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
STEP 2
Use the given values of z, x, and y to find the constant
of variation a.
Substitute 75 for z, 3 for x, and 25 for y.
–75 = a(3)(–5)
Simplify.
–75 = –15a
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
a. y varies inversely with x.
b. z varies jointly with x, y,
and r.
c. y varies inversely with the
square of x.
Equation
y= a
x
z = axyr
a
y= 2
x
d. z varies directly with y and z = ay
x
inversely with x.
e. x varies jointly with t and r
and inversely with s.
atr
x= s
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
STEP 2
Use the given values of z, x, and y to find the constant
of variation a.
7 = a(1)(2)
Substitute 7 for z, 1 for x, and 2 for y.
7 = 2a
Simplify.
7
Solve for a.
2 =a
STEP 3
Rewrite the joint variation equation with the value of a
from Step 2.
z = 7 xy
2
GUIDED PRACTICE
for Examples 5 and 6
STEP 4
Calculate z when x = – 2 and y = 5 using substitution.
7
7
z = 2 xy = 2 (– 2)(5) = – 35
ANSWER
z = 7 xy ; – 35
2
GUIDED PRACTICE
10.
for Examples 5 and 6
x = 4, y = –3, z =24
SOLUTION
STEP 1
Write a general joint variation equation.
z = axy
STEP 2
Use the given values of z, x, and y to find the constant
of variation a.
Substitute 24 for z, 4 for x, and –3 for y.
24 = a(4)(– 3)
24 = –12a
–2 =a
Simplify.
Solve for a.
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
ANSWER
z = – 2 xy ; 20
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
STEP 2
Use the given values of z, x, and y to find the constant
of variation a.
18 = a(– 2)(6)
Substitute 18 for z, – 2 for x, and 6 for y.
18 = –12a
Simplify.
– 3 =a
Solve for a.
2
STEP 3
Rewrite the joint variation equation with the value of a
from Step 2.
z = – 3 xy
2
GUIDED PRACTICE
for Examples 5 and 6
STEP 4
Calculate z when x = – 2 and y = 5 using substitution.
z = – 3 xy = – 3 (– 2)(5) = 15
2
2
ANSWER
z = – 3 xy ; 15
2
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
STEP 2
Use the given values of z, x, and y to find the constant
of variation a.
56 = a(– 6)(–4)
Substitute 56 for z, – 6 for x, and – 4 for y.
56 = 24a
7
3 =a
Simplify.
Solve for a.
STEP 3
Rewrite the joint variation equation with the value of a
from Step 2.
z = 7 xy
3
GUIDED PRACTICE
for Examples 5 and 6
STEP 4
Calculate z when x = – 2 and y = 5 using substitution.
z = 7 xy = 7 (– 2)(5) = – 70
3
3
3
ANSWER
z = 7 xy ; – 70
3
3
GUIDED PRACTICE
for Examples 5 and 6
Write an equation for the given relationship.
13. x varies inversely with y and directly with w.
SOLUTION
x= a w
y
14. p varies jointly with q and r and inversely with s.
SOLUTION
p = aqr
s