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.
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
• 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.
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.
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.
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?
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
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.
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.
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
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?
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.
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.
(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?
Direct Variation y = kx Inverse Variation xy = k