2. Direct Variation
If two quantities vary directly, their relationship can be described as:
y = kx
where x and y are the two quantities and
k is the constant of variation.

3. Direct Variation y = kx k ≠ 0
Assume y varies directly as x:
1. If y = 6 when x = 12, find x when y = 9
We can do this problem two ways:
• a. Write the equation y = kx and substitute in the first pair to find k

4. • Then write the equation y = kx and substitute in k and the given number for the 2nd pair
• b. Write the proportion
and substitute in the given information; cross multiply to find the missing value.

5. Assume y varies directly as x.
2. If y = 6 when x = 2, find x when y = 4.
3. If y = 12 when x = 3, find y when x = 5.

6. Many of the word problems you solved using a proportion were direct variation problems.
For example: Total cost varies directly as number of tickets. If 5 tickets to a play cost $30, how much would 25 tickets cost?
Solve as a proportion.

7. Total cost varies directly as number of tickets
Total cost varies directly as number of tickets. If 5 tickets to a play cost $30, how much would 25 tickets cost?
Let the number of tickets be x and the total cost be y. Then y = kx can be changed to 30 = 5k and dividing by 5, k = 6
Now we can use x and y to represent any pair of number of tickets and total price and the equation for this problem becomes y = 6x
What does the 6 represent?

8. k, the constant of variation, can also be found by taking any pair of x and y and taking the ratio of y to x:
k =
So we could have found k in the previous word problem by:
k = or k = 6

9. Distance varies directly as time
Distance varies directly as time. If distance equals 150 miles when time equals 3 hours, find time if distance equals 350 miles.
What is k? What does k represent? Write the direct variation equation for this problem.

10. Inverse Variation
If two quantities vary inversely, their relationship can be described as:
xy = k
where x and y are the two quantities and k is the constant of variation.

11. Inverse Variation xy = k y varies inversely as x
k is called the constant of variation
Do NOT use a proportion to solve inverse variation problems. Simply set up an equation:
x1y1 = x2y2

12. Assume y varies inversely as x:
If y = 6 when x = 12, find x when y = 9.
What is k? What is the inverse variation equation for this problem?

13. Assume y varies inversely as x.
2. If y = 4 when x = 6, find x when y = 12.
3. If y = 6 when x = 8, find y when x = 4.

14. If two quantities vary inversely, their product equals k.
With our distance equation D = rt, rate and time vary inversely. D is the constant of variation.

15. (rate)(time) = (rate)(time)
Rate varies inversely as time. If the rate equals 45 when time equals 4, find the rate when time equals 6 hours.
(rate)(time) = (rate)(time)
45 ● 4 = r ● 6
Now solve for r.
What is k? What does it represent?

16. Direct Variation
y = kx
Inverse Variation
xy = k