Pre-Algebra

11-5 Direct Variation

Learn to recognize direct variation by graphing tables of data and checking for constant ratios.

Pre-Algebra

11-5 Direct Variation

Pre-Algebra

11-5 Direct Variation

The graph of a direct-variation equation is always linear and always contains the point (0, 0). The variables x and y either increase together or decrease together.

Helpful Hint

Pre-Algebra

11-5 Direct Variation

Determine whether the data set shows direct variation.

A.

Example 1: Determining Whether a Data Set Varies Directly

Pre-Algebra

11-5 Direct Variation

Make a graph that shows the relationship between Adam’s age and his length.

Example 1 Continued

Pre-Algebra

11-5 Direct Variation

You can also compare ratios to see if a direct variation occurs.

223

2712=

?81

264

81 ≠ 264

The ratios are not proportional.

The relationship of the data is not a direct variation.

Example 1 Continued

Pre-Algebra

11-5 Direct Variation

Determine whether the data set shows direct variation.

B.

Example 2: Determining Whether a Data Set Varies Directly

Pre-Algebra

11-5 Direct Variation

Make a graph that shows the relationship between the number of minutes and the distance the train travels.

Example 2 Continued

Plot the points.

The points lie in a straight line.

(0, 0) is included.

Pre-Algebra

11-5 Direct Variation

You can also compare ratios to see if a direct variation occurs.

The ratios are proportional. The relationship is a direct variation.

2510

5020

7530

10040= = = Compare ratios.

Example 2 Continued

Pre-Algebra

11-5 Direct Variation

Determine whether the data set shows direct variation.

A.

Example 3

Kyle's Basketball Shots 

Distance (ft) 20 30 40

Number of Baskets 5 3 0

Pre-Algebra

11-5 Direct Variation

Make a graph that shows the relationship between number of baskets and distance.

Example 3 Continued

Num

ber

of

Bask

ets

Distance (ft)

2

3

4

20 30 40

1

5

Pre-Algebra

11-5 Direct Variation

You can also compare ratios to see if a direct variation occurs.

Example 3 Continued

520

330=

?60

150

150 60.

The ratios are not proportional.

The relationship of the data is not a direct variation.

Pre-Algebra

11-5 Direct Variation

Determine whether the data set shows direct variation.

B.

Example 4

Ounces in a Cup

Ounces (oz) 8 16 24 32

Cup (c) 1 2 3 4

Pre-Algebra

11-5 Direct Variation

Make a graph that shows the relationship between ounces and cups.

Example 4 Continued

Num

ber

of

Cup

s

Number of Ounces

2

3

4

8 16 24

1

32

Plot the points.

The points lie in a straight line.

(0, 0) is included.

Pre-Algebra

11-5 Direct Variation

You can also compare ratios to see if a direct variation occurs.

Example 4 Continued

The ratios are proportional. The relationship is a direct variation.

Compare ratios. = 1 8 = =2

163

24 432

Pre-Algebra

11-5 Direct Variation

Find each equation of direct variation, given that y varies directly with x.

3. y is 54 when x is 6

Example 3: Finding Equations of Direct Variation

Pre-Algebra

11-5 Direct Variation

4. x is 12 when y is 15

Example 4: Finding Equations of Direct Variation

y = kx

.

Pre-Algebra

11-5 Direct Variation

5. y is 8 when x is 5

Example 5: Finding Equations of Direct Variation

Pre-Algebra

11-5 Direct Variation

Find each equation of direct variation, given that y varies directly with x.

6. y is 24 when x is 4

Example 6

Pre-Algebra

11-5 Direct Variation

B. x is 28 when y is 14

Example 9

Pre-Algebra

11-5 Direct Variation

C. y is 7 when x is 3

Example 10

Pre-Algebra

11-5 Direct Variation

Lesson Review: Part 1

Find each equation of direct variation, given that y varies directly with x.

1. y is 78 when x is 3.

2. x is 45 when y is 5.

3. y is 6 when x is 5.

y = 26x

y = x19

y = x65