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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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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.


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


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.


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


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.


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


= 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.


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


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


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.


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


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


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.


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


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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