1. EXAMPLE 5 Write a joint variation equation The variable z varies jointly with x and y. Also, z = –75 when x = 3 and y = –5. Write an equation that relates x, y, and z. Then find z when x = 2 and y = 6. SOLUTION STEP 1 Write a general joint variation equation. z = axy –75 = a(3)(–5) Use the given values of z, x, and y to find the constant of variation a. STEP 2 Substitute  75 for z, 3 for x, and 25 for y. –75 = –15a Simplify. 5 = a Solve for a.

2. EXAMPLE 5 Write a joint variation equation STEP 3 Rewrite the joint variation equation with the value of a from Step 2. z = 5xy STEP 4 Calculate z when x = 2 and y = 6 using substitution. z = 5xy = 5(2)(6) = 60

3. EXAMPLE 6 Compare different types of variation Write an equation for the given relationship. Relationship Equation a. y varies inversely with x. b. z varies jointly with x, y, and r. z = axyr y = a x c. y varies inversely with the square of x. y = a x2x2 d. z varies directly with y and inversely with x. z = ay x e. x varies jointly with t and r and inversely with s. x = atr s

4. GUIDED PRACTICE for Examples 5 and 6 The variable z varies jointly with x and y. Use the given values to write an equation relating x, y, and z. Then find z when x = –2 and y = 5. 9. x = 1, y = 2, z = 7 SOLUTION STEP 1 Write a general joint variation equation. z = axy

5. GUIDED PRACTICE for Examples 5 and 6 7 = a(1)(2) Use the given values of z, x, and y to find the constant of variation a. STEP 2 Substitute 7 for z, 1 for x, and 2 for y. 7 = 2a Simplify. Solve for a. STEP 3 Rewrite the joint variation equation with the value of a from Step 2. = a 7 2 z = xy 7 2

6. GUIDED PRACTICE for Examples 5 and 6 STEP 4 Calculate z when x = – 2 and y = 5 using substitution. z = xy = (– 2)(5) = – 35 7 2 7 2 ANSWER z = xy 7 2 ; – 35

7. GUIDED PRACTICE for Examples 5 and 6 10. x = 4, y = –3, z =24 SOLUTION STEP 1 Write a general joint variation equation. z = axy 24 = a(4)(– 3) Use the given values of z, x, and y to find the constant of variation a. STEP 2 Substitute 24 for z, 4 for x, and –3 for y. 24 = –12a Simplify. Solve for a. = a – 2

8. GUIDED PRACTICE for Examples 5 and 6 STEP 3 Rewrite the joint variation equation with the value of a from Step 2. z = – 2 xy STEP 4 Calculate z when x = – 2 and y = 5 using substitution. z = – 2 xy = – 2 (– 2)(5) = 20 z = – 2 xy ; 20 ANSWER

9. GUIDED PRACTICE for Examples 5 and 6 The variable z varies jointly with x and y. Use the given values to write an equation relating x, y, and z. Then find z when x = –2 and y = 5. 11. x = –2, y = 6, z = 18 SOLUTION STEP 1 Write a general joint variation equation. z = axy

10. GUIDED PRACTICE for Examples 5 and 6 18 = a(– 2)(6) Use the given values of z, x, and y to find the constant of variation a. STEP 2 Substitute 18 for z, – 2 for x, and 6 for y. 18 = –12a Simplify. Solve for a. STEP 3 Rewrite the joint variation equation with the value of a from Step 2. 3 = a 2 – z = xy 3 2 –

11. GUIDED PRACTICE for Examples 5 and 6 STEP 4 Calculate z when x = – 2 and y = 5 using substitution. 33 z = xy = (– 2)(5) = 15 2 – 2 – ANSWER z = xy 3 2 – ; 15

12. GUIDED PRACTICE for Examples 5 and 6 The variable z varies jointly with x and y. Use the given values to write an equation relating x, y, and z. Then find z when x = –2 and y = 5. 12. x = –6, y = – 4, z = 56 SOLUTION STEP 1 Write a general joint variation equation. z = axy

13. GUIDED PRACTICE for Examples 5 and 6 56 = a(– 6)(–4) Substitute 56 for z, – 6 for x, and – 4 for y. 56 = 24a Simplify. Solve for a. Use the given values of z, x, and y to find the constant of variation a. STEP 2 STEP 3 Rewrite the joint variation equation with the value of a from Step 2. = a 7 3 z = xy 7 3

14. GUIDED PRACTICE for Examples 5 and 6 STEP 4 Calculate z when x = – 2 and y = 5 using substitution. z = xy = (– 2)(5) = 7 3 7 3 70 3 – z = xy 7 3 70 3 – ;ANSWER

15. GUIDED PRACTICE for Examples 5 and 6 Write an equation for the given relationship. 13. x varies inversely with y and directly with w. 14. p varies jointly with q and r and inversely with s. x = a y w SOLUTION p = aqr s SOLUTION