Beginning Algebra, 11th Edition

Moving along the line from the point to the point , we see that y

changes by units. This is the vertical change (rise). Similarly, xchanges by

units, which is the horizontal change (run). The slope of the line is the ratio

of to y 2 - y 1 x 2 - x 1.

x 2 - x 1

y 2 - y 1

1 x 1 , y 12 1 x 2 , y 22

SECTION 3.3 The Slope of a Line 201

x

y

y 2 – y 1 = change

in y-values (rise)

0

(x 1 , y 1 )

(x 2 , y 2 )

x 2 – x 1 = change

in x-values (run)

(x 1 , y 2 )

slope =

y 2 – y 1

x 2 – x 1

FIGURE 21

NOTE Subscript notation is used to identify a point. It does notindicate any opera-

tion. Note the difference between a nonspecific value, and which means

Read as “x 2 x-sub-two,” not“xsquared.”

x 2 , x^2 , x#x.

Slope Formula

The slopemof the line through the points and is

m 1 where x 1 Zx 22.

change in y

change in x



y 2 y 1

x 2 x 1

1 x 1 , y 12 1 x 2 , y 22

The slope gives the change in y for each unit of change in x.

Finding Slopes of Lines

Find the slope of each line.

(a)The line through and

Use the slope formula. Let and

Begin at and count grid squares in

FIGURE 22to confirm your calculation.

What happens if we let and

?

The same slope is obtained.

slope m=

y 2 - y 1

x 2 - x 1

=

7 - 1 - 22

- 4 - 1

=

9

- 5

=-

9

5

1 - 4, 7 2 = 1 x 2 , y 22

1 1, - 22 = 1 x 1 , y 12

1 - 4, 7 2

slope m=

change in y

change in x

=

y 2 - y 1

x 2 - x 1

=

- 2 - 7

1 - 1 - 42

=

- 9

5

=-

9

5

1 - 4, 7 2 = 1 x 1 , y 12 1 1, - 22 = 1 x 2 , y 22.

1 - 4, 7 2 1 1, - 22

EXAMPLE 2

y

0 x

(– 4, 7)

5

9

5

(1, –2)

m = = –

1 – (– 4) = 5

–2 – 7 = –9

–9

FIGURE 22

Substitute carefully here.