parallel, and slope can be used to determine whether nonvertical lines are parallel. For example, the lines y : 3x - 4 and y : 3x f I are parallel becausetheir slopesare equal and theiry interceptsare different.
44
Chapter1 www
Inequalities,
Equations,
and Modeling
Theorem:ParallelLines
Two nonvertical lines in the coordinate plane are parallel if and only if their
; slopes are equal.
A proof to this theorem is outlined in Exercises97 and98.
e&rr".f/t
of paratteLtines
tr Writing equatiCIns
Find the equationin slope-interceptform of the line through (1, -4) that is parallel
toy:3xt2.
Sohltion
Since y : 3x * 2 has slope 3, any line parallel to it also has slope 3. Write the
equationof the line through ( 1, -4) with slope 3 in point-slope form:
l0
Y-(-4):3(xy+4:3x-3
-10
!:3x-7
llEl
1)
Point-slopeform
Slope-interceptform
E'tl
_10
Figure1.35
Theline y : 3x - 7 goesthrough(1, -4) andis parallelto y : 3x -f 2.
ru The graphsof !1 : 3x 7 and y2: 3x + 2 in Fig. 1.35 supportthe
I
answer.
PerpendicutarLines
Two lines are perpendicular if they intersectat a right angle. Slope can be used
to determine whether lines are perpendicular. For example, lines with slopes
such as 213 and -312 are perpendicular.The slope -312 is the opposite of the
reciprocal of 213. In the following theorem we use the equivalent condition that
the product of the slopes of two perpendicular lines is -1, provided they both
have slopes.
Theorem:
PerpendicularLines
i
l|*o mes wi-lhslon.*--: d
mz oft ?atpa'ndi,cular,if
and onV ,rt:,**,r: -
l.
Figure1.36
PROOF The phrase "if and only if" means that there are two statementsto prove.
First we prove that if /1 with slope m1 and 12with slope tn2 zre perpendicular, then
rltrltt2 - - l. Assume that m1 > 0. At the intersectionof the lines draw a right triangle using a rise of ml and a run of 1, as shown in Fig 1.36.Rotate /1 (along with the
right triangle) 90 degrees so that it coincides with /2. Now use the triangle in its
new position to determine thatm2 : +
ot ffi!fl2 - - 1.
that the
The secondstatementto prove is that lflfli2 - - 1 or fti2 : +implies
perpendicular.
lines are
Start with the two intersecting lines and the two congruent
right triangles,as shown in Fig. 1.36.It takesa rotation of 90 degreesto get the vertical sidemarkedmlIo coincidewith the horizontal sidemarkedmr Sincethe triangles
are congruent, rotating 90 degreesmakes the triangles coincide and the lines coincide. So the lines are perpendicular.
1.4 wst'
e-*r""/.
LinearEquations
in TwoVariables
writing equatitrn$ sf, petrpffindicerl"ar
tinss
Z
Find the equation of the line perpendicularto the line 3x - 4y : g and containing
the point (-2,I). Write the answerin slope-interceptform.
$otrt:tiCIn
Rewrite 3x - 4y : 8 in slope-interceptform;
lY:-3x+8
.,
"v4: - x
J
-
2
S l o p eo f t h i s l i n e i s 3 / 4 .
Sincethe product of the slopesof perpendicularlines is - 1, the slope of the line that
we seekis -413. Use the slope -413 and the point (-2,l) in the point-slopeform:
4
y-l:--(x-(-2))
3
v -
10
, 4-9 - - x
|
- a
JJ
45
v - --x - -
J ' t a
JJ
-15.2
The last equation is the required equation in slope-interceptform. The graphs of
thesetwo equationsshould look perpendicular.
ru
-10
Ifyougraph
lr:1.
- 2and!2: -+, - withagraphingcalculator,the
f
graphs will not appearperpendicularin the standardviewing window becauseeach
axis has a different unit length. The graphs appearperpendicular in Fig. I .37 becausethe unit lengthswere made equal with the ZSquarefeatureof the TI-83.
I
Figure'1.37
Apptications
In Section 1.3 the distanceformula was usedto establishsome facts about geometric
figures in the coordinateplane. We can also use slope to prove that lines are parallel
or perpendicularin geometric figures.
A**r"'fl, E
|
2
3
4
B(5,0)7 8
x
The diagonatscf a rhombus
are perpendicular
Given a rhombus with vertices (0, 0), (5, 0), (3,4), and (8, 4), prove that the diagonals of this rhombus are perpendicular.(A rhombus is a four-sided figure in which
the sideshave equal length.)
$ottltisn
Figure1.38
Plot the four points A, B, C, andD as shown in Fig. 1.38.You should show that each
side of this figure has length 5, verifying that the figure is a rhombus. Find the slopes
of the diagonalsand their product:
|TIAC :
4-0
8-0
lTllC'
0-4
tTlBD :
|TIBD :
l.
;(-2)
:
-l
46
ChaPter 1 ffi,,ii
E q u a t i o n sI,n e q u a l i t i e sa ,n d M o d e l i n g
Since the product of the slopes of the diagonals is -1, the diagonals are perpenI
dicular.
If the value of one variable can be determined from the value of anothervariable, then we say that the first variable is a function of the secondvariable.Because
the area of a circle can be determinedfrom the radius by the formula A : nr2, we
say that A is a function of r. If y is determined from x by using the slope-intercept
form of the equation of a line y : mx * b, then y is a linear function of x. The
formula F : ?C + 32 expressesF as alinear function of C. We will discussfunctions in depth in Chapter 2.
A"*r""fe tr A ffinearfnrnction
From ABC Wirelessthe monthly cost for a cell phone with 100 minutes per month is
$35, or 200 minutesper month for $45. SeeFig. 1.39.The cost in dollars is a linear
function of the time in minutes.
a. Find the formula for C.
b. What is the cost for 400 minutes per month?
100
200 300 400
Time (in minutes)
500
F i g u r e1 . 3 9
Sclution
a. Firstfind the slope:
cz-cr
m:
t2 - ^
:
45-35 : l0 :
o'lo
2oo- loo
roo
The slopeis $0.10per minute.Now find b by usingC:35,t:
form C : mt I b:
m : 0.10in theslope-intercept
35:0.10(100)+b
35:10+b
25:b
So the formula is C
100, and
1.4 swri'ii LinearEquations
in TwoVariables
47
b. Uset : 400inthe formulaC : 0.10t+ 25:
C : 0 . 1 0 ( 4 0 0 )+ 2 5 : 6 5
r
Thecostfor 400minutesper monthis $65.
Note that in Example 9 we could have used the point-slope form
C - C r : m(t - t) to gettheformul aC :.0.10r + 25. Tryi t.
The situation in the next example leadsnaturally to an equation in standardform.
A-*r",/t
@ Interpreting stope
A manager for a country market will spend a total of $80 on apples at $0.25 each
and pearsat $0.50 each.Write the number of applesshe can buy as a linear function
of the number of pears.Find the slope and interpret your answer.
Solution
Let a representthe number of applesand p representthe number of pears.Write an
equation in standardform about the total amount spent:
0.25a+0.50p:80
r
025a
:::;:::
S.'vef.ra
The equationa : 320 - 2p or a : -2p + 320 expressesthe number of applesas
a function of the number of pears.Sincep is the first coordinateand a is the second
the slope is -2 applesper pear. So if the number of apples is decreasedby 2, then
the number of pearscan be increasedby I and the total is still $80. This makessense
I
becausethe pearscost twice as much as the apples.
l
I
For Thought
T r ue or F als e ?Ex p l a i n .
1. The slopeof the line through(2,2) and(3, 3) is 312.e
7. The slopeof the line y - 3 - 2x is 3. p
2. The slopeof the line through(-3, 1)and(-3,5)is 0. p
8. Every line in the coordinateplane has an equationin
standardform. t
3. Any two distinct parallel lines have equal slopes.r
4. The graph of x : 3 in the coordinateplane is the single
point (3, 0). r
5. Two lines with slopesm1andffi2&raperpendicularif
tltt : -llm2.t
6. Every line in the coordinateplane has an equationin
slope-interceptform. r'
9. The line y : 3x is parallel to the line y :
-3x. F
10. The line x - 3y : 4 containsthe point ( l, - 1) and has
slope t lZ. r
48
and Modeling
lnequalities,
Equations,
Chapter1 *xm
Find the slope of the line containing eachpair of points. (Example1)
23. Y : - 2 x + 4
4
v
I
)
t. (-2,3),(4,5)i
( - 1 , 2 ) ,( 3 ,6 ) I
3 . ( 1 , 3 ) ,( 3 , - 5 ) - 4
2
4. (2, -r), (5,-3) - ;
5 . (5 ,2 ) , ( - 3, 2) o
6. (o,o),(5,0)o
lt r\ /t l\
t' (r'
d'V't)'
8.
J
(r t\ - ^
r-1
|
2
/
'
\
o' :/
3
'
\
I
J
10. ( - 7 , 2 ) ,( - 7 , - 6 ) No slope
9. (5, - l), (5, 3) No slope
Find the equation of the line through the given pair of points. Solve
it for y if possible. (Example2)
1 1 .( - 1 ,- l ) , ( 3 , 4y) : 1 . . :
'7
13. (-2.6). (4. -l)
'Y- 6^'
1 5 . ( 3 , 5 ) ,( - 3 , 5 ) y : s
17. (4, -3), (4, 12) x : a
ll
3
(3,5)y ::, . +
12.(-2,1),
: -1" .
14.(-3,5),(2,I)y
+
1 6 . ( - 6 , 4 ) , ( 2 , 4 )y : +
1 8 . ( - 5 , 6 ) , ( - 5 , 4 )x : - s
Write an equation in slope-interceptformfor each of the lines
2 and 3)
shown. (Examples
1
19.y:'ir-t
20.Y:-xt2
Write each equation in slope-interceptform and identifi the slope
and y-intercept of the line. (Example3)
27,3x - 5Y: l0
, : 1 * - 2 , 1(,0-,z )
"5)
2e.y-3-2@-a)
y : 2 x - 5 , 2 , ( 0 ,- 5 )
t.
3r.y*1:t(*-(-3))
I I I .r)
v: t* - t,t' r\"'tl
3 3 .y - 4 : 0
y : 4 , 0 ,( 0 , 4 )
35. y - 0.4:
2 1 .v :
53
_-r2n'z
2 2 .y :
4
4
-1x +-
J
28. 2x - 2Y : I
y: x ),t,,(0,-tl2)
y + 5: -3(x - (-l))
-3, (0,-8)
! : -3x - 8,
3.
y-2:-;(x+5)
3 ( -t l l \
lt
- 3
y
-;- -
;.
-t'\.0'
34. - y - 1 5 : 0
y : 5 , 0 ,( 0 ,5 )
)
-2.6)
0 . 0 3 ( x - 1 0 0 )y : 0 . 0 3 x- 2 . 6 , 0 . 0 3( ,0 ,
-0.26)
36. y + 0.2 : 0.02(x - 3) y : 0.02x- 0.26,0.02'(0,
Use the y-intercept and slope to sketch the graph of each equation.
(Example4) T
37.y:I"-,
3 9 .Y :
-3x * 1
4r.y: -1* - t
)
38. .v : : x * 1
a
J
4 0 .v : - x * 3
a
J
4 2 .y = - ; x
z
tDue to spaceconstrictions,answersto theseexercisesmay be found in the complete Answers beginning on page A-l in the back of the book'
1.4 mreg+; Exercises
43.x-!:3
4 4 .2 x - 3 y : 6
Find the value of a in each case. (Examplesl 7)
45.y-5:0
4 6 .6 - y : 0
75. The line through (-2,3)
Find the equation of the line through the given pair of points in
standardform using only integers. See the strategyfor finding the
equation of a line on page 43. (Example5)
47. (3,0) and (0, -4)
4x-3Y:12
4 9 . ( 2 , 3 ) a n d( - 3 , - l )
4x-5Y:-7
5 1 . ( - 4 , 2 ) a n d( - 4 , 5 ) x : - 4
s s(. 2 , ) , ( -
),-r)
48. (-2,0) and (0, 3)
3x-2Y:-6
50. (4, -l) and(-2, -6)
5 x- 6 Y : 2 6
5 2 . ( - 3 , 6 ) a n d( 9 , 6 ) y : 6
'0.G,-,),(-',)
l6x - l5Y: )/
100x+184y:-477
49
and (8, 5) is perpendicularto
Y:Qx+2.-5
76. The line through (3, 4) and (7, a) has slope213. ZOIZ
77. The line through (-2, a) and (a, 3) has slope - | 12. S
78. The,line through (-1, a) and (3, -4) is parallel to y : ax.
Either prove or disprove each statement. (Jsea graph only as a
guide. Yourproof should rely on algebraic calculations. (Exampte8)
79. The points (-I,2), (2, -I), (3, 3), and (-2, -2) arethe vertices of a parallelogram.T
'u(-;,1),(;,-)
".(;,tJ,(-+,+)
80. The points (- 1, 1), (-2, -5), (2, -4), and (3, 2) arethe vertices of a parallelogram.T
Find the slope of each line described. (Examples6 and 7)
81. The points (-5, - 1), (-3, -4), (3,0), and (1, 3) arethe vertices of a rectangle.T
20x+42Y:3
3x-50Y:-ll
57. Alineparallelto y : 0.5x - 9 0.s
82. The points (-5, - 1), (1, -4), (4,2), and (- 1, 5) arethe vertices ofa square.F
58. A line parallel to 3x - 9y : 4 t lz
59. Alineperpendicularto3y - 3x:
83. The points (-5, 1), (-2, -3), and(4,2) arethe verticesof a
right triangle. r
J -l
60. A line perpendicularto2x - 3y : 8 -zlz
84. The points (-4, -3), (1, - 2),(2,3),and (-3, 2) arethevertices of a rhombus. T
61. A line perpendicularto the line x : 4 0
62. Alineparallel toy : 5 6
ru
(Jsea graphing calculator to solve eachproblem.
Write an equation in standardform using only integersfor each of
the lines described.In each case make a sketch. (Examples6 anclT)
( x - 5 ) l 3 a n d y 2 : x - 0 . 6 7 ( x+ 4 . 2 ) . D o t h e
8 5 . G r a p hl t :
lines appearto be parallel?Are the lines parallel?yes, no
63. The line with slope 2, going through (1, -2) 2x - y : 4
86. Graph y1 : 99x andy2 : -x199. Do the lines appearto be
perpendicular?Should they appearperpendicular?yes, yes
64. The line with slope -3, going through (-3,4) 3x + y : -5
65. The line through(1,4),parallel to y :
-3x3x + y : 7
6 6 . T h e l i n e t h r o u g h ( - 2 , 3 ) p a r a l l e lt o y : ) x
67. The line parallel to 5x - 7y :
5x-7Y:)3
68. The line parallelto 4x * 9y :
4x+9Y:2
69. The line perpendicularto y :
3x+2Y:g
70. The line perpendicular to y :
xt9Y:41
71. The line perpendicularto x 2x-tY:-5
72. The line perpendicular to 3x x-t3Y:g
73. The line perpendicularto.tr :
+ 6x - 2y: -g
35 and containing (6, 1)
5 and containing (-4,2)
(2, -3)
lx + S and containing
9x * 5 and containing (5, 4)
2y : 3 and containing (-3, 1)
y : 9 and containing (0, 0)
4 and containing (2,5) y : 5
74. The line perpendicularto y : 9 and containing ( - 1, 3)
x: -l
(x3 - 8)l@2 -t 2x -r 4).UseTRACEtoexamine
87. Graphy:
points on the graph. Write a linear function for the graph. Explain why this function is linear.
y : x - 2,x3 - 8 - (x - 2)(x2+ 2x + 4)
(x3 + 2x2 - 5x - 6)l@'t x - 6).Use
8 8 . G r a p hy :
TRACE to examine points on the graph. Write a linear function for the graph. Factor x3 + 2x2 - 5x - 6 completely.
y : x * l , ( x + l ) ( x + 3 ) ( x- 2 )
Solve each problem. (Examples9 and 10)
89. Celsius to Fahrenheit Formula Fahrenheit temperatureF is a
linear function of Celsius temperature C. The ordered pair
(0,32) is an orderedpair of this function because0"C is equivalent to 32F, the freezing point of water. The ordered pair
(I00,2I2) is also an orderedpair of this function because
100'C is equivalentto 2l2oF, the boiling point of water. Use
Chapter 1 +;;
50
E q u a t i o n Isn, e q u a l i t i easn,d M o d e l i n g
the two given points and the point-slope formula to write F as
a function of C. Find the Fahrenheittemperatureof an oven at
').
r Js V
n o\ .a f( _ - t l .- l 0 l " |
5
90. Cost o.fBtrsinessCards SpeedyPrinting charges$23 for 200
deluxebusinesscardsand $35 for 500 deluxebusinesscards.
Given that the cost is a linear function of the number of cards
printed find a formula for that function and find the cost of
- 15.S-13
700 businesscards.C' ., 0.0-1lr
91. VolumeDiscount Mona Kalini givesa walking tour of
Honolulu to one personfor $49.To increaseher business,she
advertisedat the National Orthodontist Conventionthat she
would lower the price by $ 1 per personfor each additional person.Write the costper personc as a function of the numberof
people n on the tour. How much does she make for a tour with
40 people? t' - -50 rr.$4(X)
92. TicketPricing At $10 per ticket,Willie Williams and the
Wranglerswill fill all 8000 seatsin the AssemblyCenter.The
managerknows that for every $l increasein the price, 500
ticketswill go unsold.Write the numberof ticketssold n, as a
function of the ticket price, p. How much money will be taken
in if the ticketsare $20 each?r =. --500/;+ 13,000,
560.000
93. Lindberghs Air Speed CharlesLindbergh estimatedthat at
the start of his historic flight the practical economicalair
speedwas 95 mph and at 4000 statutemiles from the starting
point it was 75 mph (www.charleslindbergh.com).
Assume
that the practicaleconomicalair speedS is a linear function
of the distanceD from the startingpoint as shown in the accompanying figure. Find a formula for that function.
.\0.0051J-9-5
New York
1 1 0 0m i
Atlantic Ocean
Figure for Exercise 94
95. Computersand Printers An office managerwill spenda
total of $60,000 on computersat $2000 each and printers at
$ I 500 each.Write the number of computerspurchasedas a
function of the number of printers purchased.Find and interpret the slope. (. : _,lp + 30.
incrcases
by 4 thenr.
,1.I1.7.,
e6.Carpenters
andaurynrJBecausel:1';'.n:f*Jn.o u
"u.ry,
contractorwill distribute a total of $2400 in bonusesto 9 carpentersand 3 helpers.The carpentersall get the sameamount
and the helpersall get the sameamount. Write the amount of
a helper'sbonus as a function of the amount of a carpenter's
bonus.Find and interpretthe slope.
It -. -3t + 800" 3. lf t'incrcitscs
by lBl.then/r clecreases
by 53.
For Writing/Discussion
9 7 . Equal Slopes Show that if y : mx -t bl and y : mx * b2
are equationsof lines with equalslopes,but b1 * b2,then
they have no point in common.
98. Unequal Slopes Show that the lines y : tntx * bl and
! : rftzx * b2 intersectat a point with x-coordinate
(bz - b)l@r - mz.)providedm1 * m2.Explainhowthis
exerciseand the previous exerciseprove the theoremthat two
nonvertical lines are parallel if and only if they have equal
slopes.
E
99. SummingAngles Considerthe anglesshown in the accompanying figure. Show that the degreemeasureof angleI plus
the degreemeasureof angle B is equal to the degreemeasure
of angle C.
3so
)'
12Jz 234
Distance(thousandsof miles)
Figure for Exercise 93
2
I
94. SpeedOver Newfoundland In Lindbergh's flying log he
recordedhis air speedoverNewfoundlandas 98 mph, I 100miles
into his flight. According to the formula from Exercise93,
what should have been his air speedover Newfoundland?
89.-5
mph
-'7
{'
.'i'
^l
0
Figure for Exercise99