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EXAMPLE 5

STEP 1. Write a general joint variation equation. STEP 2. Use the given values of z , x , and y to find the constant of variation a. EXAMPLE 5. Write a joint variation equation.

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EXAMPLE 5

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  1. STEP 1 Write a general joint variation equation. STEP2 Use the given values of z, x, and y to find the constant of variation a. EXAMPLE 5 Write a joint variation equation The variable zvaries jointly with xand y. Also, z= –75 when x = 3 and y = –5. Write an equation that relates x, y, and z. Then find zwhen x = 2 and y = 6. SOLUTION z = axy –75 = a(3)(–5) Substitute 75 for z, 3 for x, and 25 for y. –75 = –15a Simplify. 5 = a Solve for a.

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

  3. y = y = z = atr x = s ay a a x2 x x EXAMPLE 6 Compare different types of variation Write an equation for the given relationship. Relationship Equation a.yvaries inversely with x. b.zvaries jointly with x, y, and r. z = axyr c.y varies inversely with the square of x. d.zvaries directly with yand inversely with x. e.xvaries jointly with tand rand inversely with s.

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

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

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

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

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

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

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

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

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

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

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

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

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