Determine the slope of the line that passes through each pair of points:

(3, 5) and (7, 12) (-2, 4) and (5, 4)

(-3, 6) and (2, -6) (7, -2) and (7, 13)

Determine the value of n so that the slope of the

line through (n, 4) and (1, n) is . 21

47

m 0m

512

m undefined

3n

Algebra 1 Glencoe McGraw-Hill JoAnn Evans

Math 8H

4-2

Slope and

Direct Variation

A direct variation equation is a special type of linear equation.

Every direct variation equation will graph as a line that passes through the origin. (0,

0)

x

y

The two quantities will be represented as y and x.

Written in ratio form the ratio of y to x is .

What ratio did we study in the previous lesson? SLOPE!

In direct variation equations, the slope has a different name. It is known as the “constant of

variation”.

When two quantities have a constant ratio, they are said to have a direct variation.

xy

Slope is the ratio of the change in y to the change in

x.change in y

m (SLOPE)change in x

A direct variation equation is: kxy

In a direct variation equation k is called the constant of variation. On

the graph of a direct variation equation k is the slope of the line.

Solve the direct variation equation for y.

kxy

kxy

)x()x(

kxy

A direct variation equation represents a constant rate of change.

“k” is the constant of variation

This is a graph of the direct variation

equation y = 3x.

The constant of variation is 3.

What is the slope of the line?

12

12

xxyy

313

0103

(1, 3)

(0, 0)

The slope of the line is the same

as the constant of variation.

This is a graph of the

direct variation equation

y = x.

What is the constant of

variation?

What is the slope of the line?

12

12

xxyy

41

0401

(4, 1)

(0, 0)

The slope of the line is the same

as the constant of variation.

41

41

Remember: every direct variation equation will graph as a line that passes through

the origin.

x

y

Graph y = 5x

x

y

•

• 1. Write the slope as a ratio.

15

5

2. Plot a point at (0, 0).

3. Walk the slope. A

slope

of tells you to go

UP 5, OVER 1.

15

4. Plot the point. Connect the two

points with a line.

Graph y = x

1. The slope is already a ratio. Assign the negative to the numerator.2. Plot a point at (0, 0).

3. Walk the slope. A

slope

of tells you to go

DOWN 3, OVER 4.

43

4. Plot the point. Connect the two

points with a line.

x

y

••

43

Graph y = x

x

y

••

52

Graph y = -x

x

y

••

What is the slope?

It’s -1. Written as a

ratio, that’s . 11

Y varies directly as x. Write a direct variation equation that relates x and y.

kxy

Use this information towrite the direct variation

equation.

( 27) k( 3) 3 3

If y = -27 when x = -3, find x when y = 108.

Using the equation, answer the question.

y 9x108 9x

9 9

12 x

x equals 12 when y = 108.

k9

kxy x9y

Y varies directly as x. Write a direct variation equation that relates x and y.

kxy

Use this information towrite the direct variation

equation.

( 15) k(5) 5 5 k3

kxy x3y

If y = -15 when x = 5, find x when y = -87.

Using the equation, answer the question.

y 3x87 3x

3 3 x29

x equals 29 when y = -87.

Y varies directly as x. Write a direct variation equation that relates x and y.

kxy

Use this information towrite the direct variation

equation.

(7.5) k(.5) .5 .5

15 k

kxy y 15x

If y = 7.5 when x = 0.5, find y when x = -0.3.

Using the equation, answer the question.

y 15x

y 15 .3 y 4.5

y equals -4.5 when x = -.3.

Y varies directly as x. Write a direct variation equation that relates x and y.

kxy

Use this information towrite the direct variation

equation.

(12) k(18) 18 18

2k

3

kxy 2

y x3

If y = 12 when x = 18, find x when y = -16.

Using the equation, answer the question.

2y x

3

216 x

3

24 x

x equals 24when y = -16.

The cost of bananas varies directly with their weight. If 3 pounds of bananas cost $2.04, find the cost of 4

pounds.

c kw

2.04 k 3 3 3

.68 k

c .68w

If c = $2.04 when w = 3, find c when w = 4.

c .68w

c .68 4

c 2.72

The cost is $2.72 for 4 lb. of bananas.

Write a direct variation equation that relates the cost, c, to the weight, w. Use the equation to answer the question.

d = rt is a direct variation equation! Distance (d) varies directly as time (t).The rate (r) is the constant of variation.

A hot air balloon’s distance of ascent varies directly as the time. The balloon ascended 372

feet in six minutes. Write a direct variation equation that relates the distance, d, to the time,

t.d = rt

(372) = r(6)

62 = r The balloon’s ascent rate is 62 feet per minute.d = 62t is the direct variation

equation.

Use the direct variation equation to find how long will it take for the balloon to rise 1209 feet.

d = 62t (1209) = 62t

19.5 = t

The balloon should ascend 1209 feet in 19.5 minutes.