Direct Variation

Learn to recognize direct variation and identify the constant of proportionality.

Spiders

How many legs does a spider have? 8 legs Therefore

2 spiders have a total of 16 legs3 spiders have a total of 24 legs4 spiders have a total of 32 legsAnd so on. . .

This type of relationship is called . . .

Direct Variation

The relationship between the amount of spiders and how many legs they have can be said to vary directly!

We will be learning about equations, tables, and graphs of direct variations.

Sometimes this is called direct proportion rather than direct variation but it is the same thing.

Direct Variation

A direct variation relationship can be represented by a linear equation in the form y = kx, where k is a positive number called the constant of proportionality.

The constant of proportionality can sometimes be referred to as the constant of variation.

y = kx When two variable quantities have a constant

(unchanged) ratio, their relationship is called a direct proportion.

We say, “y varies directly as x.” k is the constant of proportionality which

means it never changes within a problem.

Finding Values Using a Direct Variation Equation

At a frog jumping contest, Edward’s frog jumped 60 inches. Bella’s frog jumped 72 inches. Jacob’s frog jumped 6.5 feet. Use the equation y = 12x, where y is the number of inches and x is the number of feet to find the missing values of the table.

Edward’s Frog Bella’s Frog Jacob’s Frog

Inches (y) 60 72 ___

Feet (x) ___ 6 6.5

Using Equations and Tables

Edward’s Frog Bella’s Frog Jacob’s Frog

Inches (y) 60 72 ___

Feet (x) ___ 6 6.5

xy 12 xy 12

x1260 5.612ySubstitute

12

12

12

60 x 78ySolve

x5 Simplify

78

5

Now You Try

Identify the constant of proportionality:

1. y = 15x

15

2. y = .72x

.72

3. y = ¼x

¼

How about solving for k?

When we are given the x and y values, we can solve for the constant of proportionality.

Example If y varies directly with x, and y = 8 when x = 12,

find k and write an equation that expresses this variation.

Steps: 8 = k × 12 Substitute numbers into y = kx 8/12 = (k × 12)/12 Divide both sides by 12 2/3 = k Simplify y = 2/3x Plug k back into the equation

Now You Try

If y varies directly as x, and x = 12 when y = 9, what is the equation that describes this direct variation?

y = kx 9 = k × 12 9/12 = k ¾ = k y = ¾x

Now You Try

If y varies directly as x, and x = 5 when y = 10, what is the equation that describes this direct variation?

y = kx 10 = k × 5 10/5 = k 2 = k y = 2x

Copy and fill out tables:

x y

1

2

3

4

5

x y

1

2

3

4

5

y = x y = 4x

1

2

3

4

5

4

8

12

16

20

Copy and fill out tables:

x y

1

2

3

4

5

x y

1

2

3

4

5

y = 10x y = ½x

10

20

30

40

50

½

1

3/2

2

5/2

Direct Variation Equation

Sometimes we will have to put an equation into y = kx form and solve for y.

Then we will be asked to identify k, the constant of proportionality.

Examples: Tell whether each

equation or relationship is a direct variation. If so identify the constant of proportionality.

1. 4y = 2x

2. ½y – ¾x = 0

3. 7y – 5 = 3x

More Examples

If y varies directly as x and y = 24 when x = 16, find y when x = 12.

Solution: Set up a proportion since the ratios of corresponding values of x to y are always the same.

1216

24 y

y 161224y16288

16

16

16

288 y

y18