1. Exercise
Solve.
5 = 3x
x =
5 3

2. Exercise
Solve.
4 = k
1 3
k = 12

3. Exercise
Solve.
y 25
3 5 =
y = 15

4. Exercise
Solve.
3 5
25 x =
x = 41
2 3

5. Exercise
If 4 = 6k, what is the value of 3k? 2

6. Direct Variation
A direct variation is formed by the variables x and y if the ratio y : x always equals a constant k, where k is a positive number.

7. Directly Proportional
Variables are directly proportional when y is said to vary directly with x.

8. Constant of Proportionality
The constant k is the constant of variation, or the constant of proportionality.

9. x hours
y miles
y x 1 30 2 60 3 90 4
120

10. Example 1
Does y vary directly with x in the following table? If so, find the constant of variation and write an equation for the direct variation. x y 1 3 3 9 5 15 7 21

11. x y 1 3 3 9 5 15 7 21 =
3 1
y x 3 =
21 7
y x 3 =
9 3
y x 3 =
y x 3 k
y = kx =
15 5
y x 3
y = 3x

12. Constant of Variation
The constant of variation is the steady rate of change. The constant k is the constant of variation, or the constant of proportionality.

13. Example 2
Indicate which equations represent a direct variation. If an equation describes a direct variation, give the constant of variation.
f(x) = 2.2x
direct variation; k = 2.2

14. y = 4x − 1
This is not a direct variation; the variable must be a multiple of x.
d = 45t
This is a direct variation; k = 45.
y = −2x
This is not a direct variation; the coefficient of x must be positive.

15. y = mx + b
y = kx

16. Example 3
Graph the direct variation y = 4x. x −1 1 y −4 4

17. y x

18. Example 4
Find k if y varies directly with x and y = 12 when x = . Write an equation for the direct variation.
1 2
y = kx
k = 24
12 = k( )
1 2
y = 24x
2(12) = k( )(2)
1 2

19. Example 5
If y varies directly with x and y = 6 when x = 2, find y when x = .
2 3
y = kx
y = 3x
y = 3( )
2 3
6 = k(2)
3 = k
y = 2

20. Example Find k if y varies directly with x and y = 14 when x = 4.

21. Example Find k if y varies directly with x and y = 15 when x = 2.

22. Example
If y varies directly with x and y = 7 when x = 1, find y when x = 6.
y = 42

23. Example
If y varies directly with x and y = 27 when x = 15, find y when x = 6.
54 5
y =

24. Example
Indicate which equations represent a direct variation. If an equation describes a direct variation, give the constant of variation. If not, explain.

25. y = 4x
Yes. k = 4.
y = 3x + 5
No. The y-intercept is not zero.
y = −4x
No. The slope is not positive.