Section 1.2

Linear Equations in Two Variables

What you should learn

• How to use slope to graph linear equations in two variables

• How to write linear equations in two variables

• How to use slope to identify parallel and perpendicular lines

• How to use linear equations in two variables to model and solve real-life problems

Slope

Slope describes the direction of a line.

Guard against 0 in the

denominator

SlopeIf x1 x2, the slope of the line

through the distinct points (x1, y1) and (x2, y2) is

12

12

xx

yy

xinchange

yinchange

run

risem

Why is this

needed?

x-axis

y-axis

Find the slope between (-3, 6) and (5, 2)

RiseRun

-48

-12

= =

Find the slope between(-3, 6) and (5, 2)

12

12

xx

yym

)3()5(

)6()2(

m8

4

2

1

Find the Slope

(5, -2)

(11, 2)

(3, 9)

12

12

xx

yym

1

2 9

11 3m

Green

511

)2(22

m

Blue

3

2 9

5 3m

Red

7

8

3

2

11

2

Find the slope between (5, 4) and (5, 2).

12

12

xx

yym

)5()5(

)4()2(

m0

2 STOP

This slope is undefined.

x-axis

y-axis

Find the slope between (5, 4) and (5, 2).

RiseRun

-20

Undefined= =

Find the slope between(5, 4) and (-3, 4).

12

12

xx

yym

)5()3(

)4()4(

m8

0

STOP

This slope is zero.

x-axis

y-axis

Find the slope between (5, 4) and (-3, 4).

RiseRun

0-8

Zero= =

From these results we can see...

• The slope of a vertical line is undefined.

• The slope of a horizontal line is 0.

x-axis

y-axis

Graph the line that goes through (1, -3) with

(1,-3)

4

3m

Using Slope to Graph

• Graph the line that contains (4, 5) and has a slope of 3/2.

(4, 5)

(6, 8)

(8, 11)

Using Slope to Graph

• Graph (0, 8)

(4, 5)

(8, 2)

38

4y x

Slope

12

12

xx

yym

Point-Slope

11 xxmyy

12

12

xx

yym

Slope

1 1( , )x yPoint

Slope Intercept

bmxy 12

12

xx

yym

(0, )bSlope

Y-intercept

CByAx Standard Form of a Line

B

Am

A

Cintercept -x

B

Cintercept -y

Parallel Lines

• Two distinct nonvertical lines are parallel if and only if their slopes are equal.

• That is,

1 2m m

Parallel

• Find the slope-intercept form of the line that passes through (2,1) parallel to

2x – 3y = 5.

Perpendicular

• Find the slope-intercept form of the line that passes through (2,1) perpendicular to

2x – 3y = 5.

Homework

• Page 21

• 1, 9-27 odd 39- 45 odd, 51-79 odd

• 87- 95 odd, 105 - 113 odd, 117