2. Direct and Inverse Variations

3. Direct Variation When we talk about a direct variation, we are talking about a relationship where as x increases, y increases or decreases at a CONSTANT RATE.

4. Direct Variation Direct variation uses the following formula:

5. Direct Variation example: if y varies directly as x and y = 10 as x = 2.4, find x when y =15. what x and y go together?

6. Direct Variation If y varies directly as x and y = 10 find x when y =15. y = 10, x = 2.4 make these y 1 and x 1 y = 15, and x = ? make these y 2 and x 2

7. Direct Variation if y varies directly as x and y = 10 as x = 2.4, find x when y =15

8. Direct Variation How do we solve this? Cross multiply and set equal.

9. Direct Variation We get: 10x = 36 Solve for x by dividing both sides by 10. We get x = 3.6

10. Direct Variation Let’s do another. If y varies directly with x and y = 12 when x = 2, find y when x = 8. Set up your equation.

11. Direct Variation If y varies directly with x and y = 12 when x = 2, find y when x = 8.

12. Direct Variation Cross multiply: 96 = 2y Solve for y. 48 = y.

13. Inverse Variation Inverse is very similar to direct, but in an inverse relationship as one value goes up, the other goes down. There is not necessarily a constant rate.

14. Inverse Variation With Direct variation we Divide our x’s and y’s. In Inverse variation we will Multiply them. x 1 y 1 = x 2 y 2

15. Inverse Variation If y varies inversely with x and y = 12 when x = 2, find y when x = 8. x 1 y 1 = x 2 y 2 2(12) = 8y 24 = 8y y = 3

16. Inverse Variation If y varies inversely as x and x = 18 when y = 6, find y when x = 8. 18(6) = 8y 108 = 8y y = 13.5

17. Let’s apply what we have learned. The pressure P of a compressed gas is inversely proportional to its volume V according to Boyle’s Law. A pressure of 40 pounds per square inch is created by 600 cubic inches of a certain gas. Find the pressure when the gas is compressed to 200 cubic inches.

18. x 1 y 1 =x 2 y 2 so, (40)(600) = x 2 (200) 200x 2 = 24000 x 2 = 120 pounds/in 2

19. Now try this one on your own. A pressure of 20 pounds per inch squared is exerted by 400 inches cubed of a certain gas. Use Boyle’s Law to find the pressure of the gas when it is compressed to a volume of 100 inches cubed.

20. Joint Variation Joint variation occurs when one quantity varies directly as the product of two or more quantities. y varies jointly as x and z, then y 1 = _y 2 x 1 z z x 2 z 2

21. Joint Variation Suppose y varies jointly as x and z. Find y when x=8 and z=3, if y=16 when z=2 and x=5. y 1 = _y 2 x 1 z z x 2 z 2 16 = _y 2 5(2) 8(3) 16(8)(3) = 5(2)(y 2 )y 2 = 38.4

22. Practice makes PERFECT!!! PRACTICE SHEET 9-4 11 – 22 ALL