Linear Equations and Lines
Linear Equations
and Lines
A linear equation in two variables is an equation that’s equivalent to an
A linearofequation
equation
the formin two variables is an equation that’s equivalent to an
equation of the form
ax+by+c=O
ax + by + c = 0
where a, b, and c arjnstant numbers, and where a and b don’t both equal
where
a, oth
b, and c 41JI
are constant
and where
a and
don’tofboth
equal
1
7
0. (If the
the polynomial
on the
leftb side
0 thennumbers,
the equal
0.
(Ifwouldn’t
they were
0 then the polynomial on the left side of the equal sign
sign
beboth
linear.)
wouldn’t be linear.)
Examples.
Examples.
• 3x + 2y 7 = 0 is a linear equation.
• 3x + 2y − 7 = 0 is a linear equation.
• 2x = —4y +3 is a linear equation. It’s equivalent to 2x + 4y —3 = 0,
•and
2x =
It’s equivalent to 2x + 4y − 3 = 0,
2x −4y
linearequation.
polynomial.
+ 4y+ 3 3isisa alinear
and 2x + 4y − 3 is a linear polynomial.
• y = 2x +5 is a linear equation. It’s equivalent to the linear equation
•y—2x5=0.
y = 2x + 5 is a linear equation. It’s equivalent to the linear equation
y − 2x − 5 = 0.
• x = y is a linear equation. It’s equivalent to x
y = 0.
• x = y is a linear equation. It’s equivalent to x − y = 0.
• x = 3 is a linear equation. It’s equivalent to x
3 = 0.
• x = 3 is a linear equation. It’s equivalent to x − 3 = 0.
• y = —2 is a linear equation. It’s equivalent to y + 2 = 0.
• y = −2 is a linear equation. It’s equivalent to y + 2 = 0.
IIaso4aInear equations are called linear Xbecause their
solutions
Linear
equations
are
called
linear
because
their
sets
of
solutions
form
straight
form%straight line5in the plane. We’ll have more to say about these lines
in
lines
in
the
plane.
We’ll
have
more
to
say
about
these
lines
in
the
rest
of
this
the rest of this chapter.
chapter.
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134
140
Vertical lines
lines
Vertical
lines
Vertical
The solutions
solutions of
of aaa linear
linearequation
equationxxx===c,c,c,where
wherecccEE∈R
constant,form
formaaa
The
solutions
of
linear
equation
where
RRisisisaaaconstant,
constant,
form
The
vertical line:
line: the
the line
line of
of all
allpoints
pointsin
inthe
theplane
planewhose
whosex-coordinates
x-coordinatesequal
equalc.c.c.
vertical
line:
the
line
of
all
points
in
the
plane
whose
x-coordinates
equal
vertical
CC
Horizontal lines
lines
Horizontal
lines
Horizontal
Ifcccisisisaaaconstant
constantnumber,
number,then
thenthe
thehorizontal
horizontalline
lineof
ofall
allpoints
pointsin
theplane
plane
If
constant
number,
then
the
horizontal
line
of
all
points
ininthe
the
plane
If
whosey-coordinates
y-coordinatesequal
equalcccisisisthe
theset
setof
ofsolutions
solutionsof
ofthe
theequation
equationyyy===c.c.c.
whose
y-coordinates
equal
the
set
of
solutions
of
the
equation
whose
Slope
Slope
Slope
Theslope
slopeof
ofaaaline
lineisisisthe
theratio
ratioof
ofthe
thechange
changein
thesecond
secondcoordinate
coordinateto
the
The
slope
of
line
the
ratio
of
the
change
ininthe
the
second
coordinate
The
totothe
the
change in
inthe
thefirst
firstcoordinate.
coordinate. In
Indifferent
differentwords,
words,ifififaaaline
linecontains
containsthe
thetwo
two
change
in
the
first
coordinate.
In
different
words,
line
contains
the
change
two
points (xi,
(x1 ,y’)
y ) and
and (x
(x2 ,y2),
y2 ), then
thenthe
theslope
slopeisisisthe
thechange
changein
they-coordinate
y-coordinate
points
(xi,
,
2
(x
the
slope
the
change
ininthe
the
points
y-coordinate
,
2
y2), then
y’)1 and
–which
whichequals
equals Y2
y − y1 –divided
dividedby
bythe
thechange
changein
thex-coordinate
x-coordinate –which
which
which
equals
divided
by
the
change
ininthe
the
x-coordinate
which
Y2 2
equalsX
x2 −x
x1 .
equals
2
X
.
1
x
equals
2
.
1
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—
—
—
—
Slopeof
ofline
linecontaining
containing(x
(x1 , y1 )and
and2
(x2y2):
,y2):
y2 ):
Slope
of
line
containing
,
1
(x
,
2
(x
Slope
,
1
,
Yl) and (x
Yl)
y −Yiy1
Yi
x
−
x1
2
2
x
2
x
Y22
Y2
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—
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—
141
135
135
21
21
s)o:
s)o:
—ç
Example:
Thecontaining
slope of the
the two
(−1, 4) and (2, −5)
ple: The slope
of the line
theline
twocontaining
points (—1,4)
andpoints
(21
—ç
equals
ple: The slope of the line—5—4
containing the−5
two− points
(—1,4) and (21
4
−9
=
= −3
2 − (−1)
3
2—(—1)3
—5—4
21
2—(—1)3
21I
s)o:
1leI
s
1leI
sope:—3
s
sope:—3
—ç
Example: The slope of the line containing
the two The
containing
points
(—1,4)
Example:
of theand
line(21
slope
equals
equals
—5—4
Lines
with
same
slope
eithermeaning
equal orthey
parallel,
s with the same
slope
arethe
either
equal
or are
parallel,
nevermeaning they never 5—4
intersect.
2—(—1)3
2—(—1)
ct.
s with the same slope are either equal or parallel, meaning they never
ct.
po..ca1leI
iies
nvpraUeI
hves
sope:—3
142
Lines with 136
the same slope are either equal
Lines
either equa
or with
same slope
arenever
parallel,
the meaning
they
intersect.
intersect.
136
zii
ZL a
Slope-intercept form for linear equations
I
In calculus, the most common form of linear equation you’ll see is y = ax+b,
where a and b are constants. For example y = 2x − 5 or y = −2x + 7.
An equation of the form y = ax + b is linear, because it’s equivalent to
the two points (—1
of bthe
line containing
They slope
Example:
y − ax − b = 0. An equation
of the form
= ax +
is called
a linear equation
in slope-intercept form. equals
5—4
2—(—1)3
Claim: The solutions of the equation y = ax + b (where
a and b are
numbers) form a line of slope a that contains the point (0, b) on the y-axis.
slope: a
slop
Proof: That (0, b) is a solution
of y =
+ b isslope
easyare
to check.
Just replace
or parallel, meanin
either equal
Lines with
theaxsame
x with 0 and y with b to see
that
intersect.
(b) = b = a(0) + b
The point (1, a + b) is also on the line for y = ax + b:
(a + b) = a + b = a(1) + b
Now that we know two points on the line, (0, b) and (1, a + b), we can find
the slope of the line. The slope is
(a + b) − b a
= =a
1−0
1
S
136
P
siopt’ 2
143
Slope
Claim: y = ax is a line of slope a that contains the point (0,0).
Claim: y = ax is a line of slope a that contains the point (0, 0).
Proof: This claim follows from the previous claim if we write y = ax as
y Proof:
= ax + 0.This
Theclaim
previous
claim
tells
that y =claim
ax + 0if awe
line
of slope
follows
from
theusprevious
write
as
y = aaxthat
contains
the
point
(0,
0).
y = ax +0. The previous claim tells us that y = ax +0 a line of slope a that
contains the point (0,0).
Point-slope form for linear equations
Claim:
L CR
2way
line containing
be the
the point
2=
R
and
having
Another Let
common
to write
linear equations
is as
a(x
− p),
e q)
(p, (y
q) −
p).’S
slope
number
a. Then
a(xa linear
whereequal
a, p, qto∈ the
R are
constants.
Thist is eqtn$-Eri
called the point-slope
(y form
q) = of
equation.
Examples
(y −
2) =
7(x −let’s
3) and
1) =
−2(x + 4).
Before writing
the include
proof for
this
claim,
look(yat− an
example.
Let’s
re
suppose that we2r4t€
Zi1ing an equation
a line
slope
Claim: Let L ⊆ R2 be the line containing
the point for
(p, q)
∈ R2whose
and having
is —2, and that passes t1,irough the point in the plane (4, 1). Then4he claim
slope equal to the number a. Then (y − q) = a(x − p) is an equation for L.
says that we can use LTf equation (y i) = 2(x 4). We may prefer to
As an example
of theasclaim,
− 1) =
−2(x
simplify
this equation
= 2x
8 or
y 1 (y
y =−2x4) is7.a line of slope −2 that
passes through the point in the plane138(4, 1).
-
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—
144
—
(z
S
ltai
211
S
2)
Or —2(x)
Or —2(x)
XaXI
P
P
L_d Xl
siopt’ 2siopt’ 2
Slope Slope
Example:
of the line contain
The slope
and (2()
points (—1,4)
The slope of the line containing the two
Example:
Now let’s
the proof
of proof
the claim.
Nowturn
let’storeturn
to the
of
the
claim.
equals
equals
5—4
Proof:Proof:
Let S Let
be the
setthe
of solutions
of y =ofax.
shown
in the inprevious
of solutions
S be
As shown
set54
the previous
y =Asax.
2-(-1
claim, claim,
S is a Sline
the inplane
whose whose
slope equals
a and athat
the the
is ainline
the2-(-1)3
plane
slope equals
and contains
that contains
point (0,
0). (0,0).
point
S
.
Slope s)ope:—3
(p.)
(0,0)
(a, a)
r
(2,-5)
We canWe
usecan
theuse
addition
function
A(p,q) to
shiftto the
horizontally
by
the addition
function
A(p,q)
theS line
shiftline
by
S horizontally
Lines
with
the
same
slope
are
either e
never
they
meaning
parallel,
or
equal
are
either
slope
same
with
the
p,Lines
and p,vertically
by
q.
This
new
line,
A
(S),
is
parallel
to
our
original
line,
(p,q) A(p,q)(S), is parallel to our original line,
and vertically by q. This new line,
intersect.
intersect.
so its slope
a. Thea. line
also
contains
the point
(p, q), (p, q),
so its also
slopeequals
also equals
TheA(p,q)
line(S)
A(pq)(S)
also contains
the point
because
(0, 0) ∈(0,0)
S and Sthus
q) =(p,Aq)(p,q)=(0,
0) ∈ A(p,q)
because
and(p,
thus
A(p,q)(0,
0)) (S).A(p,q)(S).
Because
A(p,q) (S)
has
and acontains
the point
(p, q), (p,
weq),seewethat
Because
A(p,q)
has aslope
and contains
the point
(5)slope
see that
A(p,q) (S)
is
the
line
L
from
the
claim
whose
equation
we
would
like
to
identify,
A(p,q) (5) is the line L from the claim whose equation we would like to identify,
and weand
canwe
find
equation
forA(p,q)
(S) = L using
POTS:
canthe
find
the equation
forA(p,q)(S)
= L using POTS:
ojorS
1
(pry)
—A
I
I.
136
16
(-‘a(xp)
145
139
Problem:
Problem: Find
Find an
an equation
equation for
for the
the line
line containing
containing the
the two
two points
points (1,3)
(1, 3)
and
(—2, −4).
—4)
and (−2,
3)
Solution:
Solution: First
First we’ll
we’ll find
find the
the slope
slope of
of the
the line.
line. It’s
It’s
—4—
−4 − 33 —7
−7 77
=
=
—2
−2 − 11 —3
−3 33
Second,
Second, we’ll
we’ll use
use the
the previous
previous claim
claim which
which tells
tells us
us that
that aa line
line containing
containing the
the
7
point
point (1,3)
(1, 3) and
and with
with slope
slope 3 has
has as
as an
an equation
equation
7
(y-3)=(x-1)
(y − 3) = (x − 1)
3
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140
146
—
Exercises
For #1-4, find the slope of the line that contains the two given points in
the plane.
1.) (1, 2) and (3, 4)
3.) ( 2, 3) and (2, 1)
2.) (5, 10) and (0, 0)
4.) (3, 2) and ( 6, 7)
For #5-10, identify the slope of the line that is the set of solutions of the
given linear equation in slope-intercept form.
5.) y = 2x + 3
8.) y = x + 5
6.) y =
4x + 7
9.) y = 17
7.) y =
x
10.) y = 3x
2
4
For #11-16, identify the number b with the property that (0, b) is a point
in the line of the solutions of the given linear equation. This number is called
the y-intercept.
11.) y = 3x + 8
14.) y = 9
12.) y = 2x + 9
15.) y = 5x
13.) y =
3x
7
8
16.) y = 2x + 5
For #17-20, write an equation for a line that has the given slope and
contains the given point. Write answers in point-slope form (y q) = a(x p).
17.) Slope: 4. Point: (2, 3).
19.) Slope:
18.) Slope:
20.) Slope: 7. Point: ( 4, 3).
2. Point: ( 3, 4).
147
1. Point: (1, 5).
For #21-24, provide an equation for a line that contains the given pair of
points. Write your answers in the slope-intercept form y = ax + b.
21) (8, 2) and (2, 5)
23.) ( 3, 5) and (3, 4)
22.) (7, 3) and (2, 1)
24.) (2, 8) and ( 2, 5)
Find the inverse matrices.
25.)
✓
3
1
2
2
◆
1
26.)
✓
4
2
3
1
◆
1
Find the solutions of the following equations in one variable.
27.) e2x
3
28.) log3 (x)2 + 3 = 4 log3 (x)
=0
Recall that pX : R2 ! R where pX (x, y) = x is called the projection onto
the x-axis, and that pY : R2 ! R where pY (x, y) = y is the projection onto
the y-axis. In the remaining questions, find the appropriate values.
29.) pX (3, 7)
33.) pX (1, 8)
30.) pX 2, 8
34.) pY 6, 9
31.) pY ( 2, 5)
35.) pY ( 1, 7)
32.) pY (4, 0)
36.) pX (2, 3)
148