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Unit1: Equations and
Inequalities
1.4 Rewriting Formulas and
Equations
1.4- Rewriting Formulas and Equations
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.
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
EXAMPLE 2
Rewrite a formula with three variables
Solve the formula P = 2l + w 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.
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
=
4.
2
Find h if b = 12 m
and A = 84 m2.
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
=
5.
2
Find h if b1 = 6 in.,
2 1
b2 = 8 in., and A = 70 in.2
EXAMPLE 3
Rewrite a linear equation
Solve 9x – 4y = 7 for y. Then find the value of y when
x = –5.
STEP 1 Solve the equation for y.
Write original equation.
Subtract 9x from each side.
Divide each side by –4.
EXAMPLE 4
Rewrite a nonlinear equation
Solve 2y + xy = 6 for y. Then find the value of y when
x = –3.
STEP 1 Solve the equation for y.
Write original equation.
Distributive property
Divide each side by (2 + x).
GUIDED PRACTICE
for Examples 3 and 4
Solve the equation for y. Then find the value of y when
x = 2.
6. y – 6x = 7
7. 5y – x = 13
8. 4y – xy = 28
ANSWER
ANSWER
ANSWER
y = 7 + 6x
y = 19
13
x
y= 5 + 5
y=5
y = – 3x + 6
2
y=3
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
12. 3 = 2xy – x
ANSWER
ANSWER
y = –1 – 2x
5
5
y = –1
+x
y = 32x
1
y= 1
4
13. 4y – xy = 28
ANSWER
y = 428
–x
y = 14
EXAMPLE 5
Solve a multi-step problem
Movie Rental
A video store rents new movies for one price and
older movies for a lower price, as shown at the right.
• Write an equation that represents the store’s
monthly revenue.
• Solve the revenue equation for the variable
representing the number of new movies rented.
• The owner wants $12,000 in revenue per month. How
many new movies must be rented if the number of older
movies rented is 500? 1000?
EXAMPLE 5
Solve a multi-step problem
SOLUTION
STEP 1
Write a verbal model. Then write an equation.
An equation is R = 5n1 + 3n2.
STEP 2
Solve the equation for n1.
EXAMPLE 5
Solve a multi-step problem
R = 5n1 + 3n2 Write equation.
R – 3n2 = 5n1
Subtract 3n2 from each side.
R – 3n2
= n1
5
Divide each side by 5.
STEP 3 Calculate n1 for the given values of R and n2.
If n2 = 500, then n1 = 12,000 – 3 500 = 2100.
5
If n2 = 1000, then n1 = 12,000 – 3 1000 = 1800.
5
ANSWER
If 500 older movies are rented, then 2100 new movies
must be rented. If 1000 older movies are rented, then
1800 new movies must be rented.
GUIDED PRACTICE
for Example 5
14. What If? In Example 5, how many new movies must
be rented if the number of older movies rented is 1500?
ANSWER
If 1500 older movies are rented, then 1500
new movies must be rented
GUIDED PRACTICE
for Example 5
15. What If? In Example 5, how many new movies
must be rented if customers rent no older movies at
all?
ANSWER
If 0 older movies are rented, then 2400 new
movie must be rented
GUIDED PRACTICE
for Example 5
16. Solve the equation in Step 1 of Example 5 for n2.
ANSWER
R – 5n1
n2 =
3