1. 12-7 Joint and Combined Variation Warm-up Problems 1.Find the equation of variation where y varies directly as x, and y = 6, when x = 5. 2.Find the equation of variation where y varies inversely as x, and y = 2 and x = 7.

2. Joint Variation An equation of the form z = kxy, where k is a nonzero constant, expresses joint variation.

3. Example 1 Find an equation of joint variation where V varies jointly as B and h. One set of values for the relationship is V = 35, B = 7, and h = 15. Find V when B = 18 and h = 6.

4. Try This a.Find an equation of joint variation where w varies jointly as x, y, and z. One set of values for the relationship is w = 36, x = 3, y = 5, and z = 6. Find w when x = 2, y = 8, and z = 5.

5. Combined Variation An equation of the form, where k is a nonzero constant, expresses combined variation.

6. Example 2 Find an equation of combined variation where A varies directly as b and inversely as c. One set of values is A = 4, b = 12, and c = 9. Find A when b = 7 and c = 3.

7. Try This b.Find an equation of combined variation where P varies directly as q and inversely as r. One set of values is P = 0.064, q = 16, and r = 5. Find P when q = 12 and r = 10.

8. Example 3 The volume of a pyramid varies jointly as the height of the pyramid and the area of its base. The volume of a pyramid with height 12 cm and base 5 cm² is 20 cm³. Find the volume of the Great Pyramid whose height is 147 m and whose base has an area of 52,900 m².

9. Try This The temperature inside the chamber of a piston varies jointly as the pressure and the volume. The temperature is 300 Kelvin when the volume is 200 in.³ and the pressure is 100 pounds per square inch (lb/in²). Find the temperature when the pressure is 70 lb/in.² and the volume is 400 in.³.

10. 12-5 Direct Variation Warm-up Problems Graph f(x) = x² + x – 2.

11. Chapter 12 12-5 Direct Variation

12. Direct Variation An equation of the form y = kx, where k is a nonzero constant, expresses direct variation. k is called the constant of variation.

13. Example 1 Find an equation of variation where y varies directly as x, and y = 2 when x = 1.

14. Try This Find an equation of variation where y varies directly as x. One pair of values is given. a.y = 84 when x = 12 b.y = 50 when x = 80

15. Example 2 The weight (V) of an object on Venus varies directly as its weight on (E) Earth. A person weighing 120 lb on Earth would weigh 106 lb on Venus. How much would a person weighing 150 lb on Earth weigh on Venus?

16. Try This c.The cost (c) of operating a TV varies directly as the number (n) of hours it is in operation. It costs $14.00 to operate a standard size color TV continuously for 30 days. At this rate, about how much would it cost to operate the TV for 1 day? 1 hour?