1. Pre-Algebra
11-5
Direct Variation
Learn to recognize direct variation by graphing tables of data and checking for constant ratios.

2. Pre-Algebra
11-5
Direct Variation

3. Direct Variation 11-5 Helpful Hint
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

4. Example 1B: Determining Whether a Data Set Varies Directly
Pre-Algebra
11-5
Direct Variation
Example 1B: Determining Whether a Data Set Varies Directly
Determine whether the data set shows direct variation. B.

5. Direct Variation 11-5 Example 1B Continued
Pre-Algebra
11-5
Direct Variation
Example 1B Continued
Make a graph that shows the relationship between the number of minutes and the distance the train travels.
Plot the points.
The points lie in a straight line.
(0, 0) is included.

6. Direct Variation 11-5 Example 1B Continued
Pre-Algebra
11-5
Direct Variation
Example 1B Continued
You can also compare ratios to see if a direct variation occurs. 25 10 50 20 75 30
100 40
Compare ratios. = = =
The ratios are proportional. The relationship is a direct variation.

7. Example 1A: Determining Whether a Data Set Varies Directly
Pre-Algebra
11-5
Direct Variation
Example 1A: Determining Whether a Data Set Varies Directly
Determine whether the data set shows direct variation. A.

8. Direct Variation 11-5 Example 1A Continued
Pre-Algebra
11-5
Direct Variation
Example 1A Continued
Make a graph that shows the relationship between Adam’s age and his length.

9. Direct Variation 11-5 Example 1A Continued
Pre-Algebra
11-5
Direct Variation
Example 1A Continued
You can also compare ratios to see if a direct variation occurs. 81
81 ≠ 264
The ratios are not proportional. 22 3 27 12 = ?
264
The relationship of the data is not a direct variation.

10. Example 2A: Finding Equations of Direct Variation
Pre-Algebra
11-5
Direct Variation
Example 2A: Finding Equations of Direct Variation
Find each equation of direct variation, given that y varies directly with x.
A. y is 54 when x is 6
y = kx
y varies directly with x.
54 = k  6
Substitute for x and y.
9 = k
Solve for k.
Substitute 9 for k in the original equation.
y = 9x

11. Example 2B: Finding Equations of Direct Variation
Pre-Algebra
11-5
Direct Variation
Example 2B: Finding Equations of Direct Variation
B. x is 12 when y is 15
y = kx
y varies directly with x.
15 = k  12
Substitute for x and y.
= k 5 4
Solve for k.
Substitute for k in the original equation. 5 4
y = x 5 4

12. Example 2C: Finding Equations of Direct Variation
Pre-Algebra
11-5
Direct Variation
Example 2C: Finding Equations of Direct Variation
C. y is 8 when x is 5
y = kx
y varies directly with x.
8 = k  5
Substitute for x and y.
= k 8 5
Solve for k.
Substitute for k in the original equation. 8 5
y = x 8 5

13. Pre-Algebra
11-5
Direct Variation

14. Direct Variation 11-5 Try This: Example 1B
Pre-Algebra
11-5
Direct Variation
Try This: Example 1B
Determine whether the data set shows direct variation. B.
Ounces in a Cup
Ounces (oz) 8 16 24 32
Cup (c) 1 2 3 4

15. Try This: Example 1B Continued
Pre-Algebra
11-5
Direct Variation
Try This: Example 1B Continued
Make a graph that shows the relationship between ounces and cups.
Number of Cups
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.

16. Try This: Example 1B Continued
Pre-Algebra
11-5
Direct Variation
Try This: Example 1B Continued
You can also compare ratios to see if a direct variation occurs. = 1 8 2 16 3 24 4 32
Compare ratios.
The ratios are proportional. The relationship is a direct variation.

17. Direct Variation 11-5 Try This: Example 1A
Pre-Algebra
11-5
Direct Variation
Try This: Example 1A
Determine whether the data set shows direct variation. A.
Dan's Basketball Shots 
Distance (ft) 20 30 40
Number of Baskets 5 3

18. Try This: Example 1A Continued
Pre-Algebra
11-5
Direct Variation
Try This: Example 1A Continued
Make a graph that shows the relationship between number of baskets and distance. 5 4 3
Number of Baskets 2 1 20 30 40
Distance (ft)

19. Direct Variation 11-5 Try This: Example 1A
Pre-Algebra
11-5
Direct Variation
Try This: Example 1A
You can also compare ratios to see if a direct variation occurs. 60
150 ≠ 60.
The ratios are not proportional. 5 20 3 30 = ?
150
The relationship of the data is not a direct variation.

20. Direct Variation 11-5 Try This: Example 2A
Pre-Algebra
11-5
Direct Variation
Try This: Example 2A
Find each equation of direct variation, given that y varies directly with x.
A. y is 24 when x is 4
y = kx
y varies directly with x.
24 = k  4
Substitute for x and y.
6 = k
Solve for k.
Substitute 6 for k in the original equation.
y = 6x

21. Example 3: Money Application
Pre-Algebra
11-5
Direct Variation
Example 3: Money Application
Ms. Cason has $4000 in a CD and $4000 in a money market account. The amount of interest she has earned since the beginning of the year is organized in the following table. Determine whether there is a direct variation between either of the data sets and time. If so, find the equation of direct variation.

22. Direct Variation 11-5 Example 3 Continued A. interest from CD and time
Pre-Algebra
11-5
Direct Variation
Example 3 Continued
A. interest from CD and time
interest from CD
time = 17 1
interest from CD
time
= = 17 34 2
The second and third pairs of data result in a common ratio. In fact, all of the nonzero interest from CD to time ratios are equivalent to 17.
= = = 17
interest from CD
time
= = 17 1 34 2 51 3 68 4
The variables are related by a constant ratio of 17 to 1, and (0, 0) is included. The equation of direct variation is y = 17x, where x is the time, y is the interest from the CD, and 17 is the constant of proportionality.

23. Direct Variation 11-5 Example 3 Continued
Pre-Algebra
11-5
Direct Variation
Example 3 Continued
B. interest from money market and time
interest from money market
time
= = 19 19 1
interest from money market
time
= =18.5 37 2
19 ≠ 18.5
If any of the ratios are not equal, then there is no direct variation. It is not necessary to compute additional ratios or to determine whether (0, 0) is included.

24. Pre-Algebra
11-5
Direct Variation

25. Direct Variation 11-5 Helpful Hint
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

26. Direct Variation 11-5 Try This: Example 2B B. x is 28 when y is 14
Pre-Algebra
11-5
Direct Variation
Try This: Example 2B
B. x is 28 when y is 14
y = kx
y varies directly with x.
14 = k  28
Substitute for x and y.
= k 1 2
Solve for k.
Substitute for k in the original equation. 1 2
y = x 1 2

27. Direct Variation 11-5 Try This: Example 2C C. y is 7 when x is 3
Pre-Algebra
11-5
Direct Variation
Try This: Example 2C
C. y is 7 when x is 3
y = kx
y varies directly with x.
7 = k  3
Substitute for x and y.
= k 7 3
Solve for k.
Substitute for k in the original equation. 7 3
y = x 7 3

28. Direct Variation 11-5 Try This: Example 3
Pre-Algebra
11-5
Direct Variation
Try This: Example 3
Mr. Lotkowski has $2000 in a CD and $2000 in a money market account. The amount of interest he has earned since the beginning of the year is organized in the following table. Determine whether there is a direct variation between either of the data sets and time. If so, find the equation of direct variation.

29. Try This: Example 3 Continued
Pre-Algebra
11-5
Direct Variation
Try This: Example 3 Continued
Interest
Interest from
Time (mo)
from CD ($)
Money Market ($) 1 12 15 2 30 40 3 45 4 50

30. Try This: Example 3 Continued
Pre-Algebra
11-5
Direct Variation
Try This: Example 3 Continued
A. interest from CD and time
interest from CD
time = 12 1
interest from CD
time
= = 15 30 2
The second and third pairs of data do not result in a common ratio.
If any of the ratios are not equal, then there is no direct variation. It is not necessary to compute additional ratios or to determine whether (0, 0) is included.

31. Try This: Example 3 Continued
Pre-Algebra
11-5
Direct Variation
Try This: Example 3 Continued
B. interest from money market and time
interest from money market
time
= = 15 15 1
interest from money market
time
= =20 40 2
15 ≠ 20
If any of the ratios are not equal, then there is no direct variation. It is not necessary to compute additional ratios or to determine whether (0, 0) is included.