Linear Equations and Lines

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 = ¡ª4y +3 is a linear equation. It¡¯s equivalent to 2x + 4y ¡ª3 = 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¡ª2x5=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 = ¡ª2 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.

¡ª

¡ª

¡ª

¡ª

134

140

Vertical lines

lines

Vertical

lines

Vertical

The solutions

solutions of

of aaa linear

linearequation

equationxxx===c,c,c,where

wherecccEE¡ÊR

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

¨Cwhich

whichequals

equals Y2

y ? y1 ¨Cdivided

dividedby

bythe

thechange

changein

thex-coordinate

x-coordinate ¨Cwhich

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

¡ª

¡ª

¡ª

¡ª

¡ª

¡ª

¡ª

¡ª

¡ª

¡ª

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

¡ª

¡ª

¡ª

¡ª

141

135

135

21

21

s)o:

s)o:

¡ª?

Example:

Thecontaining

slope of the

the two

(?1, 4) and (2, ?5)

ple: The slope

of the line

theline

twocontaining

points (¡ª1,4)

andpoints

(21

¡ª?

equals

ple: The slope of the line¡ª5¡ª4

containing the?5

two? points

(¡ª1,4) and (21

4

?9

=

= ?3

2 ? (?1)

3

2¡ª(¡ª1)3

¡ª5¡ª4

21

2¡ª(¡ª1)3

21I

s)o:

1leI

s

1leI

sope:¡ª3

s

sope:¡ª3

¡ª?

Example: The slope of the line containing

the two The

containing

points

(¡ª1,4)

Example:

of theand

line(21

slope

equals

equals

¡ª5¡ª4

Lines

with

same

slope

eithermeaning

equal orthey

parallel,

s with the same

slope

arethe

either

equal

or are

parallel,

nevermeaning they never 5¡ª4

intersect.

2¡ª(¡ª1)3

2¡ª(¡ª1)

ct.

s with the same slope are either equal or parallel, meaning they never

ct.

po..ca1leI

iies

nvpraUeI

hves

sope:¡ª3

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 (¡ª1

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¡ª4

2¡ª(¡ª1)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¡Ê 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)

¡Ê R2whose

and having

is ¡ª2, 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).

-

¡ª

¡ª

¡ª

¡ª

¡ª

144

¡ª

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