EXAMPLE 1
Rewrite a formula with two variables
Solve the formula C = 2πr for r. Then find the radius of
a circle with a circumference of 44 inches.
SOLUTION
STEP 1 Solve the formula for r.
C=2πr
Write circumference formula.
C
Divide each side by 2π.
2π = r
STEP 2 Substitute the given value into the rewritten formula.
C 44
7 Substitute 44 for C and simplify.
r = 2π = 2π
ANSWER The radius of the circle is about 7 inches.
GUIDED PRACTICE
for Example 1
1. Find the radius of a circle with a circumference of
25 feet.
SOLUTION
STEP 1 Solve the formula for r.
C=2πr
Write circumference formula.
C
Divide each side by 2π.
2π = r
STEP 2 Substitute the given value into the rewritten formula.
C 25
3.9 Substitute 25 for C and simplify.
r = 2π = 2π
ANSWER The radius of the circle is about 4 feet.
GUIDED PRACTICE
2.
for Example 1
The formula for the distance d between opposite
vertices of a regular hexagon is d = 2a where a is
3
the distance between opposite sides. Solve the
formula for a. Then find a when d = 10 centimeters.
SOLUTION
a=
d 3
2
When d = 10cm, a = 5 3 cm or
8.7cm
EXAMPLE 2
Rewrite a formula with three variables
Solve the formula P = 2l + 2w for w. Then find the width of
a rectangle with a length of 12 meters and a perimeter of
41 meters.
SOLUTION
STEP 1 Solve the formula for w.
P = 2l + 2w
Write perimeter formula.
P – 2l = 2w
Subtract 2l from each side.
P – 2l = w
2
Divide each side by 2.
EXAMPLE 2
Rewrite a formula with three variables
STEP 2
Substitute the given values into the rewritten formula.
41 – 2(12)
w=
2
Substitute 41 for P and 12 for l.
w = 8.5
Simplify.
ANSWER
The width of the rectangle is 8.5 meters.
GUIDED PRACTICE
for Example 2
3. Solve the formula P = 2l + 2w for l. Then find the length
of a rectangle with a width of 7 inches and a perimeter
of 30 inches.
STEP 1 Solve the formula for p = 2l + 2w for l
P = 2l + 2w Write perimeter formula.
P – 2w = 2l
P – 2w = l
2
Subtract 2w from each side.
Divide each side by 2
GUIDED PRACTICE
for Example 2
STEP 2
Substitute the given values in.
Formula for
l = P – 2w
2
30 – 2 (7)
Multiply.
=
2
= 16
2
Subtract.
=8
Divide.
ANSWER
Length of rectangle is 8 in.
GUIDED PRACTICE
for Example 2
4. Solve the formula A = lw for w. Then find the width of a
rectangle with a length of 16 meters and an area of 40
square meters.
STEP 1 Solve the formula for w
A = lw
Write perimeter formula.
A =w
l
Divide each side by l
GUIDED PRACTICE
for Example 2
STEP 2
Substitute the given values into rewrite formula.
w = A Write perimeter formula.
l
w = 40 Subtract 40 from A and 16 for l.
16
w = 2.5 Divide
Write of rectangle is 2.5 m
GUIDED PRACTICE
for Example 2
Solve the formula for the variable in red. Then use the
given information to find the value of the variable.
1 bh
A
=
5.
2
Find h if b = 12 m
and A = 84 m2.
A=
1 bh
2
Write perimeter formula.
2A = bh
Multiply each side by 2.
2A = h
b
Divide each side by b
GUIDED PRACTICE
for Example 2
Find the value of h if b = 12m and A = 84m2.
2A = h
b
2A = h
b
2(84) = h
12
h = 14
h = 14m
Find h if b = 12 m
and A = 84 m2.
Write formula.
Substitute 84 for A and 12 for b.
Simplify
GUIDED PRACTICE
for Example 2
Solve the formula for the variable in red. Then use the
given information to find the value of the variable.
1 bh
A
=
6.
Find b if h = 3 cm
2
and A = 9 cm2.
A=
1 bh
2
Write perimeter formula.
2A = bh
Multiply each side by 2.
2A = b
h
Divide each side by h
GUIDED PRACTICE
for Example 2
Solve the formula for the variable in red. Then use the
given information to find the value of the variable.
1 bh
A
=
6.
Find b if h = 3 cm
2
and A = 9 cm2.
2A = b
h
2(9) = b
3
b=6
b = 6cm
Write formula.
Substitute 9 for A and 3 for b.
Simplify
GUIDED PRACTICE
for Example 2
Solve the formula for the variable in red. Then use the
given information to find the value of the variable.
1 (b + b )h
A
=
7.
2
Find h if b1 = 6 in.,
2 1
b2 = 8 in., and A = 70 in.2
A=
1 (b + b )h
2
2 1
2A = (b1 + b2)h
h=
2A
(b1 + b2)
Write perimeter formula.
Multiply each side by 2.
Divide by (b1 + b2)
GUIDED PRACTICE
for Example 2
Solve the formula for the variable in red. Then use the
given information to find the value of the variable.
1 (b + b )h
A
=
7.
2
Find h if b1 = 6 in.,
2 1
b2 = 8 in., and A = 70 in.2
h=
2A
(b1 + b2)
Write formula.
h=
2(70)
(6 + 8)
Substitute 70 for A and 6 for b1 and 8 for b2.
h = 10
h = 10 in.
Simplify
EXAMPLE 3
Rewrite a linear equation
Solve 9x – 4y = 7 for y. Then find the value of y when
x = –5.
SOLUTION
STEP 1 Solve the equation for y.
9x – 4y = 7
Write original equation.
Subtract 9x from each side.
– 4y = 7 – 9x
y = – 7 + 9 x Divide each side by – 4.
4
4
EXAMPLE 3
Rewrite a linear equation
STEP 2 Substitute the given value into the rewritten
equation.
y = – 7 + 9 (–5) Substitute –5 for x.
4
4
Multiply.
y = – 7 – 45
4
4
y = – 13
Simplify.
CHECK
9x – 4y = 7 Write original equation.
9(– 5) – 4(– 13) =? 7 Substitute –5 for x and –13 for y.
7 = 7 Solution checks.
EXAMPLE 4
Rewrite a nonlinear equation
Solve 2y + xy = 6 for y. Then find the value of y when
x = –3.
SOLUTION
STEP 1 Solve the equation for y.
2y + x y = 6
Write original equation.
(2+ x) y = 6
Distributive property
y=
6
2+x
Divide each side by ( 2 + x).
EXAMPLE 4
Rewrite a nonlinear equation
STEP 2 Substitute the given value into the rewritten
equation.
6
y=
2 + (– 3) Substitute – 3 for x.
y=–6
Simplify.
GUIDED PRACTICE
for Examples 3 and 4
Solve the equation for y. Then find the value of y when
x = 2.
8. y – 6x = 7
STEP 1 Solve the equation for y.
y – 6x = 7
y = 7 + 6x
STEP 2 Substitute the given value into the rewritten
equation.
y = 7 + 6 (2) Substitute 2 for n.
y = 7 + 12
Multiply.
y = 19
Add.
GUIDED PRACTICE
CHECK
y – 6x = 7
19 – 6 (2) =? 7
for Examples 3 and 4
Write original equation.
Substitute 2 for x and 19 for y.
7 = 7 Solution checks.
GUIDED PRACTICE
for Examples 3 and 4
Solve the equation for y. Then find the value of y when
x = 2.
9. 5y – x = 13
STEP 1 Solve the equation for y.
5y – x = 13
5y = 13 + x
Write original equation
Add x to each side
13
x Divide each side by 5
+
y= 5
5
GUIDED PRACTICE
for Examples 3 and 4
STEP 2 Substitute the given value into the rewritten
equation.
13
2
y= 5 + 5
y=5
Substitute
2 for n.
Simplify.
CHECK
5y – x = 13
5(5) – 2 =? 7
3 = 3
Write original equation.
Substitute
2 for x and 19 for y.
Solution checks.
GUIDED PRACTICE
for Examples 3 and 4
Solve the equation for y. Then find the value of y when
x = 2.
10. 3x + 2y = 12
STEP 1 Solve the equation for y.
3x + 2y = 12
5y = 12 – 3x
Write original equation
Subtract 3x from each side
12 3x
y = 2 + 2 Divide each side by 2
= – 3x + 6
2
GUIDED PRACTICE
for Examples 3 and 4
STEP 2 Substitute the given value into the rewritten
equation.
(2)
Substitute 2 for n.
y = – 3 2 +6
–2
y = 2 +6
Multiply
y=3
Simplify
CHECK
3x – yx = 12
3(2) + 2(3) =? 12
12 = 12
Write original equation.
Substitute
3 for y and 2 for x.
Solution checks.
GUIDED PRACTICE
for Examples 3 and 4
Solve the equation for y. Then find the value of y when
x = 2.
11. 2x + 5y = –1
STEP 1 Solve the equation for y.
2x + 5y = –1
5y = –1 – 2x
Write original equation
Subtract 2x from each side
–1 2x
y = 5 – 5 Divide each side by 5
–1 2x
= 5 –5
GUIDED PRACTICE
for Examples 3 and 4
STEP 2 Substitute the given value into the rewritten
equation.
– 1 – 2(2)
y= 2
5
–1– 4
y= 5
5
y=–1
Substitute 2 for n.
Multiply
Simplify
CHECK
2x + 5y = –1
2(2) + 5(–1) =? –1
–1 = –1
Write original equation.
Substitute
2 for x and –1 for y.
Solution checks.
GUIDED PRACTICE
for Examples 3 and 4
Solve the equation for y. Then find the value of y when
x = 2.
12. 3 = 2xy – x
STEP 1 Solve the equation for y.
3 = 2xy – x
Write original equation
3 + x = 2xy
Add x to each side
3 +x = y
2x
Divide 2x to each side
GUIDED PRACTICE
for Examples 3 and 4
STEP 2 Substitute the given value into the rewritten
equation.
3+2 =y
2 (2)
5 =1 1
y= 4
4
Substitute
2 for n.
Simplify
CHECK
3= 2xy –x
3 =? 2(2)
3 = 3
(
Write original equation.
5 ) – (2) Substitute 2 for x and 5 1 for y.
4
4
Solution checks.
GUIDED PRACTICE
for Examples 3 and 4
Solve the equation for y. Then find the value of y when
x = 2.
13. 4y – xy = 28
STEP 1 Solve the equation for y.
4y – xy = 28
Write original equation
(4 – x)y = 28
Distributive property
28 = y
4–x
Divide each side by (4 – x)
GUIDED PRACTICE
for Examples 3 and 4
STEP 2 Substitute the given value into the rewritten
equation.
28
y= 4–2
Substitute
y = 14
Simplify
2 for x.
CHECK
4y –xy = 28
4(14) – (2) (14) =? 28
28 = 28
Write original equation.
Substitute
14 for y and 2 for y.
Solution checks.