Section 1.5 Rewriting Equations and Formulas 35
Rewriting Equations and Formulas
1.5
Essential QuestionEssential Question How can you use a formula for one
measurement to write a formula for a different measurement?
Using an Area Formula
Work with a partner.
a. Write a formula for the area A of
b
A = 30 in.
2
h = 5 in.
a parallelogram.
b. Substitute the given values into the
formula. Then solve the equation
for b. Justify each step.
c. Solve the formula in part (a) for b without rst substituting values into the formula.
Justify each step.
d. Compare how you solved the equations in parts (b) and (c). How are the processes
similar? How are they different?
Using Area, Circumference, and
Volume Formulas
Work with a partner. Write the indicated formula for each gure. Then write a new
formula by solving for the variable whose value is not given. Use the new formula to
nd the value of the variable.
a. Area A of a trapezoid b. Circumference C of a circle
A = 63 cm
2
b
2
= 10 cm
b
1
= 8 cm
h
C = 24 ft
r
π
c. Volume V of a rectangular prism d. Volume V of a cone
V = 75 yd
3
B = 15 yd
2
h
V = 24 m
3
h
π
B = 12 m
2
π
Communicate Your AnswerCommunicate Your Answer
3. How can you use a formula for one measurement to write a formula for a
different measurement? Give an example that is different from those given
in Explorations 1 and 2.
REASONING
QUANTITATIVELY
To be pro cient in math,
you need to consider the
given units. For instance,
in Exploration 1, the
area A is given in square
inches and the height h
is given in inches. A unit
analysis shows that the
units for the base b are
also inches, which
makes sense.
hsnb_alg1_pe_0105.indd 35hsnb_alg1_pe_0105.indd 35 2/4/15 3:05 PM2/4/15 3:05 PM
36 Chapter 1 Solving Linear Equations
1.5
Lesson
Rewriting a Literal Equation
Solve the literal equation 3y + 4x = 9 for y.
SOLUTION
3y + 4x = 9 Write the equation.
3y + 4x − 4x = 9 − 4x Subtract 4
x
from each side.
3y = 9 − 4x Simplify.
3y
—
3
=
9 − 4x
—
3
Divide each side by 3.
y = 3 −
4
—
3
x
Simplify.
The rewritten literal equation is y = 3 −
4
—
3
x.
What You Will LearnWhat You Will Learn
Rewrite literal equations.
Rewrite and use formulas for area.
Rewrite and use other common formulas.
Rewriting Literal Equations
An equation that has two or more variables is called a literal equation. To rewrite a
literal equation, solve for one variable in terms of the other variable(s).
Rewriting a Literal Equation
Solve the literal equation y = 3x + 5xz for x.
SOLUTION
y = 3x + 5xz Write the equation.
y = x(3 + 5z) Distributive Property
y
—
3 + 5z
=
x(3 + 5z)
—
3 + 5z
Divide each side by 3 + 5
z
.
y
—
3 + 5z
= x
Simplify.
The rewritten literal equation is x =
y
—
3 + 5z
.
In Example 2, you must assume that z ≠ −
3
—
5
in order to divide by 3 + 5z. In general, if
you have to divide by a variable or variable expression when solving a literal equation,
you should assume that the variable or variable expression does not equal 0.
Monitoring ProgressMonitoring Progress
Help in English and Spanish at BigIdeasMath.com
Solve the literal equation for y.
1. 3y − x = 9 2. 2x − 2y = 5 3. 20 = 8x + 4y
Solve the literal equation for x.
4. y = 5x − 4x 5. 2x + kx = m 6. 3 + 5x − kx = y
literal equation, p. 36
formula, p. 37
Previous
surface area
Core Vocabulary
Core Vocabu
l
l
a
r
r
y
REMEMBER
Division by 0 is unde ned.
hsnb_alg1_pe_0105.indd 36hsnb_alg1_pe_0105.indd 36 2/4/15 3:05 PM2/4/15 3:05 PM
Section 1.5 Rewriting Equations and Formulas 37
Using a Formula for Area
You own a rectangular lot that is 500 feet deep. It has an area of 100,000 square feet.
To pay for a new water system, you are assessed $5.50 per foot of lot frontage.
a. Find the frontage of your lot.
b. How much are you assessed for the new water system?
SOLUTION
a. In the formula for the area of a rectangle, let the width w represent the lot frontage.
A = w Write the formula for area of a rectangle.
A
—
= w Divide each side by to solve for
w
.
100,000
—
500
= w
Substitute 100,000 for
A
and 500 for .
200 = w Simplify.
The frontage of your lot is 200 feet.
b. Each foot of frontage costs $5.50, and
$5.50
—
1 ft
⋅
200 ft = $1100.
So, your total assessment is $1100.
Monitoring ProgressMonitoring Progress
Help in English and Spanish at BigIdeasMath.com
Solve the formula for the indicated variable.
7. Area of a triangle: A =
1
—
2
bh; Solve for h.
8. Surface area of a cone: S =
π
r
2
+
π
r ; Solve for .
Rewriting a Formula for Surface Area
The formula for the surface area S of a rectangular prism is S = 2 w + 2 h + 2wh.
Solve the formula for the length .
SOLUTION
S = 2 w + 2 h + 2wh Write the equation.
S − 2wh = 2 w + 2 h + 2wh − 2wh Subtract 2
wh
from each side.
S − 2wh = 2 w + 2 h Simplify.
S − 2wh = (2w + 2h) Distributive Property
S − 2wh
—
2w + 2h
=
(2w + 2h)
——
2w + 2h
Divide each side by 2
w
+ 2
h
.
S − 2wh
—
2w + 2h
=
Simplify.
When you solve the formula for , you obtain =
S − 2wh
—
2w + 2h
.
Rewriting and Using Formulas for Area
A formula shows how one variable is related to one or more other variables.
A formula is a type of literal equation.
h
w
500 ft
w
frontage
hsnb_alg1_pe_0105.indd 37hsnb_alg1_pe_0105.indd 37 2/4/15 3:05 PM2/4/15 3:05 PM
38 Chapter 1 Solving Linear Equations
Mercury
427°C
Venus
864°F
Rewriting and Using Other Common Formulas
Rewriting the Formula for Temperature
Solve the temperature formula for F.
SOLUTION
C =
5
—
9
(
F − 32) Write the temperature formula.
9
—
5
C =
F − 32 Multiply each side by
9
—
5
.
9
—
5
C
+ 32 = F − 32 + 32 Add 32 to each side.
9
—
5
C + 32 = F
Simplify.
The rewritten formula is F =
9
—
5
C + 32.
Using the Formula for Temperature
Which has the greater surface temperature: Mercury or Venus?
SOLUTION
Convert the Celsius temperature of Mercury to degrees Fahrenheit.
F =
9
—
5
C + 32
Write the rewritten formula from Example 5.
=
9
—
5
(
427) + 32 Substitute 427 for
C
.
= 800.6 Simplify.
Because 864°F is greater than 800.6°F, Venus has the greater surface temperature.
Monitoring ProgressMonitoring Progress
Help in English and Spanish at BigIdeasMath.com
9. A fever is generally considered to be a body temperature greater than 100°F. Your
friend has a temperature of 37°C. Does your friend have a fever?
Core Core ConceptConcept
Common Formulas
Temperature F = degrees Fahrenheit, C = degrees Celsius
C =
5
—
9
(F − 32)
Simple Interest I = interest, P = principal,
r = annual interest rate (decimal form),
t = time (years)
I = Prt
Distance d = distance traveled, r = rate, t = time
d = rt
hsnb_alg1_pe_0105.indd 38hsnb_alg1_pe_0105.indd 38 2/4/15 3:05 PM2/4/15 3:05 PM
Section 1.5 Rewriting Equations and Formulas 39
Using the Formula for Simple Interest
You deposit $5000 in an account that earns simple interest. After 6 months, the
account earns $162.50 in interest. What is the annual interest rate?
SOLUTION
To nd the annual interest rate, solve the simple interest formula for r.
I = Prt Write the simple interest formula.
I
—
Pt
= r
Divide each side by
Pt
to solve for
r
.
162.50
—
(5000)(0.5)
= r
Substitute 162.50 for
I
, 5000 for
P
, and 0.5 for
t
.
0.065 = r Simplify.
The annual interest rate is 0.065, or 6.5%.
Solving a Real-Life Problem
A truck driver averages 60 miles per hour while delivering freight to a customer. On
the return trip, the driver averages 50 miles per hour due to construction. The total
driving time is 6.6 hours. How long does each trip take?
SOLUTION
Step 1 Rewrite the Distance Formula to write expressions that represent the two trip
times. Solving the formula d = rt for t, you obtain t =
d
—
r
. So,
d
—
60
represents
the delivery time, and
d
—
50
represents the return trip time.
Step 2 Use these expressions and the total driving time to write and solve an
equation to nd the distance one way.
d
—
60
+
d
—
50
= 6.6
The sum of the two trip times is 6.6 hours.
11d
—
300
= 6.6
Add the left side using the LCD.
11d = 1980 Multiply each side by 300 and simplify.
d = 180 Divide each side by 11 and simplify.
The distance one way is 180 miles.
Step 3 Use the expressions from Step 1 to nd the two trip times.
So, the delivery takes 180 mi ÷
60 mi
—
1 h
= 3 hours, and the return trip takes
180 mi ÷
50 mi
—
1 h
= 3.6 hours.
Monitoring ProgressMonitoring Progress
Help in English and Spanish at BigIdeasMath.com
10. How much money must you deposit in a simple interest account to earn $500 in
interest in 5 years at 4% annual interest?
11. A truck driver averages 60 miles per hour while delivering freight and 45 miles
per hour on the return trip. The total driving time is 7 hours. How long does each
trip take?
COMMON ERROR
The unit of t is years. Be
sure to convert months
to years.
1 yr
—
12 mo
⋅
6 mo = 0.5 yr
hsnb_alg1_pe_0105.indd 39hsnb_alg1_pe_0105.indd 39 2/4/15 3:05 PM2/4/15 3:05 PM
40 Chapter 1 Solving Linear Equations
Exercises
1.5
Dynamic Solutions available at BigIdeasMath.com
In Exercises 3–12, solve the literal equation for y.
(See Example 1.)
3. y − 3x = 13 4. 2x + y = 7
5. 2y − 18x = −26 6. 20x + 5y = 15
7. 9x − y = 45 8. 6x − 3y = −6
9. 4x − 5 = 7 + 4y 10. 16x + 9 = 9y − 2x
11. 2 +
1
—
6
y = 3x + 4 12. 11 −
1
—
2
y = 3 + 6x
In Exercises 13–22, solve the literal equation for x.
(See Example 2.)
13. y = 4x + 8x 14. m = 10x − x
15. a = 2x + 6xz 16. y = 3bx − 7x
17. y = 4x + rx + 6 18. z = 8 + 6x − px
19. sx + tx = r 20. a = bx + cx + d
21. 12 − 5x − 4kx = y 22. x − 9 + 2wx = y
23. MODELING WITH MATHEMATICS The total cost
C (in dollars) to participate in a ski club is given by
the literal equation C = 85x + 60, where x is the
number of ski trips you take.
a. Solve the equation for x.
b. How many ski trips do
you take if you spend
a total of $315? $485?
24. MODELING WITH MATHEMATICS The penny
size of a nail indicates the length of the nail.
The penny size d is given by the literal
equation d = 4n − 2, where n is the
length (in inches) of the nail.
a. Solve the equation for n.
b. Use the equation from part (a) to nd
the lengths of nails with the following
penny sizes: 3, 6, and 10.
ERROR ANALYSIS In Exercises 25 and 26, describe and
correct the error in solving the equation for x.
25.
12 2x 2(y x)
2x 2(y x) 12
x (y x) 6
✗
26.
10 ax 3b
10 x(a 3b)
10
—
a 3b
x
✗
In Exercises 27–30, solve the formula for the indicated
variable. (See Examples 3 and 5.)
27. Pro t: P = R − C; Solve for C.
28. Surface area of a cylinder: S = 2
π
r
2
+ 2
π
rh;
Solve for h.
29. Area of a trapezoid: A =
1
—
2
h(b
1
+ b
2
); Solve for b
2
.
30. Average acceleration of an object: a =
v
1
− v
0
—
t
;
Solve for v
1
.
Monitoring Progress and Modeling with MathematicsMonitoring Progress and Modeling with Mathematics
1. VOCABULARY Is 9r + 16 =
π
—
5
a literal equation? Explain.
2. DIFFERENT WORDS, SAME QUESTION Which is different? Find “both” answers.
Solve 3x + 6y = 24 for x.
Solve 24 − 3x = 6y for x.
Solve 6y = 24 − 3x for y in terms of x.
Solve 24 − 6y = 3x for x in terms of y.
Vocabulary and Core Concept CheckVocabulary and Core Concept Check
n
hsnb_alg1_pe_0105.indd 40hsnb_alg1_pe_0105.indd 40 2/3/16 10:27 AM2/3/16 10:27 AM
Section 1.5 Rewriting Equations and Formulas 41
31. REWRITING A FORMULA A common statistic used in
professional football is the quarterback rating. This
rating is made up of four major factors. One factor is
the completion rating given by the formula
R = 5
(
C
—
A
− 0.3
)
where C is the number of completed passes and A is
the number of attempted passes. Solve the formula
for C.
32. REWRITING A FORMULA Newton’s law of gravitation
is given by the formula
F = G
(
m
1
m
2
—
d
2
)
where F is the force between two objects of masses
m
1
and m
2
, G is the gravitational constant, and d is
the distance between the two objects. Solve the
formula for m
1
.
33. MODELING WITH MATHEMATICS The sale price
S (in dollars) of an item is given by the formula
S = L − rL, where L is the list price (in dollars)
and r is the discount rate (in decimal form).
(See Examples 4 and 6.)
a. Solve the formula for r.
Sale price:
$18
Sale price:
$18
b. The list price of the shirt
is $30. What is the
discount rate?
34. MODELING WITH MATHEMATICS The density d of a
substance is given by the formula d =
m
—
V
, where m is
its mass and V is its volume.
Density: 5.01g/cm
3
Volume: 1.2 cm
3
Pyrite
a. Solve the formula for m.
b. Find the mass of the pyrite sample.
35. PROBLEM SOLVING You deposit $2000 in an account
that earns simple interest at an annual rate of 4%. How
long must you leave the money in the account to earn
$500 in interest? (See Example 7.)
36. PROBLEM SOLVING A ight averages 460 miles per
hour. The return ight averages 500 miles per hour
due to a tailwind. The total ying time is 4.8 hours.
How long is each ight? Explain. (See Example 8.)
37. USING STRUCTURE An athletic facility is building an
indoor track. The track is composed of a rectangle and
two semicircles, as shown.
x
r
r
a. Write a formula for the perimeter of the
indoor track.
b. Solve the formula for x.
c. The perimeter of the track is 660 feet, and r is
50 feet. Find x. Round your answer to the
nearest foot.
38. MODELING WITH MATHEMATICS The distance
d (in miles) you travel in a car is given by the two
equations shown, where t is the time (in hours) and
g is the number of gallons of gasoline the car uses.
d 55t
d 20g
a. Write an equation that relates g and t.
b. Solve the equation for g.
c. You travel for 6 hours. How many gallons of
gasoline does the car use? How far do you travel?
Explain.
hsnb_alg1_pe_0105.indd 41hsnb_alg1_pe_0105.indd 41 2/4/15 3:06 PM2/4/15 3:06 PM
42 Chapter 1 Solving Linear Equations
39. MODELING WITH MATHEMATICS One type of stone
formation found in Carlsbad Caverns in New Mexico
is called a column. This cylindrical stone formation
connects to the ceiling and the oor of a cave.
column
stalagmite
a. Rewrite the formula for the circumference of a
circle, so that you can easily calculate the radius
of a column given its circumference.
b. What is the radius (to the nearest tenth of a foot)
of a column that has a circumference of 7 feet?
8 feet? 9 feet?
c. Explain how you can nd the area of a
cross section of a column when you know its
circumference.
40. HOW DO YOU SEE IT? The rectangular prism shown
has bases with equal side lengths.
b
b
a. Use the gure to write a formula for the surface
area S of the rectangular prism.
b. Your teacher asks you to rewrite the formula
by solving for one of the side lengths, b or .
Which side length would you choose? Explain
your reasoning.
41. MAKING AN ARGUMENT Your friend claims that
Thermometer A displays a greater temperature than
Thermometer B. Is your friend correct? Explain
your reasoning.
yg
yg
y
g
−10
0
20
10
30
40
50
60
70
80
90
100
F
Thermometer A
Thermometer B
42. THOUGHT PROVOKING Give a possible value for h.
Justify your answer. Draw and label the gure using
your chosen value of h.
A = 40 cm
2
8 cm
h
MATHEMATICAL CONNECTIONS In Exercises 43 and 44,
write a formula for the area of the regular polygon.
Solve the formula for the height h.
43 .
b
h
center
44.
b
h
center
REASONING In Exercises 45 and 46, solve the literal
equation for a.
45. x =
a + b + c
—
ab
46 . y = x
(
ab
—
a − b
)
Maintaining Mathematical ProficiencyMaintaining Mathematical Proficiency
Evaluate the expression. (Skills Review Handbook)
47. 15 − 5 + 5
2
48. 18
⋅
2 − 4
2
÷ 8 49. 3
3
+ 12 ÷ 3
⋅
5 50. 2
5
(5 − 6) + 9 ÷ 3
Solve the equation. Graph the solutions, if possible. (Section 1.4)
51.
x − 3
+ 4 = 9 52.
3y − 12
− 7 = 2 53. 2
2r + 4
= −16 54. −4
s + 9
= −24
Reviewing what you learned in previous grades and lessons
hsnb_alg1_pe_0105.indd 42hsnb_alg1_pe_0105.indd 42 2/4/15 3:06 PM2/4/15 3:06 PM
