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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 1Write 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.
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
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 = ax
c. y varies inversely with the square of x.
y =ax2
d. z varies directly with y and inversely with x.
z =ayx
e. x varies jointly with t and r and inversely with s.
x = atrs
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
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.
= a72
z = xy72
GUIDED PRACTICE for Examples 5 and 6
STEP 4
Calculate z when x = – 2 and y = 5 using substitution.
z = xy = (– 2)(5) = – 3572
72
ANSWER z = xy72 ; – 35
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
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 ; 20ANSWER
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
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 = a2–
z = xy32
–
GUIDED PRACTICE for Examples 5 and 6
STEP 4
Calculate z when x = – 2 and y = 5 using substitution.3 3z = xy = (– 2)(5) = 152
–2
–
ANSWER z = xy32
– ; 15
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
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.
= a73
z = xy73
GUIDED PRACTICE for Examples 5 and 6
STEP 4
Calculate z when x = – 2 and y = 5 using substitution.
z = xy = (– 2)(5) =73
73
703
–
z = xy73
703
–;ANSWER
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 =ay w
SOLUTION
p =aqrs
SOLUTION
