Big Ideas Math Answers 8th Grade Chapter 1 Equations are provided in a comprehensive manner for better understanding. Solve all the questions from Big Ideas Math Book 8th Grade Chapter 1 Equations Answer Key and become an expert in equations topic. Thus, you can find answers for any kind of question framed on the concept and clear your tests or assessments with higher scores. Hence Download Big Ideas Math Answers Grade 8 Chapter 1 Equations pdf and learn the fundamentals in an easy manner.

## Big Ideas Math Book 8th Grade Answer Key Chapter 1 Equations

The list of topics in Equations is given in the below section. The solutions are prepared as per the list of the topics in Big Ideas Math Book 8th Grade Chapter 1 Equations. To help you out with chapter 4 Multiply Whole Numbers we have jotted down all the exercise problems in your Big Ideas Math Textbook. Just tap on the respective link you wish to prepare and learn the fundamentals included within it easily. Here you can learn the topics like Solving Simple Equations, Solving Multi-step Equations, Solving Equations with Variables on Both Sides, Rewriting Equations and Formulas.

performance

Lesson: 1 Solving Simple Equations

Lesson: 2 Solving Multi-step Equations

Lesson: 3 Solving Equations with Variables on Both Sides

Lesson: 4 Rewriting Equations and Formulas

Chapter: 1 – Equations

### STEAM Video/performance

STEAM Video

A half marathon is a race that is 13.1 miles long. How can a runner develop a routine to help train for a half marathon?

Watch the STEAM Video “Training for a Half Marathon.” Then answer the following questions.

Question 1.
Alex and Enid are training for a half marathon. They run four days each week, as shown in the table. How far do they have to run on Saturday to average 4.75 miles per running day in Week Nine?

They have to run 13.1 miles on Saturday in Week Nine.

Explanation:
The average on Saturday = 4.7 miles per running day
Sum of distance ran miles/number of weeks = 4.7
(7.0 + 7.0 + 8.5 + x)/4 = 4.7
(22.5 + x)/4 = 4.7
22.5 + x = 4.7 x 4
22.5 + x = 9.4
x = 9.4 – 22.5
x = -13.1
They have to run 13.1 miles on Saturday in Week Nine.

Question 2.
Assuming they meet their goal on Saturday in Week Nine, what is the average number of miles per running day over the 4 weeks in the table?

The average number of miles per running day over the 4 weeks in the table is 2.875 miles per day.

Explanation:
Given the running details on week 4 is
Monday – 2.0, Wednesday – 2.1, Friday – 1.9, Saturday – 5.5
The average number of miles per running day of week 4 = (2.0 + 2.1 + 1.9 + 5.5)/4
= 11.5/4
= 2.875 miles per day.

Target Heart Rates

After completing this chapter, you will be able to use the concepts you learned to answer the questions in the STEAM Video Performance Task. You will be given information about a person’s heart rate.

You will be asked to ﬁnd the range of a person’s target heart rate. What factors might affect the range of a person’s target heart rate?

### Getting Ready for Chapter 1

Chapter Exploration

Question 1.
Work with a partner. Use algebra tiles to model and solve each equation.

c. x – 4 = 1
d. x + 5 = -2
e. -7 = x + 4
f. x + 6 = 7
g. -5 + x = -3
h. -4 = x – 4

a. x = -6
b. x = -1
c. x = 5
d. x = -7
e. x = -11
f. x = 1
g. x = 2
h. x = 0

Explanation:
a. x + 3 = -3
Subtract 3 from both sides
x + 3 – 3 = -3 – 3
x = -6
b. -3 = x – 2
-3 + 2 = x – 2 + 2
-1 = x
c. x – 4 = 1
x – 4 + 4 = 1 + 4
x = 5
d. x + 5 = -2
Subtract 5 from each side
x + 5 – 5 = -2 – 5
x = -7
e. -7 = x + 4
Subtract 4 from each side
-7 – 4 = x + 4 – 4
-11 = x
f. x + 6 = 7
Subtract 6 from each side
x + 6 – 6 = 7 – 6
x = 1
g. -5 + x = -3
-5 + x + 5 = -3 + 5
x = 2
h. -4 = x – 4
-4 + 4 = x – 4 + 4
0 = x

Question 2.
WRITE GUIDELINES
Work with a partner. Use your models in Exercise 1 to summarize the algebraic steps that you can use to solve an equation.

Write the given equation as it is.
And add or subtract the same quantity on both sides of the equation that makes variable one side of the equation.
Simplify the equation to get the value variable.

Vocabulary

The following vocabulary term is deﬁned in this chapter. Think about what the term might mean and record your thoughts.

### Lesson 1.1 Solving Simple Equations

EXPLORATION 1

Work with a partner.
a. You have used the following properties in a previous course. Explain the meaning of each property.

• Subtraction Property of Equality
• Multiplication Property of Equality
• Division Property of Equality

b. Which property can you use to solve each of the equations modeled by the algebra tiles? Solve each equation and explain your method.

c. Write an equation that can be solved using one property of equality. Exchange equations with another pair and ﬁnd the solution.

1.1 Lesson

Try It

Solve the equation. Check your solution.

Question 1.
b + 2 = -5

b = -7

Explanation:
Given equation is b + 2 = -5
Subtract 2 from each side
b + 2 – 2 = -5 – 2
b = -7
Putting b = -7 in b + 2 = -5
-7 + 2 = -5

Question 2.
-3 = k + 3

k = -6

Explanation:
Given equation is -3 = k + 3
Subtract 3 from each side
-3 -3 = k + 3 – 3
-6 = k
Putting k = -6 in -3 = k + 3
-3 = -6 + 3

Question 3.

t = -1/2

Explanation:
Given equation is t – 1/4 = -3/4
t – 1/4 + 1/4 = -3/4 + 1/4
t = (-3 + 1)/4
t = -2/4
t = -1/2
Putting t = -1/2 in t – 1/4 = -3/4
-1/2 – 1/4 = (-2 – 1)/4
= -3/4

Try It

Solve the equation. Check your solution.

Question 4.

y = -28

Explanation:
Given equation is y/4 = -7
Multiply each side by 4
y/4 x 4 = -7 x 4
y = -28
Putting y = -28 in y/4 = -7
-28/4 = -7

Question 5.

z = -9

Explanation:
Given equation is -2z/3 = 6
Multiply each side by 3
-2z/3 x 3 = 6 x 3
-2z = 18
Divide each side by -2
-2z/-2 = 18/-2
z = -9
Putting z = -9 in -2z/3 = 6
[-2(-9)]/3 = 18/3 = 6

Question 6.
0.09w = 1.8

w = 20

Explanation:
Given equation is 0.09w = 1.8
Divide each side by 0.09
0.09w/0.09 = 1.8/0.09
w = 20
putting w = 20 in 0.09w = 1.8
0.09 x 20 = 1.8

Question 7.
6π = πx

x = 6

Explanation:
Given equation is 6π = πx
Divide each side by π
6π/π = πx/π
6 = x
Putting x = 6 in 6π = πx
6π = π(6)

Try It

Solve the equation. Check your solution.

Question 8.

p = 13

Explanation:
Given equation is p – 8 ÷ 1/2 = -3
p – 16 = -3
p – 16 + 16 = -3 + 16
p = 13
Putting p = 13 in p – 8 ÷ 1/2 = -3
13 – 8 ÷ 1/2 = -3
13 – 16 = -3

Question 9.
q + | -10 | = 2

q = -8

Explanation:
q + | -10 | = 2
Mod of negative 10 is 10
q + 10 = 2
Subtract 10 from each side
q + 10 – 10 = 2 – 10
q = -8
Putting q = -8 in q + | -10 | = 2
-8 + | -10 | = -8 + 10 = 2

Self-Assessment for Concepts & Skills
Solve each exercise. Then rate your understanding of the success criteria in your journal.

WRITING
Are the equations equivalent? Explain.

Question 10.
x + 3 = 4 and x = 1

Yes, equations are equivalent.

Explanation:
Given equations are x + 3 = 4 and x = 1
x = 4 – 3 & x = 1
x = 1 & x = 1

Question 11.

No, equations are not equivalent.

Explanation:
Given equations are -y/5 = 2 and y = 10
-y = 2 x 5 & y = 10
-y = 10 & y = 10
y = -10 & y = 10

Question 12.
OPEN-ENDED
Write an equation that you can use the Division Property of Equality to solve.

3x + 6 = 36

Explanation:
3x + 6 = 36
Subtract 6 from each side
3x + 6 – 6 = 36 – 6
3x = 30
Dvide each side by 3
3x/3 = 30/3
x = 10

SOLVING EQUATIONS
Solve the equation. Check your solution.

Question 13.
-5 = w – 3

w = -2

Explanation:
-5 = w – 3
-5 + 3 = w – 3 + 3
-2 = w
Putting w = -2 in -5 = w – 3
-5 = -2 – 3

Question 14.

n = -12

Explanation:
Given equation is -2/3 n = 8
multiply each side by 3
-2/3 n x 3 = 8 x 3
-2n = 24
Divide each side by -2
-2n/-2 = 24/-2
n = -12
Putting n = -12 in -2/3 n = 8
-2/3 (-12) = -2 x -4 = 8

Question 15.

p = 33

Explanation:
Given equation is p – 9 ÷ 1/3 = 6
p – 27 = 6
p – 27 + 27 = 6 + 27
p = 33
Putting p = 33 in p – 9 ÷ 1/3 = 6
33 – 9 ÷ 1/3 = 33 – 27
= 6

Question 16.
q + | 3 | = -5

q = -8

Explanation:
q + | 3 | = -5
q + 3 = -5
Subtract 3 from each side
q + 3 – 3 = -5 – 3
q = -8
Putting q = -8 in q + | 3 | = -5
-8 + | 3 | = -8 + 3
= -5

Question 17.
WHICH ONE DOESN’T BELONG?
Which equation does not belong with the other three? Explain your reasoning.

x – 3 = 9 does not belong with the other three.

Explanation:
x – 2 = 4
x = 4 + 2
x = 6
x – 3 = 9
x = 9 + 3
x = 12
x – 5 = 1
x = 1 + 5
x = 6
x – 6 = 0
x = 6
So, x – 3 = 9 does not belong with the other three.

Self-Assessment for Problem Solving

Solve each exercise. Then rate your understanding of the success criteria in your journal.

Question 18.
A shipwreck is 300 meters away from a diving station. An undersea explorer travels away from the station at a speed of 2 meters per second. The explorer is x meters away from the station and will reach the shipwreck in 100 seconds. What is the value of x?

The value of x is 100.

Explanation:
Given that,
The distance from shipwreck to the diving station = 300 m
The distance formula is d = speed x time
The distance from the explorer to the shipwreck = (2 m/s) x (100s)
= 200 m
He is also x meters away from the diving station
So, the distance between the diving station and shipwreck is x + 20 m
x + 20 = 300
Subtract 200 from both sides
x + 200 – 200 = 300 – 200
x = 100

Question 19.
You conduct an inventory for a hardware store and count 40 rolls of duct tape. Your manager wants to keep 7 boxes of duct tape in stock. If each box holds 8 rolls of duct tape, how many boxes should you order? Justify your answer.

I should order 2 boxes of duct tapes

Explanation:
Number of rolls of duct tape available = 40
Number of rolls of duct tape each box can hold = 8 rols
Number of boxes of duct tape the manager wants to keep in stock = 7
Let x represent the number of boxes of duct tape available
Then 8x = 40
Divide each side by 8
8x/8 = 40/8
x = 5
5 boxes of duct tapes are available in stock.
Number of boxes I order = Number of boxes in the stock – Number of boxes available = 7 – 5
= 2
So, I should order 2 boxes of duct tapes

Question 20.
DIG DEEPER!
Your ﬁtness tracker overestimates the number of steps you take by 5%. The tracker indicates that you took 7350 steps today. Write and solve an equation to ﬁnd the actual number of steps you took today.

Your actual number of steps is 7000.

Explanation:
Let x be the actual number of steps
Since the fitness tracker overestimates, the sum of the number of actual steps and 5% of its equal to 7350
Then, the equation will be
x + 0.05x = 7350
1.05x = 7350
Divide each side by 1.05
1.05x/1.05 = 7350/1.05
x = 7000
So, your actual number of steps is 7000.

### Solving Simple Equations Homework & Practice 1.1

Review & Refresh

Evaluate the expression.

Question 1.
(32 – 8) + 4

(32 – 8) + 4 = 5

Explanation:
Given expression is (32 – 8) + 4
= (9 – 8) + 4
= 1 + 4
= 5

Question 2.
1 + 5 × 32

1 + 5 × 3² = 41

Explanation:
Given expression is 1 + 5 × 3²
= 1 + 5 x 9
= 1 + 40
= 41

Question 3.
4 × 3 + 102

4 × 3 + 10² = 112

Explanation:
Given expression is 4 × 3 + 10²
= 4 x 3 + 10 x 10
= 12 + 100
= 112

Identify the terms, coefficients, and constants in the expression.

Question 4.
11q + 2

The constant = 2
Terms = p
Coefficients = 11

Explanation:
Given that,
11q + 2
The constant = 2
Terms = p
Coefficients = 11

Question 5.
h + 9 + g

The constant = 9
Terms = h, g
Coefficients = 1, 1

Explanation:
Given that,
h + 9 + g
The constant = 9
Terms = h, g
Coefficients = 1, 1

Question 6.
6m2 + 7n

The constant = 0
Terms = m², n
Coefficients = 6, 7

Explanation:
Given that,
6m² + 7n
The constant = 0
Terms = m², n
Coefficients = 6, 7

Write the phrase as an expression.

Question 7.
the quotient of 22 and a number a

22/a

Explanation:
Given that,
the quotient of 22 and a number a
= 22/a

Question 8.
the difference of a number t and 9

t – 9

Explanation:
Given that,
the difference of a number t and 9
t – 9

Concepts, Skills, &Problem Solving
USING PROPERTIES OF EQUALITY
Which property of equality can you use to solve the equation modeled by the algebratiles? Solve the equation and explain your method. (See Exploration 1, p. 3.)

Question 9.

x = 4

Explanation:
x – 5 = -1
x – 5 + 5 = -1 + 5
x = 4

Question 10.

x = 3

Explanation:
x + x + x = + 1 +1 +1 +1 +1 +1 +1 +1 +1
3x = 9
Divide each side by 3
3x/3 = 9/3
x = 3

SOLVING EQUATIONS USING ADDITION OR SUBTRACTION
Solve the equation. Check your solution.

Question 11.
x + 12 = 7

x = -5

Explanation:
Given equation is
x + 12 = 7
Subtract 12 from both sides
x + 12 – 12 = 7 – 12
x = -5
Putting x = -5 in x + 12 = 7
-5 + 12 = 7

Question 12.
g – 16 = 8

g = 24

Explanation:
Given equation is
g – 16 = 8
g – 16 + 16 = 8 + 16
g = 24
Putting g = 24 in g – 16 = 8
24 – 16 = 8

Question 13.
-9 + p = 12

p = 21

Explanation:
Given equation is -9 + p = 12
-9 + p + 9 = 12 + 9
p = 21
putting p = 21 in -9 + p = 12
-9 + 21 = 12

Question 14.
2.5 + y = -3.5

y = -5.5

Explanation:
Given equation is
2.5 + y = -3.5
Subtract 2.5 from each side
2.5 + y -2.5 = -3.5 – 2.5
y = -5.5
putting y = -5.5 in 2.5 + y = -3.5
2.5 – 5.5 = -3.5

Question 15.
x – 8π = π

x = 9π

Explanation:
Given equation is
x – 8π = π
x – 8π + 8π = π + 8π
x = 9π
Putting x = 9π in x – 8π = π
9π – 8π = π

Question 16.
4π = w – 6π

w = 10π

Explanation:
Given equation is
4π = w – 6π
4π + 6π= w – 6π + 6π
10π = w
putting w = 10π in 4π = w – 6π
4π = 10π – 6π

Question 17.

d = 2/3

Explanation:
Given equation is 5/6 = 1/6 + d
Subtract 1/6 from each side
5/6 -1/6 = 1/6 + d – 1/6
(5 – 1)/6 = d
d = 4/6
d = 2/3
Putting d = 2/3 in 5/6 = 1/6 + d
5/6 = 1/6 + 2/3
= (1 + 4)/6 = 5/6

Question 18.

r = -7/24

Explanation:
Given equation is 3/8 = r + 2/3
Subtract 2/3 from each side
3/8 – 2/3 = r + 2/3 – 2/3
(9 – 16)/24 = r
-7/24 = r
Putting r = -7/24 in 3/8 = r + 2/3
3/8 = -7/24 + 2/3
= (-7 + 16)/24 = 9/24
= 3/8

Question 19.
n – 1.4 = -6.3

n = -4.9

Explanation:
Given equation is n – 1.4 = -6.3
n – 1.4 + 1.4 = -6.3 + 1.4
n = -4.9
Putting n = -4.9 in n – 1.4 = -6.3
-4.9 – 1.4 = -6.3

Question 20.
MODELING REAL LIFE
A discounted concert ticket costs $14.50 less than the original price p. You pay$53 for a discounted ticket. Write and solve an equation to ﬁnd the original price.

53 = p − 14.5
The original price was $67.50. Explanation: Let c represent what you paid for the ticket, the discounted price. We’re told that the discount price, which you pay, is$ 14.50 less than the original price, which will be represented by p.
Algebraically, this means that
c = p –  14.5
Furthermore, we’re told you pay $53, i.e c = 53 53 = p − 14.5 Now, solve for p p = 53 + 14.5 p = 67.50 So, the original price was$67.50.

Question 21.
PROBLEM SOLVING
A game of bowling has ten frames. After ﬁve frames, your friend’s bowling score is 65 and your bowling score is 8 less than your friend’s score.
a. Write and solve an equation to ﬁnd your score.
b. By the end of the game, your friend’s score doubles and your score increases by 80. Who wins the game? Explain.

a) x + 8= 65
My score is 57
b) I won the game.

Explanation:
a) Friends bowling score = 65
My score = 8 less than friends sore
Let us take x as my score
Equation is x + 8= 65
Subtrcat 8 from each side
x + 8 – 8 = 65 – 8
x = 57
My score is 57
b) Friend’s score is doubled = 65 x 2 = 130
My score is increased by 80 = 57 + 80 = 137
So, I won the game.

SOLVING EQUATIONS USING MULTIPLICATION OR DIVISION
Solve the equation. Check your solution.

Question 22.
7x = 35

x = 5

Explanation:
Given equation is 7x = 35
divide each side by 7
7x/7 = 35/7
x = 5
putting x = 5 in 7x = 35
7(5) = 35

Question 23.
4 = -0.8n

n = -5

Explanation:
Given equation is 4 = -0.8n
Divide each side by -0.8
4/-0.8 = -0.8n/-0.8
-5 = n
Putting n = -5 in 4 = -0.8n
4 = -0.8(-5)

Question 24.
6 = –$$\frac{w}{8}$$

w = -48

Explanation:
Given equation is 6 = -w/8
Multiply each side by 8
6 x 8 = -w/8 x 8
48 = -w
Multiply each side by (-1)
48 x -1 = -w x -1
w = -48
putting w = -48 in 6 = -w/8
6 = -(-48)/8 = 48/8

Question 25.

m = 7.3π

Explanation:
Given equation is m/π = 7.3
multiply each side by π
m/π x π = 7.3 x π
m = 7.3π
Putting m = 7.3π in m/π = 7.3
7.3π/π = 7.3

Question 26.
-4.3g = 25.8

g = -6

Explanation:
Given equation is -4.3g = 25.8
divide each side by -4.3
-4.3g/-4.3 = 25.8/-4.3
g = -6
putting g = -6 in -4.3g = 25.8
-4.3(-6) = 25.8

Question 27.

k = 5/3

Explanation:
Given equation is 3/2 = (9/10) k
Multiply each side by 10/9
3/2 x (10/9) = (9/10) k x (10/9)
5/3 = k
Putting k = 5/3 in 3/2 = (9/10) k
3/2 = (9/10) x (5/3)

Question 28.
-7.8x = -1.56

x = 0.2

Explanation:
Given equation is -7.8x = -1.56
Divide each side by -7.8
-7.8x/-7.8 = -1.56/-7.8
x = 0.2
Putting x = 0.2 in -7.8x = -1.56
-7.8(0.2) = -1.56

Question 29.

p = -7/3

Explanation:
Given equation is -2 = (6/7)p
Multiply each side by (7/6)
-2 x (7/6) = (6/7)p x (7/6)
-7/3 = p
putting p = -7/3 in -2 = (6/7)p
-2 = (6/7) x (-7/3)

Question 30.
3πd = 12π

d = 4π

Explanation:
Given equation is 3πd = 12π
Divide each side by 3π
3πd/3π = 12π/3π
d = 4π
Putting d = 4π in 3πd = 12π
3π x 4π = 12π

Question 31.
YOU BE THE TEACHER

Wrong

Explanation:
-1.5 + k = 8.2
k = 8.2 + 1.5
k = 9.7

Question 32.
STRUCTURE
A gym teacher orders 42 tennis balls. The tennis balls come in packs of 3. Which of the following equations represents the number x of packs?

3x = 42

Explanation:
A gym teacher orders 42 tennis balls
Each pack has 3 balls
To find the number of packages of tennis balls, take the total number of tennis balls and divide that number by the number of packages.
42/3 = 14
3x = 42

Question 33.
MODELING REAL LIFE
You clean a community park for 6.5 hours. You earn $42.25. How much do you earn per hour? Answer: The amount I earn per hour is$6.50

Explanation:
The amount I earned for cleaning a community park = $42.25 Time = 6.5 hours Let the amount I earned per hour is x Equation is 6.5x = 42.25 Divide each side by 6.5 6.5x/6.5 = 42.25/6.5 x = 6.5 The amount I earn per hour is$6.50

Question 34.
MODELING REAL LIFE
A rocket is scheduled to launch from a command center in 3.75 hours. What time is it now?

The time is 7:35

Explanation:
A rocket is scheduled to launch at 11:20 AM from a command center in 3.75 hours.
11(20/60) – 3(3/4) = 11(1/3) – 3(3/4)
= 34/3 – 15/4
= (136 – 45)/12
= 91/12 = 7(7/12)
The time is 7:35

Question 35.
MODELING REAL LIFE
After earning interest, the balance of an account is $420. The new balance is $$\frac{7}{6}$$ of the original balance. How much interest did it earn? Answer: You earned$60 as interest.

Explanation:
x/420 = 6/7
x = 420 x 6/7
x = 360 x 6
x = 360
The amountt of interest = new balance – original balance
= 420 – 360 = 60
So, you earned $60 as interest. Question 36. MODELING REAL LIFE After a cleanup, algae covers 2 miles of a coastline. The length of the coastline1covered after the cleanup is of the previous length. How many miles of the coast did the algae previously cover? Answer: Miles of the coast the algae previously covered is 6 miles. Explanation: Number of miles the algae covers after cleanup = 2 miles Let x as the miles algae previously covered Equation is (1/3)x = 2 x = 2 x 3 x = 6 Miles of the coast the algae previously covered is 6 miles. Question 37. PROBLEM SOLVING Cedar Point, an amusement park, has some of the tallest roller coasters in the United States. The Mantis is 165 feet shorter than the Millennium Force. What is the height of the Mantis? Answer: The height of Mantis is 145 feet. Explanation: Height of the Millennium Force = 310 Feet The mantis is shorter than Millennium Force by 165 feet Let us take x as the height of the mantis x = 310 – 165 x = 145 feet The height of Mantis is 145 feet. SOLVING AN EQUATION Solve the equation. Check your solution. Question 38. -3 = h + 8 ÷ 2 Answer: h = -7 Explanation: Given equation is -3 = h + 8 ÷ 2 -3 = h + 4 Subtract 4 from each side -3 – 4 = h + 4 – 4 -7 = h Putting h = -7 in -3 = h + 8 ÷ 2 -3 = -7 + 8 ÷ 2 = -7 + 4 Question 39. 12 = w – | -7 | Answer: w = 19 Explanation: Given equation is 12 = w – | -7 | 12 = w – 7 Add 7 to each side 12 + 7 = w – 7 + 7 19 = w Putting w = 19 in 12 = w – | -7 | 12 = 19 – | -7 | = 19 – 7 Question 40. q + | 6.4 | = 9.6 Answer: q = 3.2 Explanation: Given equation is q + | 6.4 | = 9.6 q + 6.4 = 9.6 Subtract 6.4 from each side q + 6.4 – 6.4 = 9.6 – 6.4 q = 3.2 Putting q = 3.2 in q + | 6.4 | = 9.6 3.2 + | 6.4 | = 3.2 + 6.4 = 9.6 Question 41. d – 2.8 ÷ 0.2 = -14 Answer: d = 0 Explanation: Given equation is d – 2.8 ÷ 0.2 = -14 d – 14 = -14 Add 14 to each side d – 14 + 14 = -14 + 14 d = 0 Put d = 0 in d – 2.8 ÷ 0.2 = -14 0 – 2.8 ÷ 0.2 = -14 Question 42. Answer: x = -3/9 Explanation: Given equation is 8/9 = x + (1/3) x 7 8/9 = x + 7/3 Subtract 7/3 from each side 8/9 -7/3 = x + 7/3 -7/3 (8 – 21)/9 = x -13/9 = x Put x = -3/9 in 8/9 = x + (1/3) x 7 8/9 = -13/9 + (1/3) x 7 = -13/9 + 7/3 = (-13 + 21)/3 Question 43. Answer: p = -1/12 Explanation: Given equation is p – 1/4 . 3 = -5/6 p – 3/4 = -5/6 Add 3/4 to each side p – 3/4 + 3/4 = -5/6 + 3/4 p = (-10 + 9)/12 p = -1/12 put p = -1/12 in p – 1/4 . 3 = -5/6 -1/12 – 1/4 . 3 = -1/12 – 3/4 = (-1 – 9)/12 = -10/12 = -5/6 Question 44. GEOMETRY The volume V of the prism is 1122 cubic inches. Use the formula V = Bh to ﬁnd the height h of the prism. Answer: The height of the prism is 12 inches. Explanation: Prism volume = 1122 cubic inches Base b = 93.5 sq in Volume formula V = Bh 1122 = 93.5 x h h = 1122/93.5 h = 12 inches SOLVING AN EQUATION Write and solve an equation to ﬁnd the value of x. Question 45. The angles are complementary. Answer: x = 45° Explanation: Complementary angles mean the sum of angles is 90 degrees. x + 45 = 90 x = 90 – 45 x = 45° Question 46. The angles are supplementary. Answer: x° = 130° Explanation: Two angles are called supplementary when their measures add up to 180 degrees. So, x° + 50° = 180° x° = 180° – 50° x° = 130° Question 47. CRITICAL THINKING Which of the operations +, −, ×, ÷, are inverses of each other? Explain. Answer: addition (+), Subtraction (-) are inverse to each other. Multiplication (x), division (÷) are inverse to each other. Because, when you move the quantity which is having + sign from one side to the other side of the equation, it automatically converts to – sign. In the same way other operations also. Question 48. LOGIC Without solving, determine whether the solution of -2x = -15 is greater than or less than -15. Explain. Answer: The solution is greater than -15 Explanation: The solution is greater than -15 Here both sides of the equation have – sign. They are eliminated When you divide 15 by 2 you will get obviously answer greater than -15. Question 49. OPEN-ENDED Write a subtraction equation and a division equation so that each has a solution of 2. Justify your answer. Answer: The subtraction equation is x + 4 = 6 The division equation is 2x = 4 Explanation: The subtraction equation is x + 4 = 6 x = 6 – 4 x = 2 The division equation is 2x = 4 x = 4/2 x = 2 Question 50. MODELING REAL LIFE Ants of a particular species can carry 50 times their body weight. It takes 32 ants of that species to carry the cherry shown. About how much does each ant weigh? Answer: The weight of each ant is 3 mg Explanation: Weight of the cherry = 4800 mg Number of Ants that carry the cherry = 32 Ants of a particular species can carry 50 times their body weight. Let w is the weight of each ant w. 50 . 32 = 4800 1600w = 4800 w = 4800/1600 w = 3 The weight of each ant is 3 mg Question 51. REASONING One-fourth of the girls and one-eighth of the boys in a grade retake their school pictures. The photographer retakes pictures for 16 girls and 7 boys. How many students are in the grade? Answer: Total number of students = 120 Explanation: Let the total number of girls be g, hence according to the question only one-fourth of a total number of girls(g) have retaken their pictures which is 16. (1/4)g = 16 g = 16 x 4 g= 64 According to the question, only one-eighth of the total number of boys(b) have retaken their pictures which is 7. (1/8)b = 7 b = 7 x 8 b = 56 Total number of students = boys + girls = 56 + 64 = 120 Question 52. DIG DEEPER! You use a crowdfunding website to raise money. The website keeps 5% of each donation. Five of your friends each donate the same amount. The total funding you receive is$47.50. How much does each friend donate?

Each friend donate $10 Explanation: Assume x as the amount each friend donate the website keeps 5% of each donation The total funding you receive is$47.50.
4.75x = 47.50
x = 47.50/4.75
x = 10
Each friend donate $10 Question 53. CRITICAL THINKING A neighbor pays you and two friends$90 to paint her garage. You divide the money three ways in the ratio 2 : 3 : 5.
a. How much does each person receive?
b. What is one possible reason the money is not divided evenly?

a. They receive $18, &=$27, $45 b. The possible reason for unequal distribution is that the ratios are different for each of them. Explanation: a. let the common multiplier for each of the people be x So, each one gets the value of 2x, 3x, 5x As per the question, the total amount paid is$90
So, 2x + 3x + 5x = 90
10x = 90
x = 90/10
x = 9
Hence, person receives 9(2) = $18, 9(3) =$27, 9(5) = $45 b. The possible reason for unequal distribution is that the ratios are different for each of them. ### Lesson 1.2 Solving Multi-step Equations EXPLORATION 1 Work with a partner. Find each angle measure in each ﬁgure. Use equations to justify your answers. 1.2 Lesson Try It Solve the equation. Check your solution. Question 4. -4n – 8n + 17 = 23 Answer: n = -1/2 Explanation: Given equation is -4n – 8n + 17 = 23 -12n + 17 = 23 -12n = 23 – 17 -12n = 6 n = -6/12 n = -1/2 Put n = -1/2 in -4n – 8n + 17 = 23 -4(-1/2) – 8(-1/2) + 17 = 2 + 4 + 17 = 23 Question 5. 10 = 3n + 20 – n Answer: n = -5 Explanation: Given equation is 10 = 3n + 20 – n 10 = 2n + 20 10 – 20 = 2n -10 = 2n n = -10/2 n = -5 put n = -5 in 10 = 3n + 20 – n 10 = 3(-5) + 20 – (-5) = -15 + 20 + 5 = -15 + 25 Try It Solve the equation. Check your solution. Question 6. -3(x + 2) + 5x = -9 Answer: x = -3/2 Explanation: Given equation is -3(x + 2) + 5x = -9 -3x – 6 + 5x = -9 2x – 6 = -9 2x = -9 + 6 2x = -3 x = -3/2 Put x = -3/2 in -3(x + 2) + 5x = -9 -3((-3/2) + 2) + 5(-3/2) = 9/2 – 6 – 15/2 = (9 – 15 – 12)/2 = -18/2 = -9 Question 7. 5 + 1.5(2d – 1) = 0.5 Answer: d = -1 Explanation: Given equation is 5 + 1.5(2d – 1) = 0.5 5 + 3d – 1.5 = 0.5 3d + 3.5 = 05 3d = 0.5 – 3.5 3d = -3 d = -3/3 d = -1 Put d = -1 in 5 + 1.5(2d – 1) = 0.5 5 + 1.5(2(-1) – 1) = 5 + 1.5(-2 – 1) = 5 + 1.5(-3) = 5 – 4.5 = 0.5 Self-Assessment for Concepts & Skills Solve each exercise. Then rate your understanding of the success criteria in your journal. SOLVING AN EQUATION Solve the equation. Check your solution. Question 8. -5x + 1 = 31 Answer: x = -6 Explanation: Given equation is -5x + 1 = 31 Subtract 1from each side -5x + 1 -1 = 31 – 1 -5x = 30 Divide each side by -5 -5x/-5 = 30/-5 x = -6 Put x = -6 in -5x + 1 = 31 -5(-6) + 1 = 30 + 1 = 31 Question 9. $$\frac{1}{3}$$x – 9 = -12 Answer: x = -9 Explanation: Given equation is (1/3)x – 9 = -12 (1/3)x = -12 + 9 (1/3)x = -3 x = -3 x 3 x = -9 Put x = -9 in (1/3)x – 9 = -12 (1/3)(-9) – 9 = -3 – 9 = -12 Question 10. -n – 6n + 4 = 53 Answer: n = -7 Explanation: Given equation is -n – 6n + 4 = 53 -7n + 4 = 53 -7n = 53 – 4 -7n = 49 n = -49/7 n = -7 put n = -7 in -n – 6n + 4 = 53 -(-7) – 6(-7) + 4 = 7 + 42 + 4 = 53 Question 11. 14 = 6n + 6 – 2n Answer: n = 2 Explanation: Given equation is 14 = 6n + 6 – 2n 14 = 4n + 6 14 – 6 = 4n 8 = 4n 8/4 = n n = 2 Put n = 2 in 14 = 6n + 6 – 2n 14 = 6(2) + 6 – 2(2) = 12 + 6 – 4 = 18 – 4 Question 12. -8(x + 1) + 2x = -32 Answer: x = 4 Explanation: Given equation is -8(x + 1) + 2x = -32 -8x – 8 + 2x = -32 -6x – 8 = -32 -6x = -32 + 8 -6x = -24 x = -24/-6 x = 4 Put x = 4 in -8(x + 1) + 2x = -32 -8(4 + 1) + 2(4) = -8(5) + 8 = -40 + 8 = -32 Question 13. 3 + 4.5(2d – 3) = 7.5 Answer: d = 2 Explanation: Given equation is 3 + 4.5(2d – 3) = 7.5 3 + 9d – 13.5 = 7.5 9d – 10.5 = 7.5 9d = 7.5 + 10.5 9d = 18 d = 18/9 d = 2 Put d = 2 in 3 + 4.5(2d – 3) = 7.5 3 + 4.5(2(2) – 3) = 3 + 4.5(4 – 3) = 3 + 4.5(1) = 3 + 4.5 = 7.5 Question 14. WRITING Write the sentence as an equation, then solve. Answer: 2 + 3x = 17, x = 5 Explanation: 2 + 3x = 17 Subtract 2 from each side 2 + 3x – 2 = 17 – 2 3x = 15 Divide each side by 3 3x/3 = 15/3 x = 5 Question 15. OPEN-ENDED Explain how to solve the equation 2(4x – 11) + 9 = 19. Answer: x = 4 Explanation: Given equation is 2(4x – 11) + 9 = 19. Expand brackets 8x – 22 + 9 = 19 Simpify 8x – 13 = 19 Add 13 to each side 8x – 13 + 13 = 19 + 13 8x = 32 Divide each side by 8 8x/8 = 32/8 x = 4 Question 16. CRITICAL THINKING How can you solve 3(x + 2) = 9 without distributing the 3? Answer: x = 1 Explanation: 3(x + 2) = 9 Divide both sides by 3 3(x + 2)/3 = 9/3 x + 2 = 3 Subtract 2 from both sides x + 2 -2 = 3 – 2 x = 1 Self-Assessment for Problem Solving Solve each exercise. Then rate your understanding of the success criteria in your journal. Question 17. Find the number x of action ﬁgures that a small business needs to produce on Friday so that the mean number of action ﬁgures produced per day is 50. Answer: On friday the number of action figures that a sma business should produce is 53. Explanation: Mean = 50 (55 + 45 + 53 + 44 + x)/5 = 50 (197 + x)/5 = 50 197 + x = 50 x 5 197 + x = 250 x = 250 – 197 x = 53 On friday the number of action figures that a sma business should produce is 53. Question 18. DIG DEEPER! A hard drive is 80% full and has 12,000 MB of free space. One minute of video uses 60 MB of storage. How many minutes of the video should be deleted so that the hard drive is 75% full? Answer: 50 minutes video should be deleted so that the hard drive is 75% full Explanation: Let x be the capacity of the hard drive. A hard drive is 80% full and has 12,000 MB of free space 0.8x + 12000 = x 12000 = x – 0.8x 0.2x = 12000 x = 12000/0.2 x = 60,000 MB One minute of video uses 60 MB of storage. Let t be the number of minutes of video that should be deleted 60t = 0.8x – 0.75x 60t = 0.05x 60t = 0.05 x 60000 60t = 3000 t = 3000/60 t = 50 50 minutes video should be deleted so that the hard drive is 75% full Question 19. A teacher spends$354 on costumes and microphones for six cast members in a play. Each cast member receives a costume that costs $38 and a microphone that costs$c. What did the teacher spend on each microphone? Justify your answer.

The teacher spends $21 on each microphone Explanation: The amount the teacher spends on six costumes and microphone =$354
Each cast member receives a costume that costs $38 and a microphone that costs$c.
6(38 + c) = 354
38 + c = 354/6
38 + c = 59
c = 59 – 38
c = 21
The teacher spends $21 on each microphone ### Solving Multi-step Equations Homework & Practice 1.2 Review & Refresh Solve the equation. Question 1. y + 8 = 3 Answer: y = -5 Explanation: Given equation is y + 8 = 3 y + 8 – 8 = 3 – 8 y = -5 Question 2. h – 1 = 7.2 Answer: h = 8.2 Explanation: Given equation is h – 1 = 7.2 Add 1 to each side h – 1 + 1 = 7.2 + 1 h = 8.2 Question 3. 5 = -2n Answer: n = -5/2 Explanation: Given equation is 5 = -2n Divide each side by -2 5/-2 = -2n/-2 -5/2 = n Question 4. -3.3m = -1.1 Answer: m = 1/3 Explanation: Given equation is -3.3m = -1.1 Divide each side by -3.3 -3.3m/-3.3 = -1.1/-3.3 m = 1/3 Write the decimal as a fraction or mixed number in simplest form. Question 5. -0.2 Answer: -0.2 = -1/5 Explanation: Given decimal value is -0.2 Fraction form of the decimal = -2/10 = -1/5 Question 6. 3.82 Answer: 3.82 = 191/50 Explanation: Given decimal value is 3.82 Fraction form of the decimal = 382/100 = 191/50 Question 7. -0.454 Answer: -0.454 = -217/500 Explanation: Given decimal value is -0.454 Fraction form of the decimal = -454/1000 = -217/500 Question 8. -0.125 Answer: -0.125 = -1/8 Explanation: Given decimal value is -0.125 Fraction form of the decimal = -125/1000 = -1/8 Concepts, Skills, &Problem Solving FINDING ANGLE MEASURES Find each angle measure in the ﬁgure. Use equations to justify your answers. (See Exploration 1, p. 11.) Question 9. Answer: Angles of a triangle are 20°, 24°, 136°. Explanation: The sum of angles in triangle = 180° 4y + 20 + (y – 10)= 180° 4y + 20 + y – 10= 180 5y + 10 = 180 5y = 180 – 10 5y = 170 y = 170/5 y = 34° So, ranges are 4 x 34 = 136°, (34 – 10) = 24°, 20 Question 10. Answer: The angles of the triangle are 37.5°, 75°, 67.5°. Explanation: The sum of angles in triangle = 180° x + 2x + (x + 30) = 180 4x + 30 = 180 4x = 180 – 30 4x = 150 x = 150/4 x = 37.5° The measure of each angle is 37.5°, 2 x 37.5 = 75°, (37.5 + 30) = 67.5° SOLVING AN EQUATION Solve the equation. Check your solution. Question 11. 10x + 2 = 32 Answer: x = 3 Explanation: Given equation is 10x + 2 = 32 10x = 32 – 2 10x = 30 x = 30/10 x = 3 Put x = 3 in 10x + 2 = 32 10(3) + 2 = 30 + 2 = 32 Question 12. 19 – 4c = 17 Answer: c = 1/2 Explanation: Given equation is 19 – 4c = 17 Add 4c to each side 19 – 4c + 4c = 17 + 4c 19 = 17 + 4c Subtract 17 from each side 19 – 17 = 17 + 4c – 17 2 = 4c c = 2/4 c = 1/2 Put c = 1/2 in 19 – 4c = 17 19 – 4(1/2) = 19 – 2 = 17 Question 13. 5x + 2x + 4 = 18 Answer: x = 2 Explanation: Given equation is 5x + 2x + 4 = 18 7x + 4 = 18 7x = 18 – 4 7x = 14 x = 14/7 x = 2 Put x = 2 in 5x + 2x + 4 = 18 5(2) + 2(2) + 4 = 10 + 4 + 4 = 18 Question 14. 2 = -9n + 22 – n Answer: n = 2 Explanation: Given equation is 2 = -9n + 22 – n 2 = -10n + 22 2 – 22 = -10n -20 = -10n n = -20/-10 n = 2 Put n = 2 in 2 = -9n + 22 – n 2 = -9(2) + 22 – 2 = -18 + 22 – 2 = -20 + 22 Question 15. 1.1x + 1.2x – 5.4 = -10 Answer: x = -2 Explanation: Given equation is 1.1x + 1.2x – 5.4 = -10 2.3x – 5.4 = -10 2.3x = -10 + 5.4 2.3x = -4.6 x = -4.6/2.3 x = -2 Put x = -2 in 1.1x + 1.2x – 5.4 = -10 1.1(-2) + 1.2(-2) – 5.4 = -2.2 – 2.4 – 5.4 = -10 Question 16. Answer: h = -9 Explanation: Given equation is (2/3)h – (1/3)h + 11 = 8 (2 – 1)/3 h + 11 = 8 (1/3)h + 11 = 8 (1/3)h = 8 – 11 (1/3)h = -3 h = -3 x 3 h = -9 Put h = -9 in (2/3)h – (1/3)h + 11 = 8 (2/3)(-9) – (1/3)(-9) + 11 = -6 + 3 + 11 = 14 – 6 = 8 Question 17. 6(5 – 8v) + 12 = -54 Answer: v = 2 Explanation: Given equation is 6(5 – 8v) + 12 = -54 30 – 48v + 12 = -54 42 – 48v = -54 42 + 54 = 48v 96 = 48v v = 96/48 v = 2 Put v = 2 in 6(5 – 8v) + 12 = -54 6(5 – 8(2)) + 12 = 6(5 – 16) + 12 = 6(-11) + 12 = -66 + 12 = -54 Question 18. 21(2 – x) + 12x = 44 Answer: x = -2/9 Explanation: Given equation is 21(2 – x) + 12x = 44 42 – 21x + 12x = 44 42 – 9x = 44 42 – 44 = 9x -2 = 9x x = -2/9 Put x = -2/9 in 21(2 – x) + 12x = 44 21(2 – (-2/9)) + 12(-2/9) = 21(2 + 2/9) – 24/9 = 42 + 42/9 – 24/9 = 42 + (42 – 24)/9 = 42 + 18/9 = 42 + 2 = 44 Question 19. 8.5 = 6.5(2d – 3) + d Answer: d = 2 Explanation: Given equation is 8.5 = 6.5(2d – 3) + d 8.5 = 13d – 19.5 + d 8.5 = 14d – 19.5 8.5 + 19.5 = 14d 28 = 14d d = 28/14 d = 2 Put d = 2 in 8.5 = 6.5(2d – 3) + d 8.5 = 6.5(2(2) – 3) + 2 = 6.5(4 – 3) + 2 = 6.5 x 1 + 2 = 6.5 + 2 Question 20. Answer: x = -6 Explanation: Given equation is -1/4 (x + 2) + 5 = -x -x/4 – 2.(1/4) + 5 = -x -x/4 – 1/2 + 5 = -x -1/2 + 5 = -x + x/4 (-1 + 10)/2 = (-4x + x)/4 9/2 = -3x/4 9/2 x (-4/3) = x x = -6 Put x = -6 in -1/4 (x + 2) + 5 = -x -1/4 (-6 + 2) + 5 = -(-6) -1/4(-4) + 5 = 6 1 + 5 = 6 YOU BE THE TEACHER Your friend solves the equation. Is your friend correct? Explain your reasoning. Question 21. Answer: Wrong Explanation: -2(7 – y) + 4 = -4 -14 + 2y + 4 = -4 -10 + 2y = -4 2y = -4 + 10 2y = 6 y = 3 Question 22. Answer: Correct Explanation: 3(y – 1) + 8 = 11 3y – 3 + 8 = 11 3y + 5 = 11 3y = 11 – 5 3y = 6 y = 2 Question 23. STRUCTURE The cost C (in dollars) of making watches is represented C = 15n + 85. How many watches are made when the cost is$385?

20 watches can be manufactured.

Explanation:
C = 15n + 85
385 = 15n + 85
15n = 385 – 85
15n = 300
n = 300/15
n = 20
So, 20 watches can be manufactured.

Question 24.
MODELING REAL LIFE
The height of the house is 26 feet. What is the height x of each story?

Height of each story is 10 ft.

Explanation:
The height of the house is 26 feet
x + x + 6 = 26
2x + 6 = 26
2x = 26 – 6
2x = 20
x = 20/2
x = 10 ft
Height of each story is 10 ft.

Question 25.
MODELING REAL LIFE
After the addition of an acid, a solution has a volume of 90 milliliters. The volume of the solution is 3 milliliters greater than 3 times the volume of the solution before the acid was added. What was the original volume of the solution?

The original volume of the solution is 29 milliliters

Explanation:
Volume of the solution = 90 milliliters
Let us take x as the original volume of the solution
3x + 3 = 90
3x= 90 – 3
3x = 87
x = 87/3
x = 29
So, the original volume of the solution is 29 milliliters

Question 26.
PROBLEM SOLVING
A grocer prepares free samples of a salad to give out during the day. By lunchtime, the grocer has given out 5 fewer than half the total number of samples. How many samples did the grocer prepare if she gives out 50 samples before lunch?

The grocer prepares 110 samples

Explanation:
Let x be the total number of samples the grocer prepares
Total number of samples a grocer given out = 50
(1/2) x – 5 = 50
(1/2)x = 50 + 5
1/2 x = 55
x = 55 x 2
x = 110
So, the grocer prepares 110 samples.

Question 27.
GEOMETRY
What is the length of the missing base of the trapezoid?

The missing base is 6 in.

Explanation:
The trapezoid area formula is (a + b)/2 x height
Given trapezoid area = 21 sq. in
(8 + x)/2 x 3 = 21
(8 + x)/2 = 21/3
(8 + x)/2 = 7
(8 + x) = 7 x 2
8 + x = 14
x = 14 – 8
x = 6
The missing base is 6 in.

Question 28.
MODELING REAL LIFE
You order two servings of pancakes and a fruit cup. The cost of the fruit cup is $1.50. You leave a 15% tip. Your total bill is$11.50. How much does one serving of pancakes cost?

The cost of one serving of pancake is $4.25 Explanation: The total cost excluding the tip would be a sum of 1.5 for a fruit cup and twice of price of each pancake servings(b), hence the price should be 1.5 + 2b. The customer pays 15% tip that means 0.15 times the bill. 1.15(1.5 + 2b) = 11.5 1.5 + 2b = 11.5/1.15 1.5 + 2b = 10 2b = 10 – 1.5 2b = 8.5 b = 8.5/2 b = 4.25 The cost of one serving of pancake is$4.25

Question 29.
PROBLEM SOLVING
How many people must attend the third show so that the average attendance per show is 3000?

3500 people must attend the third show so that the average attendance per show is 3000.

Explanation:
The average attendance = 3000 per show
(2580 + 2920 + x)/3 = 3000
5500 + x = 3000 x 3
5500 + x = 9000
x = 9000 – 5500
x = 3500
3500 people must attend the third show so that the average attendance per show is 3000.

Question 30.
DIG DEEPER!
Divers in a competition are scored by an international panel of judges. The highest and the lowest scores are dropped. The total of the remaining scores is multiplied by the degree of difficulty of the dive. This product is multiplied by 0.6 to determine the ﬁnal score.
a. A diver’sﬁnal score is 77.7. What is the degree of difficulty of the dive?

b. CRITICAL THINKING
The degree of difficulty of a dive is 4.0. The diver’s ﬁnal score is 97.2. Judges award half or whole points from 0 to 10. What scores could the judges have given the diver?

a. The degree of difficulty of dive is 3.5
b. The scores are 7.5, 8.0, 8.0, 8.0, 8.0, 8.5, 9.0

Explanation:
a. Let x be the degree of difficulty
By eliminating the scores6.5 and 8.5, we ge
(7 + 7 + 7.5 + 7.5 + 8). x . 0.6 = 77.7
37 . 0.6x = 77.7
22.2x = 77.7
x = 77.7/22.2
x = 7/2
x = 3.5
The degree of difficulty of dive is 3.5
b. Let s be the sum of the scores given by 5 judges
s x 4 x 0.6 = 97.2
2.4s = 97.2
s = 97.2/2.4
s = 40.5
40.5/5 = 8.1
So, the average score is 8.1
The scores are 7.5, 8.0, 8.0, 8.0, 8.0, 8.5, 9.0

### Lesson 1.3 Solving Equations with Variables on Both Sides

EXPLORATION 1
Finding Missing Measures in Figures
Work with a partner.
a. If possible, ﬁnd the value of x so that the value of the perimeter (in feet) is equal to the value of the area (in square feet) for each ﬁgure. Use an equation to justify your answer.

b. If possible, ﬁnd the value of y so that the value of the surface area (in square inches) is equal to the value of the volume (in cubic inches) for each ﬁgure. Use an equation to justify your answer.

c. How are the equations you used in parts (a) and (b) different from equations used in previous sections? Explain how to solve this type of equation.

1.3 Lesson

Try It

Solve the equation. Check your solution.

Question 1.
-3x = 2x + 20

x = -4

Explanation:
Given equation is -3x = 2x + 20
-3x – 2x = 20
-5x = 20
x = -20/5
x = -4
Put x = -4 in -3x = 2x + 20
-3(-4) = 2(-4) + 20
12 = -8 + 20

Question 2.
2.5y + 6 = 4.5y – 1

y = 3.5

Explanation:
Given equation is 2.5y + 6 = 4.5y – 1
6 + 1 = 4.5y – 2.5y
7 = 2y
y = 7/2
y = 3.5
Put y = 3.5 in 2.5y + 6 = 4.5y – 1
2.5(3.5) + 6 = 4.5(3.5) – 1
8.75 + 6 = 15.75 – 1
14.75 = 14.75

Try It

Solve the equation. Check your solution.

Question 3.
6(4 – z) =2z

z = 3

Explanation:
Given equation is 6(4 – z) =2z
24 – 6z = 2z
24 = 2z + 6z
8z = 24
z = 24/8
z = 3
Put z = 3 in 6(4 – z) =2z
6(4 – 3) = 2(3)
6(1) = 6

Question 4.
5(w – 2) = -2(1.5w + 5)

w = 0

Explanation:
Given equation is 5(w – 2) = -2(1.5w + 5)
5w – 10 = -3w – 10
5w + 3w = -10 + 10
8w = 0
w = 0
Put w = 0 in 5(w – 2) = -2(1.5w + 5)
5(0 – 2) = -2(1.5(0) + 5)
5(-2) = -2(5)
= -10 = -10

Try It

Solve the equation.

Question 5.
2x + 1 = 2x – 1

The equation 2x + 1 = 2x – 1 is never true. So, it does not have a soution

Explanation:
Given equation is 2x + 1 = 2x – 1
2x + 1 = 2x – 1
2x – 2x = -1 – 1
0 = -2

Question 6.
6(5 – 2v) = -4(3v + 1)

The equation 6(5 – 2v) = -4(3v + 1) is never true. So, it does not have a soution

Explanation:
Given equation is 6(5 – 2v) = -4(3v + 1)
30 – 12v = -12v – 4
30 – 12v + 12v = -4
30 – 0 = -4
30 = -4
The equation 6(5 – 2v) = -4(3v + 1) is never true. So, it does not have a soution

Try It

Solve the equation

Question 7.

The equation 1/2 (6t – 4) = 3t – 2 is always true. So, it has infinitely many solutions.

Explanation:
Given equation is 1/2 (6t – 4) = 3t – 2
3t – 2 = 3t – 2
The equation 1/2 (6t – 4) = 3t – 2 is always true. So, it has infinitely many solutions.

Question 8.

The equation 1/3 (2b + 9) = 2/3 (b + 9/2) is always true. So, it has infinitely many solutions.

Explanation:
Given equation is 1/3 (2b + 9) = 2/3 (b + 9/2)
2b + 9 = 2/3 (b + 9/2) x 3
2b + 9 = 2(b + 9/2)
2b + 9 = 2b + 9
The equation 1/3 (2b + 9) = 2/3 (b + 9/2)is always true. So, it has infinitely many solutions.

Try It

Question 9.
WHAT IF?
The diameter of the purple circle is 3x. What is the area of each circle?

The areas of circles are 25π, 36π.

Explanation:
The diameter of the purple circle is 3x
3x = 2r
r = 3x/2
Two circles are identical
x + 2 = 3x/2
3x/2 – x = 2
x/2 = 2
x = 4
The radius of green circle is (1 + 4) = 5
The radius of purple circle is 3(4)/2 = 6
The area of each circle = πr²
= π (5)² = 25π
The area of second circle = π (6)² = 36π

Self-Assessment for Concepts & Skills
Solve each exercise. Then rate your understanding of the success criteria in your journal.

Question 10.
OPEN-ENDED
Write an equation with variables on both sides that has a single solution of -1. Explain how to solve your equation.

5x + 6= 6x + 7

Explanation:
Write an equation with variables on both sides that has a single solution of -1
5x + 6= 6x + 7
Subtract 5x from each side
5x + 6 – 5x = 6x + 7 – 5x
6 = x + 7
Subtract 7 from each side
6 – 7 = x+ 7 – 7
-1 = x

STRUCTURE
Without solving, determine whether the equation has one solution, no solution, or inﬁnitely many solutions. Justify your answer.

Question 11.
3(x – 1) = -3

The equation has only one solution

Explanation:
Given equation is 3(x – 1) = -3
(x – 1) = -1
x = -1 + 1
x = 0

Question 12.
6x + 6 = 6(x + 1)

The equation has infinitely many solutions

Explanation:
Given equation is 6x + 6 = 6(x + 1)
The equation has infinitely many solutions
6x + 6 = 6x + 6

Question 13.
z + 1 = z + 6

The equation has no solution

Explanation:
Given equation is z + 1 = z + 6
The equation has no solution
1 = 6

SOLVING AN EQUATION
Solve the equation. Check your solution, if possible.

Question 14.
-7x = x + 24

x = -3

Explanation:
Given equation is -7x = x + 24
-7x – x = 24
-8x = 24
x = -24/8
x = -3
Put x = -3 in -7x = x + 24
-7(-3) = -3 + 24
= 21

Question 15.
8(3 – z) = 4z

z = 2

Explanation:
Given equation is 8(3 – z) = 4z
24 – 8z = 4z
24 = 4z + 8z
24 =12z
z = 24/2
z = 2
Put z = 2 in 8(3 – z) = 4z
8(3 – 2) = 4(2)
8(1) = 8

Question 16.
2(t – 3) = 2t – 6

The equation has infinitely many solutions.

Explanation:
Given equation is 2(t – 3) = 2t – 6
2t – 6 = 2t – 6
The equation has infinitely many solutions.

Question 17.
WRITING AND SOLVING AN EQUATION
The squares are identical. What is the area of each square?

The areas of squares are 256 sq units, 256 sq units.

Explanation:
What is the area of each square?
So, 3x + 7 = 5x + 1
7 – 1 = 5x – 3x
6 = 2x
x = 6/2
x = 3
The side of first square = 3(3) + 7 = 9 + 7
= 16
First square area = side²
= 16² = 256 sq units
Second square side length = 5(3) + 1 = 15 + 1 = 16
Second sqaure area = 16² = 256 sq units

Self-Assessment for Problem Solving

Solve each exercise. Then rate your understanding of the success criteria in your journal.

Question 18.
Your cousin renews his apartment lease and pays a new monthly rent. His new rent is calculated by applying a discount of $50 to his original rent and then applying a 10% increase to the discounted amount. What was your cousin’s original monthly rent when his new rent is 5% greater? Answer: Cousin’s original monthly rent is$1100.

Explanation:
Let us take rent as x
Then, discounted rent = x – 50
The increase applied is 10%
So, final rent is (x – 50) + 10% = (x – 50)1.1
x + 5% = 1.05x
1.1(x – 50) = 1.05x
1.1x – 1.05x = 55
0.05x = 55
x = 55/0.05
x = 1100
Therefore, cousin’s original monthly rent is $1100. Question 19. DIG DEEPER! You and your friend race on a trail that is 10 miles long. In each situation, does your friend pass you before the end of the trail? Justify your answer. a. You have a four-mile head start and jog at 6 miles per hour. Your friend bikes at 8 miles per hour. b. You have a ﬁve-mile head start and run at 7 miles per hour. Your friend bikes at 17 miles per hour. Answer: a) Your friend will pass you after the end of the trial. b) Your friend will pass you before the end of the trail. Explanation: a) Let x be the time after which your friend passes you In x hours you jog the distance 6x, your friend bikes 8x 8x = 4 + 6x 8x – 6x = 4 2x = 4 x = 4/2 x = 2 So in 2 hours, your friend bikes 2. 8 = 16 miles Therefore, he will pass you after the end of the trial because 16 > 10. b) In x hours you jog the distance 7x, while your friend bikes 17x 17x = 5 + 7x 17x – 7x = 5 10x = 5 x = 5/10 x = 0.5 In 0.5 hours, your friend bikes 0.5 x 17 = 8.5 miles So he will pass you before the end of the trail because 8.5 < 10. ### Solving Equations with Variables on Both Sides Homework & practice 1.3 Review & Refresh Solve the equation. Check your solution. Question 1. -9z + 2 = 11 Answer: z = -1 Explanation: Given equation is -9z + 2 = 11 -9z + 2 – 2 = 11 – 2 -9z = 9 z = -9/9 z = -1 Put z = -1 in -9z + 2 = 11 -9(-1) + 2 = 9 + 2 = 11 Question 2. -3n – 4n – 17 = 25 Answer: n = -6 Explanation: Given equation is -3n – 4n – 17 = 25 -7n – 17 = 25 -7n = 25 + 17 -7n = 42 n = -42/7 n = -6 Put n = -6 in -3n – 4n – 17 = 25 -3(-6) – 4(-6) – 17 = 18 + 24 – 17 = 42 – 17 = 25 Question 3. -2(x + 3) + 5x = -39 Answer: x = -11 Explanation: Given equation is -2(x + 3) + 5x = -39 -2x – 6 + 5x = -39 3x – 6 = -39 3x = -39 + 6 3x = -33 x = -33/3 x = -11 Put x = -11 in -2(x + 3) + 5x = -39 -2(-11 + 3) + 5(-11) = -2(-8) – 55 = 16 – 55 = 3-9 Question 4. -15 + 7.5(2d – 1) = 7.5 Answer: d = 2 Explanation: Given equation is -15 + 7.5(2d – 1) = 7.5 -15 + 15d – 7.5 = 7.5 -22.5 + 15d = 7.5 15d = 7.5 + 22.5 15d = 30 d = 30/15 d = 2 Put d = 2 in -15 + 7.5(2d – 1) = 7.5 -15 + 7.5(2(2) – 1) = -15 + 7.5(4 – 1) = -15 + 7.5(3) = -15 + 22.5 = 7.5 Find the volume of the solid. Question 5. Answer: Volume = 27 cubic cm Explanation: Length = 3 cm, width = 2 cm height = 4.5 cm Solid volume formula is length x width x height Volume V = 3 x 2 x 4.5 = 27 cubic cm Question 6. Answer: Volume is 15.75 cubic cm. Explanation: Given that, b = 4.5, h = 2, l = 3.5 The volume of Triangular Prism = ½ × b × h × l V = ½ × 4.5 × 2 × 3.5 = 4.5 x 1 x 3.5 = 15.75 cubic cm Question 7. Answer: Volume is 24 cubic in Explanation: Given that, Area of pentagon = 18 in² Height h = 4 in The volume of the pentagonal pyramid = 1/3 x base area x height = 1/3 x 18 x 4 = 6 x 4 = 24 cubic in Concepts, Skills, &Problem Solving FINDING MISSING MEASURES IN FIGURES If possible, ﬁnd the value of so that the value of the surface area (in square inches) is equal to the value of the volume (in cubic inches). Use an equation to justify your answer. (See Exploration 1, p. 17.) Question 8. Answer: 2(14x + 33) = 33x, x = 66/5 Explanation: Given that, length l =x in, width w = 11 in, height h = 3 in Surface area of rectangular prism = Volume of the rectangular prism 2(wl + hl + hw) = whl 2(11x + 3x + 33) = 11 . 3. x 2(14x + 33) = 33x 28x + 66 = 33x 66 = 33x – 28x 66 = 5x x = 66/5 Question 9. Answer: 2(36 + 13x) = 36x, x = 7.2 Explanation: Given that, length l = 9 in, width w = 4 in, height h = x in The surface area of rectangular prism = Volume of the rectangular prism 2(wl + hl + hw) = whl 2(36 + 9x + 4x) = 9 . 4 . x 2(36 + 13x) = 36x 72 + 26x = 36x 72 = 36x – 26x 10x = 72 x = 72/10 x = 7.2 Question 10. Answer: 2(9x + 18) = 18x, It does not have a solution Explanation: Given that, length l = 6 in, width w = 3 in, height h = x in The surface area of rectangular prism = Volume of the rectangular prism 2(wl + hl + hw) = whl 2(3x + 6x + 18) = 6 . 3 . x 2(9x + 18) = 18x 18x + 18 = 18x 18x – 18x = 18 0 = 18 SOLVING AN EQUATION Solve the equation. Check your solution. Question 11. m – 4 = 2m Answer: m = -4 Explanation: Given equation is m – 4 = 2m 2m – m = -4 m = -4 Put m = -4 in m – 4 = 2m -4 – 4 = 2(-4) -8 = -8 Question 12. 3k – 1 = 7k + 2 Answer: k = -3/4 Explanation: Given equation is 3k – 1 = 7k + 2 -1 – 2 = 7k – 3k -3 = 4k k = -3/4 Put k = -3/4 in 3k – 1 = 7k + 2 3(-3/4) – 1 = 7(-3/4) + 2 -9/4 – 1 = -21/4 + 2 (-9 – 4)/4 = (-21 + 8)/4 -13/4 = -13/4 Question 13. 6x = 5x + 22 Answer: x = 22 Explanation: Given equation is 6x = 5x + 22 6x – 5x = 22 x = 22 Put x = 22 in 6x = 5x + 22 6(22) = 5(22) + 22 132 = 110 + 22 Question 14. -24 – 8p = 4p Answer: p = -2 Explanation: Given equation is -24 – 8p = 4p -24 = 4p + 8p -24 = 12p p = -24/12 p = -2 Put p = -2 in -24 – 8p = 4p -24 – 8(-2) = 4(-2) -24 + 16 = -8 Question 15. 12(2w – 3) = 6w Answer: w = 2 Explanation: Given equation is 12(2w – 3) = 6w 24w – 36 = 6w 24w – 6w = 36 18w = 36 w = 36/18 w = 2 Put w = 2 in 12(2w – 3) = 6w 12(2w – 3) = 6w 12(2(2)- 3) = 6(2) 12(4 – 3) = 12 12 = 12 Question 16. 2(n – 3) = 4n + 1 Answer: n = -7/2 Explanation: Given equation is 2(n – 3) = 4n + 1 2n – 6 = 4n + 1 -6 – 1 = 4n – 2n -7 = 2n n = -7/2 Put n = -7/2 in 2(n – 3) = 4n + 1 2(-7/2 – 3) = 4(-7/2) + 1 2(-7 – 6)/2 = 2(-7) + 1 -13 = -14 + 1 Question 17. 2(4z – 1) = 3(z + 2) Answer: z = 8/5 Explanation: Given equation is 2(4z – 1) = 3(z + 2) 8z – 2 = 3z + 6 8z – 3z = 6 +2 5z = 8 z = 8/5 Put z = 8/5 in 2(4z – 1) = 3(z + 2) 2(4(8/5) – 1) = 3(8/5 + 2) 64/5 – 2 = 24/5 + 6 (64 – 10)/5 = (24 + 30)/5 54/5 = 54/5 Question 18. 0.1x = 0.2(x + 2) Answer: Explanation: Given equation is 0.1x = 0.2(x + 2) 0.1x = 0.2x + 0.4 0.2x – 0.1x = -0.4 0.1x = -0.4 x = -0.4/0.1 x = -4 Put x = -4 in 0.1x = 0.2(x + 2) 0.1(-4) = 0.2(-4 + 2) -0.4 = 0.2(-2) Question 19. Answer: d = 14 Explanation: Given equation is (1/6)d + 2/3 = 1/4 (d – 2) (1/6)d + 2/3 = d/4 – 2/4 (1/6)d + 2/3 = d/4 – 1/2 d/6 – d/4 = -1/2 – 2/3 (4d – 6d)/24 = (-3 – 4)/6 -2d/24 = -7/6 d = -7/6 x (-24/2) d = 14 Put d = 14 in (1/6)d + 2/3 = 1/4 (d – 2) (1/6)14 + 2/3 = 1/4 (14 – 2) 7/3 + 2/3 = 1/4 (12) 9/3 = 3 Question 20. YOU BE THE TEACHER Your friend solves the equation shown. Is your friend correct? Explain your reasoning. Answer: Wrong Explanation: 3x – 4 = 2x + 1 3x – 2x = 1 + 4 x = 5 Question 21. MODELING REAL LIFE Write and solve an equation to ﬁnd the number of miles you must drive to have the same cost for each of the car rentals. Answer: The number of miles I should drive to have the same cost for each of the car rentals is 40 miles Explanation: Let x represent the number of miles 20 + 0.5x = 30 + 0.25x 20 – 30 = 0.25x – 0.5x -10 = -0.25x x = 10/0.25 x = 40 The number of miles I should drive to have the same cost for each of the car rentals is 40 miles SOLVING AN EQUATION Solve the equation. Check your solution, if possible. Question 22. x + 6 = x Answer: The equation has no solution Explanation: Given equation is x + 6 = x x – x = -6 0 = -6 The equation has no solution Question 23. 3x – 1 = 1 – 3x Answer: x = 1/3 Explanation: Given equation is 3x – 1 = 1 – 3x 3x + 3x = 1 + 1 6x = 2 x = 2/6 x = 1/3 Put x = 1/3 in 3x – 1 = 1 – 3x 3(1/3) – 1 = 1 – 3(1/3) 1 – 1 = 1 – 1 Question 24. 3x + 15 = 3(x + 15) Answer: The equation has no solution Explanation: Given equation is 3x + 15 = 3(x + 15) 3x + 15 = 3x + 45 3x – 3x = 45 – 15 0 = 30 The equation has no solution Question 25. 4x – 9 = 3.5x – 9 Answer: x = 0 Explanation: Given equation is 4x – 9 = 3.5x – 9 4x – 3.5x = -9 + 9 -0.5x = 0 x = 0 Put x = 0 in 4x – 9 = 3.5x – 9 4(0) – 9 = 3.5(0) – 9 -9 = -9 Question 26. Answer: The equation has infinitely many solutions Explanation: Given equation is 1/3 (9x + 3) = 3x + 1 3x + 1 = 3x + 1 3x – 3x = 1 – 1 0 = 0 The equation has infinitely many solutions Question 27. 5x – 7 = 4x – 1 Answer: x = 6 Explanation: Given equation is 5x – 7 = 4x – 1 5x – 4x = 7 – 1 x = 6 Put x = 6 in 5x – 7 = 4x – 1 5(6) – 7 = 4(6) – 1 30 – 7 = 24 – 1 23 = 23 Question 28. Answer: The equation has no solution Explanation: Given equation is ½ x + ½ x = x + 1 x = x + 1 x – x = 1 0 = 1 The equation has no solution Question 29. 2x + 4 = -(-7x + 6) Answer: x = 2 Explanation: Given equation is 2x + 4 = -(-7x + 6) 2x + 4 = 7x – 6 4 + 6 = 7x – 2x 10 = 5x x = 10/5 x = 2 Put x = 2 in 2x + 4 = -(-7x + 6) 2(2) + 4 = -(-7(2) + 6) 4 + 4 = -(-14 + 6) 8 = -(-8) Question 30. 5.5 – x = -4.5 – x Answer: The equation has no solution Explanation: Given equation is 5.5 – x = -4.5 – x 5.5 + 4.5 = -x + x 10 = 0 Question 31. -3(2x – 3) = -6x + 9 Answer: The equation has infinitely many solutions Explanation: Given equation is -3(2x – 3) = -6x + 9 -6x + 9 = -6x + 9 -6x + 6x = 9 – 9 The equation has infinitely many solutions Question 32. Answer: The equation has no solution Explanation: Given equation is 10x – 8/3 – 4x= 6x 6x – 8/3 = 6x 6x – 6x = 8/3 0 = 8/3 The equation has no solution Question 33. 6(7x + 7) = 7(6x + 6) Answer: The equation has infinitely many solutions Explanation: Given equation is 6(7x + 7) = 7(6x + 6) 42x + 42 = 42x + 42 The equation has infinitely many solutions Question 34. YOU BE THE TEACHER Your friend solves the equation shown. Is your friend correct? Explain your reasoning. Answer: The equation has infinitely many solutions Explanation: -4(2n – 3) = 12 – 8n -8n + 12 = 12 – 8n -8n + 8n = 12 – 12 0 = 0 The equation has infinitely many solutions Question 35. OPEN-ENDED Write an equation with variables on both sides that has no solution. Explain why it has no solution. Answer: 5(2x – 3) = 10x – 15 Explanation: The equation is 5(2x – 3) = 10x – 15 10x – 15 = 10x – 15 10x – 10x = -15 + 15 0 = 0 For any values of x, the equation satisies. So, The equation has infinitely many solutions Question 36. MODELING REAL LIFE A cable television provider charges$75 for installation and $39.96 per month for a basic entertainment package.A satellite television provider others free installation and charges$13.32 per month for service for each television. Your neighbor subscribes to the cable provider the same month you subscribe to the satellite provider. After how many months is your neighbor’s total cost the same as your total cost when you own three televisions?

Never my neighbor’s total cost, my total cost will be same.

Explanation:
Let x as the number of months
My satellite cost for 3 televisions is 13.32x * 3 = 39.96x
My friends cable cost = 75 + 39.96x
75 + 39.96x = 39.96x
75 = 39.96x – 39.96x
75 = 0
75 is never equal to zero.
So, never my neighbor’s total cost, my total cost will be same.

Question 37.
MODELING REAL LIFE
A pizza parlor makes 52 pizza crusts the ﬁrst week of summer and 180 pizza crusts each subsequent week. A diner makes 26 pizza crusts the ﬁrst week of summer and 90 pizza crusts each subsequent week. In how many weeks will the total number of pizza crusts made by the pizza parlor be twice the total number of pizza crusts made by the diner?

The pizza parlor always makes twice the pizza crust than the dinner.

Explanation:
Let x is the number of weeks
Pizza parlor = 52 + 180x
Dinner = 26 + 90x
The number of pizza crusts made by the pizza parlor be twice the total number of pizza crusts made by the diner
52 + 180x = 2(26 + 90x
52 + 180x = 52 + 180x
52 – 52 = 180x – 180x
0 = 0
Infinite solutions.
So, the pizza parlor always makes twice the pizza crust than the dinner.

Question 38.
PRECISION

The given triangle is not equilateral triangle

Explanation:
Equilateral triangle have all sides of equal length
2x + 5.2 = 3x + 1.2
2x – 3x = 1.2 – 5.2
-x = -4
x = 4
3x + 1.2 = 2x + 6.2
3x – 2x = 6.2 – 1.2
x = 5
So, the given triangle is not equilateral triangle

GEOMETRY
Find the perimeter of the regular polygon.

Question 39.

Perimeter = 3

Explanation:
Perimeter is the sum of all sides of the polygon
As it is a regular polygon all sides are equal
5 – 2x = -4x + 9
5 – 9 = -4x + 2x
-4 = -2x
x = 4/2
x = 2
One side = 5 – 2(2) = 5 – 4 = 1
Perimeter = 1 + 1 + 1 = 3

Question 40.

Perimeter = 7.5

Explanation:
Perimeter is the sum of all sides of the polygon
As it is a regular polygon all sides are equal
3(x – 1) = 5x – 6
3x – 3 = 5x – 6
5x – 3x = -3 + 6
2x = 3
x = 3/2 = 1.5
One side = 5(1.5) – 6
= 7.5 – 6 = 1.5
Perimeter = 5(1.5) = 7.5

Question 41.

Perimeter = 344/3

Explanation:
Perimeter is the sum of all sides of the polygon
As it is a regular polygon all sides are equal
4/3 x – 1/3 = x + 7
4/3 x – x = 7 + 1/3
x = (21 + 1)/3
x = 22/3
One side = 22/3 + 7
= (22 + 21)/3 = 43/3
Perimeter = 8(43/3)
= 344/3

Question 42.
PRECISION
The cost of mailing a DVD in an envelope using Company B is equal to the cost of mailing a DVD in a box using Company A. What is the weight of the DVD with its packing material? Round your answer to the nearest hundredth.

The weight of the DVD with its packing material is around 0.19 lb.

Explanation:
Let x is the weight of the DVD
Cost of mailing a DVD in a box using company A is 2.50x + 2.25
Cost of mailing a DVD in a box using company B 8.50x + 1.10
$8.50x +$1.10 = $2.50x +$2.25
8.50x – 2.50x = 2.25 – 1.10
6x = 1.15
x = 1.15/6
x = 0.19
The weight of the DVD with its packing material is around 0.19 lb.

Question 43.
WRITING
Would you solve the equation 0.25x + 7 = $$\frac{1}{3}$$x – 8 using fractions or decimals ? Explain.

x = 180

Explanation:
0.25x + 7 = 1/3 x – 8
0.25x – 1/3 x = -8 – 7
25/100x – 1/3 x = -15
1/4 x – 1/3 x = -15
(3 – 4)/12 x = -15
-1/12 x = -15
x = -15 x -12
x = 180

Question 44.
NUMBER SENSE
The weight of an object is equal to $$\frac{3}{4}$$ of its own weight plus $$\frac{3}{4}$$ of a pound. How much does the object weigh? Explain.

The object weight is 3 lb.

Explanation:
Let us take x as the object weight
x = 3/4 x + 3/4
4x = 3x + 3
4x – 3x = 3
x = 3
The object weight is 3 lb.

Question 45.
STRUCTURE
Fill in the blanks in three different ways to create an equation that has one solution, no solution, and infinitely many solutions.

7x + 3x + 10 = -2(10x + 10) has one solution.
7x + 3x + 10 = -2(-5x – 5) has infinite solutions
7x + 3x + 10 = -2(-5x + 1) has no solution.

Explanation:
Given that,
7x + 3x + 10 = -2(_ x + _)
One solution
7x + 3x + 10 = -2(10x + 10)
10x + 10 = -20x – 20
10x + 20x = -20 – 10
30x = -30
x = -30/3
x = -1
Infinite solutions
7x + 3x + 10 = -2(-5x – 5)
10x + 10 = 10x + 10
10x – 10x = 10 – 10
0 = 0
No solution
7x + 3x + 10 = -2(-5x + 1)
10x + 10 = 10x – 2
10 = -2

Question 46.
MODELING REAL LIFE
The volume of red blood cells in a blood sample is equal to the total volume of the sample minus the volume of plasma. What is the total volume x of blood drawn?

The volume of blood is 5.95 mL.

Explanation:
The volume of red blood cells in a blood sample is equal to the total volume of the sample minus the volume of plasma.
45% = sample volume – 5.5
0.45 = sample volume – 5.5
sample volume = 045 + 5.5
= 5.95

Question 47.
PROBLEM SOLVING
One serving of oatmeal provides 16% of the ﬁber you need daily. You must get the remaining 21 grams of ﬁber from other sources. How many grams of ﬁber should you consume daily? Justify your answer.

The total amount of fiber you need daily is 25 grams.

Explanation:
Let x is the total amount of fiber you needed daily.
Since one serving of oatmeal provides 16% of the fiber you need daily, then 0.16x grams is the amount of fiber you get from oatmeal
If you get another 21 grams of fiber from other sources, then 0.16x + 21 grams is the total amount of fiber you need daily
x and 0.16x + 21 both represent the total amount of fiber you need daily
So 0.16x + 21 = x
21 = x – 0.16x
21 = 0.84x
x = 21/0.84
x = 25
The total amount of fiber you need daily is 25 grams.

Question 48.
DIG DEEPER!
The ﬂoor of a six-foot-wide hallway is painted as shown, using equal amounts of white and black paint.
a. How long is the hallway?
b. Can this same hallway be painted with the same pattern, but using twice as much black paint as white paint? Explain.

a. The total length of the hallway = 9x + 4
b. Yes

Explanation:
a. The total length of the hallway = The length of white paints + lngth of black paints
= 5(x) + 4(x + 1)
= 5x + 4x + 4 = 9x + 4

Question 49.
PRECISION
Consider the equation c = ax – bx, where a, b, and c are whole numbers. Which of the following result in values a, b and c so that the original equation has exactly one solution? Justify your answer.

a ≠  b, c = 0

Explanation:
c = ax – bx,
Substitute a ≠  b, c = 0 values in above equation
c = ax – bx
0 = x(a – b)

### Lesson 1.4 Rewriting Equations and Formulas

EXPLORATION

Work with a partner.
a. Write a formula for the height h of each ﬁgure. Explain your method.

• A parallelogram with area A and base b
• A rectangular prism with volume V, length l, and width w
• A triangle with area A and base b

b. Write a formula for the length l of each ﬁgure. Explain your method.

• A rectangle with perimeter P and width w
• A rectangular prism with surface area S, width w, and height h.

c. Use your formulas in parts (a) and (b) to ﬁnd the missing dimension of each ﬁgure.

1.4 Lesson

Try It

Solve the equation for y.

Question 1.
5y – x = 10

y = (10 + x)/5

Explanation:
Given equation is 5y – x = 10
5y – x + x = 10 + x
5y = 10 + x
Divide both sides by 5
5y/5 = (10 + x)/5
y = (10 + x)/5

Question 2.
4x – 4y = 1

y = (4x – 1)/4

Explanation:
Given equation is 4x – 4y = 1
Subtract 4x from both sides
4x – 4y – 4x  = 1 – 4x
-4y = 1 – 4x
Divide both sides by -4
-4y/-4 = (1-4x)/-4
y = (4x – 1)/4

Question 3.
12 = 6x + 3y

y = (12 – 6x)/3

Explanation:
Given equation is 12 = 6x + 3y
Subtract 6x from each side
12 – 6x = 6x + 3y – 6x
12 – 6x = 3y
Divide each side by 3
(12 – 6x)/3 = 3y/3
y = (12 – 6x)/3

Try It

Solve the formula for the red variable.

Question 4.
Area of rectangle: A = bh

b = A/h

Explanation:
Given that,
Area of the rectangle A = bh
Divide each side by h
A/h = bh/h
A/h = b
Area/height = base

Question 5.
Simple interest: I = Prt

P = I/rt

Explanation:
Given that,
Simple interest: I = Prt
Divide each side by rt
I/rt = Prt/rt
P = I/rt

Question 6.
Surface area of cylinder: S = 2πr2 + 2πrh

h = (S – 2πr²)/2πr

Explanation:
Given that,
Surface area of cylinder: S = 2πr² + 2πrh
Subtract 2πr² from each side
S – 2πr² = 2πr² + 2πrh – 2πr²
S – 2πr² = 2πrh
Divide each side by 2πr
(S – 2πr²)/2πr = h

Try It

Question 7.
Solve the formula F = $$\frac{9}{5}$$C + 32 for C. Justify your answer.

C = 5(F – 32)/9

Explanation:
Given that,
F = (9/5)C + 32
Subtract 32 from each side
F – 32 = (9/5)C + 32 – 32
F – 32 = (9/5)C
Multiply each side by (5/9)
(F – 32) x (5/9) = (9/5)C x (5/9
5(F – 32)/9 = C

Self-Assessment for Concepts & Skills
Solve each exercise. Then rate your understanding of the success criteria in your journal.

Question 8.
REWRITING A Formula
The formula for the circumference of a circle is C = 2πr. Solve the formula for r.

r = C/2π

Explanation:
Given that
The formula for the circumference of a circle is C = 2πr
Divide each side by 2π
C/ 2π = 2πr/2π
C/2π = r

Question 9.
DIFFERENT WORDS, SAME QUESTION
Which is different? Find “both” answers.

2y – 4x = 6

Explanation:
1. 4x = 6 + 2y
4x – 6 = 2y
(4x – 6)/2 = y
y = [2(2x – 3)]/2
y = (2x – 3)
2. 6 = 4x – 2y
6 – 4x = -2y
y = (4x – 6)/2
y = (2x – 3)
3. 2y – 4x = -6
2y = -6 + 4x
y = (4x – 6)/2
y = 2x – 3
4. 2y – 4x = 6
2y = 6 + 4x
y = 3 + 2x

Self-Assessment for Problem Solving

Solve each exercise. Then rate your understanding of the success criteria in your journal.

Question 10.
Room temperature is considered to be 70°F. The temperature outside is currently 23°C. Is this greater than or less than room temperature?

The outside temperature is greater than room temperature

Explanation:
Room temperature is considered to be 70°F. The temperature outside is currently 23°C
F = (9/5) C + 32
Put C = 23°
F = (9/5)23 + 32
F = 41.1+ 32
F = 73.4°
C = 23° is greater than F = 70°
So the outside temperature is greater than room temperature

Question 11.
DIG DEEPER!
A bird ﬂies at a top speed of 20,000 meters per hour. The bird ﬂies 30,000 meters without stopping.
a. For how many hours did the bird ﬂy if it ﬂew at top speed?
b. In part(a), did you rewrite a formula to ﬁnd the number of hours the bird ﬂew, or did you use another approach? Explain.

a) The bird flew 1.5 hours at Top speed
b) Yes, I rewrite a formula to find the number of hours the bird flew.

Explanation:
Speed of the bird = 20000 meters per hour
Distance travelled = 30000 meters
a) Distance = speed x time
d = s x t
t = d/s
t = 30000/20000
t = 1.5
The bird flew 1.5 hour at Top speed
b) Yes, I rewrite a formula to find the number of hours the bird flew.

Question 12.
A ball pit is in the shape of a cylinder with a lateral surface area of 245 square feet. The diameter of the ball pit is 312 inches. What is the height of the ball pit? Justify your answer.

The height of the ball pit is 3 feet.

Explanation:
Diameter of the ball pit = 312 inches = 26 feet
Radius of the ball pit = 26/2 = 13 feet
Lateral surface area = 245 sq. ft
Surface area = 2πrh
245 = 2 x 3.14 x 13 x h
245 = 81.64h
h = 245/81.64
h = 3 feet
The height of the ball pit is 3 feet

### Rewriting Equations and Formulas Homework & Practice 1.4

Solve the equation. Check your solution, if possible.

Question 1.
-2x = x + 15

x = -5

Explanation:
Given equation is -2x = x + 15
-2x – x = 15
-3x = 15
x = -15/3
x = -5
Put x = -5 in -2x = x + 15
-2(-5) = -5 + 15
10 = 10

Question 2.
4(z – 3) = 2z

z = 6

Explanation:
Given equation is 4(z – 3) = 2z
4z – 12 = 2z
4z – 2z = 12
2z = 12
z = 12/2
z = 6
Put z = 6 in 4(z – 3) = 2z
4(6 – 3) = 2(6)
4(3) = 12

Question 3.
x – 8 = x – 1

The equation has no solution

Explanation:
Given equation is x – 8 = x – 1
x – x = -1 + 8
0 = 7
The equation has no solution

Question 4.
5(4 + t) = 5t + 20

The equation has infinitely many solutions

Explanation:
Given equation is 5(4 + t) = 5t + 20
20 + 5t = 5t + 20
20 + 5t – 20 = 5t
5t = 5t
The equation has infinitely many solutions

Find the unit rate.

Question 5.
60 miles in 5 hours

Speed = 12 miles per 1 hour

Explanation:
60 miles in 5 hours
Speed = 60 miles in 5 hours
= 60/5 = 12 miles per 1 hour

Question 6.
$8.50 : 5 ounces Answer:$8.50 : 5 ounces = 1.7 per ounce

Explanation:
$8.50 : 5 ounces Unit rate =$8.50/5
= 1.7 per ounce

Question 7.
9 pounds per 6 crates

9 pounds per 6 crates = 1.5

Explanation:
9 pounds per 6 crates
Unit rate = 9/6
= 1.5

Concepts, Skills, &Problem Solving
REWRITING FORMULA
Solve the formula for the height of the ﬁgure. Then use the new formula to ﬁnd the height. (See Exploration 1, p. 25.)

Question 8.

Height = 6mm

Explanation:
A = ½bh
36 = ½ x 12 x h
36 = 6h
h = 36/6
h = 6 mm

Question 9.

Height = 6 inch

Explanation:
Volume V = Base area x height
h = V/B
h = 36/6
h = 6 inch

IDENTIFYING LITERAL EQUATIONS
Is the equation a literal equation? Explain.

Question 10.
y = 4

y = 4 is not a liter equation.

Explanation:
y = 4 is not a liter equation. A literal equation is an equation that has two or more variables.

Question 11.
t + 8y = 7

t + 8y = 7 is a literal equation.

Explanation:
t + 8y = 7 is a literal equation because it has two variables t, and y.

Question 12.
z = 4x + 9y

z = 4x + 9y is a literal equation.

Explanation:
z = 4x + 9y is a literal equation because it has 3 variables x, y , and z.

REWRITING AN EQUATION
Solve the equation for y.

Question 13.

y = (12 – x)/3

Explanation:
Given equation is (1/3)x + y = 4
Subtract each side by (1/3)x
(1/3)x + y – (1/3)x = 4 – (1/3)x
y = 4 – (1/3)x
y = (12 – x)/3

Question 14.

y = (7 – 3x)/5

Explanation:
Given equation is 3x + (1/5)y = 7
Subtract 3x from each side
3x + (1/5)y – 3x = 7 – 3x
(1/5)y = 7 – 3x
Multiply each side by 5
(1/5)y x 5 = (7 – 3x)/5
y = (7 – 3x)/5

Question 15.
6 = 4x + 9y

y = (6 – 4x)/9

Explanation:
Given equation is 6 = 4x + 9y
Subtract 4x from each side
6 – 4x = 4x + 9y – 4x
6 – 4x = 9y
Divide each side by 9
(6 – 4x)/9 = 9y/9
y = (6 – 4x)/9

Question 16.
π = 7x – 2y

y = (7x – π)/2

Explanation:
Given equation is π = 7x – 2y
Subtract 7x from each side
π – 7x = 7x – 2y – 7x
π – 7x = -2y
Divide eachside by -2
(π – 7x)/-2 = -2y/-2
(7x – π)/2 = y

Question 17.
4.2x – 1.4y = 2.1

y = 3(2x – 1)/2

Explanation:
Given equation is 4.2x – 1.4y = 2.1
Subtract 4.2x from each side
4.2x – 1.4y – 4.2x = 2.1 – 4.2x
-1.4y = 2.1 – 4.2x
Divide each side by -1.4y
-1.4y/-1.4 = (2.1 – 4.2x)/-1.4
y = (4.2x – 2.1)/1.4
= 2.1(2x – 1)/1.4
y = 3(2x – 1)/2

Question 18.
6y – 1.5x = 8

y = (8 – 1.5x)/6

Explanation:
Given equation is 6y – 1.5x = 8
6y – 1.5x – 1.5x = 8 – 1.5x
6y = 8 – 1.5x
Divide each side by 6
6y/6 = (8 – 1.5x)/6
y = (8 – 1.5x)/6

Question 19.
YOU BE THE TEACHER
Your friend rewrites the equation 2x – y = 5. Is your friend correct? Explain your reasoning.

Wrong

Explanation:
2x – y = 5
Subtract 2x from each side
2x – y – 2x = 5 – 2x
-y = -2x + 5
y = 5 – 2x

REWRITING A FORMULA
Solve the formula for the red variable.

Question 20.
d = rt

t = d/r

Explanation:
Given that,
d = rt
t = d/r

Question 21.
e = mc2

m = e/c²

Explanation:
Given that,
e = mc²
e/c² = m

Question 22.
R – C = P

C = R – P

Explanation:
Given that,
R – C = P
-C = P – R
C = R – P

Question 23.
P = a + b + c

a = P – b – c

Explanation:
Given that,
P = a + b + c
P – b – c = a
a = P – b – c

Question 24.
B = 3$$\frac{V}{h}$$

V = Bh/3

Explanation:
Given that,
B = 3(V/h)
Multiply each side by h
Bh = 3(V/h) x h
Bh = 3V
Divide each side by 3
Bh/3 = 3V/3
V = Bh/3

Question 25.
D = $$\frac{m}{V}$$

V = m/D

Explanation:
Given that,
D = m/V
Multiply each side by V
DV = (m/V)V
DV = m
Divide each side by D
DV/D = m/D
V = m/D

Question 26.
MODELING REAL LIFE
The formula K = C + 273.15 converts temperatures from degrees Celsius to Kelvin K.
a. Convert 200 degrees Celsius to Kelvin.
b. Solve the formula for C.
c. Convert 300 Kelvin to degrees Celsius.

a. K = 473.15
b. C = K – 273.15
c. C = 26.85°

Explanation:
Given formula is K = C + 273.15
a) Convert 200 degrees Celsius to Kelvin.
K = C + 273.15
Put C = 200
K = 200 + 273.15
K = 473.15
b) Solve the formula for C
K = C + 273.15
C = K – 273.15
c) Convert 300 Kelvin to degree Celsius
C = 300 – 273.15
C = 26.85°

Question 27.
PROBLEM SOLVING
The formula for simple interest is I = Prt.
a. Solve the formula for t, when r is the simple interest per year.
b. Use the new formula to ﬁnd the value of in the table.

a. t = I/Pr
b. t =1/20

Explanation:
The formula for simple interest is I = Prt.
a. Solve the formula for t, when r is the simple interest per year.
I = Prt
Divide each side by Pr
I/Pr = Prt/Pr
t = I/Pr
b. Given that,
I = $75, P =$500, r = 5%, t = ?
t = I/Pr
Put above values in the formula
t = 75/(500 x 5)
= 1/20

Question 28.
GEOMETRY
Use the triangle shown.
a. Write a formula for the perimeter P of the triangle.
b. Solve the formula for b.
c. Use the new formula to ﬁnd b when a is 10 feet and c is 17 feet.

a. P = a + b + c
b. b = P – a – c
c. b = 15 feet

Explanation:
a. Write a formula for the perimeter P of the triangle.
We know that the perimeter is nothing but the sum of the outer edges of the triangle.
Perimeter = a + b + c
P = a + b + c
b. Solve the formula for b.
P = a + b + c
Subtract a, c from each side
P – a – c = a + b + c – a – c
P – a – c = b
c. Given that,
P = 42 feet, a = 10 ft, c = 17 ft
b = P – a – c
b = 42 – 10 – 17
b = 42 – 27
b = 15 feet

Question 29.
REASONING
The formula converts temperatures from degrees Fahrenheit F to Kelvin K.
a. Solve the formula for F.
b. The freezing point of helium is 0.95 Kelvin. What is this temperature in degrees Fahrenheit?
c. The temperature of dry ice is -78.5°C. Which is colder, dry ice or liquid nitrogen?

a. F = 9/5 (K – 273.15) + 32
b. F = = -454.96
c. liquid nitrogen is colder than dry ice

Explanation:
K = 5/9 (F – 32) + 273.15
a. Solve the formula for F.
K = 5/9 (F – 32) + 273.15
Subtract 237.15 from each side
K – 273.15 = 5/9 (F – 32)
Multiply each side by 9
9(K – 273.15) = 5(F – 32)
Divide each side by 5
9(K – 273.15)/5 = 5(F – 32)/5
9/5 (K – 273.15) = F – 32
Subtract 32 from each side
9/5 (K – 273.15) + 32 = F – 32 + 32
9/5 (K – 273.15) + 32 = F
b. The freezing point of helium is 0.95 Kelvin. What is this temperature in degrees Fahrenheit?
F = 9/5 (K – 273.15) + 32
Put K = 0.95
F = 9/5 (0.95  – 273.15) + 32
F = 9/5 (-272.2) + 35
= 9(-54.44) + 35
= -489.96 + 35
= -454.96
c. The temperature of dry ice is -78.5°C. Which is colder, dry ice or liquid nitrogen?
liquid nitrogen is colder than dry ice

Question 30.
MODELING REAL LIFE
In which city is the water temperature higher?

The water temperature is higher in Portland.

Explanation:
By observing the image we can say that in Portland the water temperature is higher.

Question 31.
GEOMETRY
The volume of a square pyramid with a height of 30 feet is 360 cubic feet. What are the side lengths of the base? Justify your answer.

The base side length is 6 feet.

Explanation:
Given that,
height h = feet
Volume V = 360 cubic feet
Volume V = Bh/3
3V = Bh
3V/h = B
B = (3 x 360)/30
B = 36
Side² = 36
Side = √(36)
Side = 6 feet
The base side length is 6 feet.

Question 32.
DIG DEEPER!
The Navy Pier Ferris Wheel in Chicago has a circumference that is 56% of the circumference of the ﬁrst Ferris wheel built in 1893.
a. What is the radius of the Navy Pier Ferris Wheel?
b. What was the radius of the ﬁrst Ferris wheel?
c. The ﬁrst Ferris wheel took 9 minutes to make a complete revolution. How fast was the wheel moving?

a. The radius of the Navy Pier Ferris Wheel is 70 feet
b. The radius of the ﬁrst Ferris wheel is 125 feet
c. The wheel is moving with 87.2 ft/min

Explanation:
a. Circle circumference = 2πr
439 = 2πr
r = 439/2π
r = 439/(2 x 3.14) = 439/6.28
r = 70 feet
The radius of the Navy Pier Ferris Wheel is 70 feet
b. New circumference = 56% of old circumference
439.6 = 0.56 x C
C = 439.6/0.56
C = 785
Old circumference = 2πr(old)
785 = 2πr(old)
r(old) = 785/2π
= 125 feet
The radius of the ﬁrst Ferris wheel is 125 feet
c. Old circumference/ time
Cold / Time = 785/9
= 87.2 ft/min
The wheel is moving with 87.2 ft/min

### Equations Connecting Concepts

Connecting Concepts

Problem-Solving Strategies
Using an appropriate strategy will help you make sense of problems as you study the mathematics in this course. You can use the following strategies to solve problems that you encounter.

Using the Problem-Solving Plan

Question 1.
The battery life of a one-year-old cell phone is 75% of its original battery life. When the battery is charged to 50% of its capacity, it dies after 4$$\frac{1}{2}$$ hours. Find the original battery life of the phone. Justify your answer.

Understand the problem.
You know how long a cell phone battery lasts when it is charged to 50% of its capacity. You also know that the battery life of the phone is 75% of its original battery life. You are asked to ﬁnd the original battery life of the phone.
Make a plan.
First, ﬁnd the battery life of the one-year-old cell phone. Then use this information to write and solve an equation for the original battery life of the phone.
Solve and check.
Use the plan to solve the problem. Then check your solution.

The original battery life of the phone is 12 hours.

Explanation:
Let x be the battery life of the one-year-old cell phone.
(100% – 50%)x = 4.5
(1 – 0.5)x = 4.5
0.5x = 4.5
x = 4.5/0.5
x = 9 hours
Let take y as the original battery life
One year old battery life = 75% of original battery life
9 = 0.75y
y = 9/0.75
y = 12 hours
75% of original battery life is 75% x 12 = 0.75 x 12 = 9
50% of the battery life of the one year old cell phone is 0.50 x 9 = 4.5
The original battery life of the phone is 12 hours.

Question 2.
The triangular prism shown has a volume of 132 cubic centimeters. Find the height of the prism. Justify your answer.

Height h = 5.5 cm

Explanation:
The triangular prism shown has a volume of 132 cubic centimeters.
a = 6 cm, b = 8 cm, h = ?
V = 0.5 x b x a x h
132 = 0.5 x 8 x 6 x h
132 = 24 x h
h = 132/24
h = 5.5
Height h = 5.5 cm

Target Heart Rates
At the beginning of this chapter, you watched a STEAM Video called “Training for a Half Marathon.” You are now ready to complete the performance task related to this video, available at BigIdeasMath.com. Be sure to use the problem-solving plan as you work through the performance task.

### Equations Chapter Review

Review Vocabulary

Write the deﬁnition and give an example of the vocabulary term.

Graphic Organizers

You can use an Information Frame to help organize and remember a concept. Here is an example of an Information Frame for solving equations with variables on both sides.

Choose and complete a graphic organizer to help you study the concept.

1. solving simple equations using addition
2. solving simple equations using subtraction
3. solving simple equations using multiplication
4. solving simple equations using division
5. inverse operations
6. literal equation

Chapter Self-Assessment

As you complete the exercises, use the scale below to rate your understanding of the success criteria in your journal.

1.1 Solving Simple Equations (pp. 3–10)

Solve the equation. Check your solution.

Question 1.
y + 8 = -11

y = -19

Explanation:
Given equation is y + 8 = -11
y + 8 – 8 = -11 – 8
y = -19
Put y = -19 in y + 8 = -11
-19 + 8 = -11

Question 2.
3.2 = -0.4n

n = -8

Explanation:
Given equation is 3.2 = -0.4n
3.2/-0.4 = -0.4n/-0.4
-8 = n
Put n = -8 in 3.2 = -0.4n
3.2 = -0.4(-8)

Question 3.

t = 12π

Explanation:
Given equation is -t/4 = -3π
-t = -3π x 4
-t = -12π
t = 12π
Put t = 12π in -t/4 = -3π
-12π/4 = -3π

Question 4.
v – | 2.4 | = 5.7

v = 8.1

Explanation:
Given equation is v – | 2.4 | = 5.7
v – 2.4 = 5.7
v = 5.7 + 2.4
v = 8.1
put v = 8.1 in v – | 2.4 | = 5.7
8.1 – | 2.4 | = 5.7
8.1 – 2.4 = 5.7

Question 5.
-6 = -2 + w

w = -4

Explanation:
Given equation is -6 = -2 + w
-6 + 2 = w
-4 = w
Put w = -4 in -6 = -2 + w
-6 = -2 – 4

Question 6.

x = -1/4

Explanation:
Given equation is x – 2/3 = -11/12
x = -11/12 + 2/3
x = (-11 + 8)/12
x = -3/12
x = -1/4
Put x = -1/4 in x – 2/3 = -11/12
-1/4 – 2/3 = -11/12
(-3 – 8)/12 = -11/12

Question 7.
The boiling point of a liquid is the temperature at which the liquid becomes a gas. The boiling point of mercury is about $$\frac{41}{200}$$ of the boiling point of lead. Write and solve an equation to ﬁnd the boiling point of lead.

41/200 x = 357
The boiling point of lead is 1741.46°C

Explanation:
The boiling point of mercury = 357°C
The boiling point of lead is x
41/200 x = 357
x = 357 x (200/41)
x = 1741.46
The boiling point of lead is 1741.46°C

Question 8.
Write an equation that you can use the Addition Property of Equality to solve.

x – 6 = 8

Explanation:
x – 6 = 8
x – 6 + 6 = 8 + 6
x = 14

Question 9.
To solve $$\frac{2}{5}$$x = 14, you multiply both sides of the equation by $$\frac{5}{2}$$. Your friend divides both sides of the equation by $$\frac{2}{5}$$. Who is correct? Explain.

Multiply both sides of the equation by 5/2

Explanation:
2/5 x = 14
Multiply both sides of the equation by 5/2
2/5 x (5/2) = 14(5/2)
x = 7 x 5
x = 35

Question 10.
Write and solve an equation to ﬁnd the value of x.

x = 45°

Explanation:
(x + 10)° = 55°
x + 10 – 10 = 55 – 10
x = 45°

Question 11.
The circumference C of a circle is 24π inches. Use the formula C = 2πr to ﬁnd the radius r of the circle.

The radius r of the circle is 12 inches.

Explanation:
The circumference C of a circle is 24π inches
C = 2πr
24π = 2πr
24π/2π = r
12 = r
The radius r of the circle is 12 inches

1.2 Solving Multi-Step Equations (pp. 11–16)

Solve the equation. Check your solution.

Question 12.
3n + 12 = 30

n = 6

Explanation:
Given equation is 3n + 12 = 30
3n = 30 – 12
3n = 18
n = 18/3
n = 6
Put n = 6 in 3n + 12 = 30
3(6) + 12 = 18 + 12 = 30

Question 13.
2(3 – p) – 17 = 41

p = -26

Explanation:
Given equation is 2(3 – p) – 17 = 41
6 – 2p – 17 = 41
-2p – 11 = 41
-2p = 41 + 11
-2p = 52
p = 52/-2
p = -26
Put p = -26 in 2(3 – p) – 17 = 41
2(3 – (-26)) – 17 = 2(3 + 26) – 17
= 2(29) – 17 = 58 – 17 = 41

Question 14.
-14x + 28 + 6x = -44

x = 9

Explanation:
Given equation is -14x + 28 + 6x = -44
-8x + 28 = -44
-8x = -44 – 28
-8x = -72
x = 72/8
x = 9
put x = 9 in -14x + 28 + 6x = -44
-14(9) + 28 + 6(9) = -126 + 28 + 54
= -44

Question 15.
1.06(12.95 + x) = 31.27

x = 16.55

Explanation:
Given equation is 1.06(12.95 + x) = 31.27
12.95 + x = 31.27/1.06
12.95 + x = 29.5
x = 29.5 – 12.95
x = 16.55
Put x = 16.55 in 1.06(12.95 + x) = 31.27
1.06(12.95 + 16.55) = 1.06(29.5) = 31.27

Question 16.
The sum of the angle measures of a quadrilateral x is 360°. Find the value of x. Then ﬁnd the angle measures of the quadrilateral.

x = 120°

Explanation:
The sum of the angle measures of a quadrilateral is 360°
x° + x° + ½x° + ½x° = 360°
2x° + x° = 360°
3x° = 360°
x = 360/3
x = 120°

Question 17.
The equation P = 2.5m + 35 represents the price (in dollars) of a bracelet, where is the cost of the materials (in dollars). The price of a bracelet is $115. What is the cost of the materials? Answer: The cost of the materials is$32.

Explanation:
The equation P = 2.5m + 35 represents the price (in dollars) of a bracelet
The price of a bracelet is $115$115 = 2.5m + 35
115 – 35 = 2.5m
80 = 2.5m
80/2.5 = m
32 = m
The cost of the materials is $32. Question 18. A 455-foot fence encloses a pasture. What is the length of each side of the pasture? Answer: The length of each side of the pasture is 50 ft, 150 ft, 180 ft, 75 ft. Explanation: A 455-foot fence encloses a pasture x + 3x + 1.5x + 180 = 455 5.5x + 180 = 455 5.5x = 455 – 180 5.5x = 275 x = 275/5.5 x = 50 The length of each side of the pasture is 50 ft, 150 ft, 180 ft, 75 ft. 1.3 Solving Equations with Variables on Both Sides (pp. 17–24) Solve the equation. Check your solution, if possible. Question 19. 3(x – 4) = -2(4 – x) Answer: x = 4 Explanation: Given equation is 3(x – 4) = -2(4 – x) 3x – 12 = -8 + 2x 3x – 2x = -8 + 12 x = 4 Put x = 4 in 3(x – 4) = -2(4 – x) 3(4 – 4) = -2(4 – 4) 3(0) = -2(0) 0 = 0 Question 20. 4 – 5k = -8 – 5k Answer: The equation has no solution Explanation: Given equation is 4 – 5k = -8 – 5k 4 + 8 = -5k + 5k 12 = 0 The equation has no solution Question 21. 5m – 1 = 4m + 5 Answer: m = 6 Explanation: Given equation is 5m – 1 = 4m + 5 5m – 4m = 5 + 1 m = 6 Put m = 6 in 5m – 1 = 4m + 5 5(6) – 1 = 4(6) + 5 30 – 1 = 24 + 5 29 = 29 Question 22. 3(5p – 3) = 5(p – 1) Answer: p = 2/5 Explanation: Given equation is 3(5p – 3) = 5(p – 1) 15p – 9 = 5p – 5 15p – 5p = -5 + 9 10p = 4 p = 4/10 p = 2/5 Put p = 2/5 in 3(5p – 3) = 5(p – 1) 3(5(2/5) – 3) = 5(2/5 – 1) 3(2 – 3) = 5(2 – 5)/5 3(-1) = -3 Question 23. 0.4n + 0.1 = 0.5(n + 4) Answer: n = -19 Explanation: Given equation is 0.4n + 0.1 = 0.5(n + 4) 0.4n + 0.1 = 0.5n + 2 0.1 – 2 = 0.5n – 04n -1.9 = 0.1n -1.9/0.1 = n n = -19 Put n = -19 in 0.4n + 0.1 = 0.5(n + 4) 0.4(-19) + 0.1 = 0.5(-19 + 4) -7.6 + 0.1 = 0.5(-15) -7.5 = -7.5 Question 24. 7t + 3 = 8 + 7t Answer: The equation has no solution Explanation: Given equation is 7t + 3 = 8 + 7t 7t – 7t = 8 – 3 0 = 5 The equation has no solution Question 25. Answer: The equation has no solution Explanation: Given equation is 1/5 (15b – 7) = 3b – 9 (15b – 7) = 5(3b – 9) 15b – 7 = 15b – 45 15b – 15b = -45 + 7 0 = -38 The equation has no solution Question 26. Answer: The equation has infinitely many solutions Explanation: Given equation is 1/6 (12z – 18) = 2z – 3 1/6 (6(2z – 3)) = 2z – 3 2z – 3 = 2z – 3 2z = 2z – 3 + 3 2z = 2z The equation has infinitely many solutions Question 27. The side lengths of an isosceles triangle are (3x + 1) inches, (4x + 5) inches, and (2x + 7) inches. Find the perimeters of two possible triangles. Answer: The perimeters of two possible triangles are 11x + 11, 12x + 19. Explanation: The side lengths of an isosceles triangle are (3x + 1) inches, (4x + 5) inches, and (2x + 7) inches. The sides of first triangle are (3x + 1), (4x + 5), (4x + 5) Perimeter = sum of all sides P = 3x + 1 + 4x + 5 + 4x + 5 = 11x + 11 The sides of the second triangle are (4x + 5), (2x + 7), (2x + 7) Perimeter = (4x + 5) + (2x + 7) + (2x + 7) = 12x + 19 Question 28. A shuttle company charges$3.25 plus $0.55 per mile. A taxi company charges$2.50 plus $0.60 per mile. After how many miles will both companies charge the same amount? Answer: After 15 miles both company charge the same amount. Explanation: Assume x as the miles 3.25 + 0.55x = 2.50 + 0.60x 3.25 – 2.50 = 0.60x – 0.55x 0.75 = 0.05x x = 0.75/0.05 x = 15 After 15 miles both company charge the same amount. Question 29. You begin the year with$25 in a savings account and $50 in a checking account. Each week you deposit$5 into the savings account and $10 into the checking account. In how many weeks is the amount in the checking account twice the amount in the savings account? Answer: Every week the amount in checking twice the amount in savings. Explanation: You begin the year with$25 in a savings account and $50 in a checking account. 50 + 10x Savings is 25 + 5x The amount in checking twice the amount in savings 50 + 10x = 2(25 + 5x) 50 + 10x = 50 + 10x Every week the amount in checking twice the amount in savings. Rewriting Equations and Formulas (pp. 25–30) Question 30. 6y + x = 8 Answer: y = (8 – x)/6 Explanation: Given equation is 6y + x = 8 6y = 8 – x y = (8 – x)/6 Question 31. Answer: y = 30x – 45 Explanation: Given equation is 10x – 1/3 y = 15 -1/3 y = 15 – 10x y = (15 – 10x)-3 y = -45 + 30x y = 30x – 45 Question 32. 20 = 5x + 10y Answer: y = (20 – 5x)/10 Explanation: Given equation is 20 = 5x + 10y 20 – 5x = 10y (20 – 5x)/10 = y Question 33. The formula converts a temperature from Kelvin K to Fahrenheit F. a. Solve the formula for K. b. Convert 240°F to Kelvin. Round your answer to the nearest hundredth. Answer: a. K = 5/9 (F – 32) + 273.15 b. K = 400 Explanation: F = 9/5 (K – 273.15) + 32 a. F – 32 = 9/5 (K – 273.15) 5(F – 32) = 9(K – 273.15) 5/9 (F – 32) = K – 273.15 5/9 (F – 32) + 273.15 = K b. K = 5/9 (F – 32) + 273.15 Put F = 240°F K = 5/9 (240 – 32) + 273.15 = 5/9(208) + 273.15 = 115.5 + 273.15 = 388.7 Rounding 388 to nearest 100 So, K = 400 Question 34. Use the trapezoid shown. a. Write the formula for the area A of a trapezoid. b. Solve the formula for h. c. Use the new formula to ﬁnd the height of the trapezoid. Answer: a. A = (a + b) h/2 b. h = 2A/(a + b) c. h = 6 cm Explanation: a. Area of trapezoid formula is A = (a + b) h/2 b. Area of trapezoid is A = (a + b) h/2 2A = (a + b)h h = 2A/(a + b) c. A = 72 cm², a = 8 cm, b = 16 cm h = (2 x 72)/(8 + 16) = 144/24 h = 6 cm Question 35. The equation for a line in slope-intercept form is y = mx + b. Solve the equation for x. Answer: x = (y – b)/m Explanation: The equation for a line in slope-intercept form is y = mx + b y = mx + b y – b = mx (y – b)/m = x Question 36. The formula for the volume of a cylinder is V = πr2h, where r is the radius of the circular base and is the height of the cylinder. a. Solve the formula for h. b. Use the new formula to ﬁnd the height of the cylinder. Answer: a. h = V/πr² b. The height h = 3.84 in Explanation: The formula for the volume of a cylinder is V = πr²h a. Solve the formula for h. V = πr²h Divide each side by πr² V/πr² = πr²h/πr² V/πr² = h b. Height h = V/πr² Volume = 6π cubic inches radius = 1.25 inch h = V/πr² = 6π / π(1.25)² = 6/1.5625 = 3.84 The height h = 3.84 in ### Equations Practice Test Practice Test Solve the equation. Check your solution, if possible. Question 1. 4 + y = 9.5 Answer: y = 5.5 Explanation: Given equation is 4 + y = 9.5 y = 9.5 – 4 y = 5.5 Put y = 5.5 in 4 + y = 9.5 4 + 5.5 = 9.5 Question 2. –$$\frac{x}{9}$$ = -8 Answer: x = 72 Explanation: Given equation is -x/9 = -8 -x = -8 * 9 -x = -72 x = 72 Put x = 72 in -x/9 = -8 -72/9 = -8 Question 3. z – $$\frac{2}{3}$$ = $$\frac{1}{8}$$ Answer: z = 19/24 Explanation: Given equation is z – 2/3 = 1/8 z = 1/8 + 2/3 z = (3 + 16)/24 z = 19/24 Put z = 19/24 in z – 2/3 = 1/8 19/24 – 2/3 = (19 – 16)/24 = 3/24 = 1/8 Question 4. 15 = 9 – 3a Answer: a = -2 Explanation: Given equation is 15 = 9 – 3a 15 – 9 = -3a 6 = -3a 6/-3 = a -2 = a put a = -2 in 15 = 9 – 3a 15 = 9 – 3(-2) = 9 + 6 Question 5. 4(b + 5) – 9 = -7 Answer: b = -9/2 Explanation: Given equation is 4(b + 5) – 9 = -7 4b + 20 – 9 = -7 4b + 11 = -7 4b = -7 – 11 4b = -18 b = -18/4 b = -9/2 Put b = -9/2 in 4(b + 5) – 9 = -7 4(-9/2 + 5) – 9 = 4(-9 + 10)/2 – 9 = 4/2 – 9 = 2 – 9 = -7 Question 6. 9j – 8 = 8 + 9j Answer: The equation has no solution Explanation: Given equation is 9j – 8 = 8 + 9j 9j = 8 + 9j + 8 9j – 9j = 16 0 = 16 The equation has no solution Question 7. 3.8n – 13 = 1.4n + 5 Answer: n = 7.5 Explanation: Given equation is 3.8n – 13 = 1.4n + 5 3.8n – 1.4n = 5 + 13 2.4n = 18 n = 18/2.4 n = 7.5 Put n = 7.5 in 3.8n – 13 = 1.4n + 5 3.8(7.5) – 13 = 1.4(7.5) + 5 28.5 – 13 = 10.5 + 5 15.5 = 15.5 Question 8. 9(8d – 5) + 13 = 12d – 2 Answer: d = 1/2 Explanation: Given equation is 9(8d – 5) + 13 = 12d – 2 72d – 45 + 13 = 12d – 2 72d – 12d – 32 = -2 60d = -2 + 32 60d = 30 d = 30/60 d = 1/2 Put d = 1/2 in 9(8d – 5) + 13 = 12d – 2 9(8(1/2) – 5) + 13 = 12(1/2) – 2 9(4 – 5) + 13 = 6 – 2 9(-1) + 13 = 4 -9 + 13 = 4 Question 9. Answer: t = -4 Explanation: Given equation is 1/4 t + 4 = 3/4(t + 8) 1/4 t + 4 = 3/4t + 8(3/4) 1/4 t + 4 = 3/4t + 6 3/4 t – 1/4 t = 4 – 6 2/4 t = -2 1/2 t = -2 t = -2 * 2 t = -4 Put t = -4 in 1/4 t + 4 = 3/4(t + 8) 1/4 (-4) + 4 = 3/4(-4 + 8) -1 + 4 = 3/4(4) 3 = 3 Question 10. The sum of the angle measures of a triangle is 180°. Find the value of x. Then ﬁnd the angle measures of the triangle. Answer: x = 57.33° The angles of traingle are 57.33°, 114.66°, 65.33° Explanation: The sum of the angle measures of a triangle is 180°. x + 2x + x + 8 = 180 3x + 8 = 180 3x = 180 -8 3x = 172 x = 172/3 x = 57.33 The angles of traingle are 57.33, 114.66, 65.33 Question 11. A formula for the perimeter of a rectangle P = 2l + 2w. a. Solve the formula for w. b. Use the new formula to ﬁnd the width w (in meters) of a rectangle with a perimeter of 2 meters and a length of 40 centimeters. Answer: a. w = (P – 2l)/2 b. Width is 15.6 m Explanation: P = 2l + 2w a. P – 2l = 2w (P – 2l)/2 = w b. P = 32 m = 3200 cm, l = 40 cm w = (P – 2l)/2 w = (3200- 2(40))/2 = (3200 – 80)/2 = 3120/2 = 1560 cm = 15.6 m Width is 1560 cm Question 12. Solve 0.5 = 0.4y – 0.25 for y. Answer: y = 1.875 Explanation: 0.5 = 0.4y – 0.25 0.5 + 0.25 = 0.4y 0.75 = 0.4y 0.75/0.4 = y 1.875 = y Question 13. Your basketball team wins a game by 13 points. The opposing team scores 72 points. Explain how to ﬁnd your team’s score. Answer: Your team’s score is 85 points Explanation: Your basketball team wins a game by 13 points. The opposing team scores 72 points. Let your team score is x x – 72 = 13 x = 13 + 72 x = 85 Your team’s score is 85 points Question 14. You are biking at a speed of 18 miles per hour. You are 3 miles behind your friend, who is biking at a speed of 12 miles per hour. Write and solve an equation to ﬁnd the amount of time it takes for you to catch up to your friend. Answer: It will take 1/2 hour for you to catch up with your friend. Explanation: Let t be the number of hours you and your friends ride bikes Distance = rate x time and you are riding the bike at a rate of 18 miles per hour, then the distance you ride your bike is 18t miles Your friend is riding the bike at 12 miles per hour, then your friend rides a distance of 12t miles. You start 3 miles behind your friend so that distance you ride to catch up to your friend is 3 more than the distance your friend rides. Therefore, the distance you ride = distance your friend rides + 3 18t = 12t + 3 18t – 12t = 3 6t = 3 t = 3/6 t = 1/2 It will take 1/2 hour for you to catch up with your friend. Question 15. Two scientists are measuring the temperatures of lava. One scientist records a temperature of 1725°F. The other scientist records a temperature of 950°C. Which is the greater temperature? Answer: The greater temperature is 950°C. Explanation: One scientist records a temperature of 1725°F. The other scientist records a temperature of 950°C. C = 5/9 (F – 32) Put F = 1725 C = 5/9 (1725 – 32) = 5/9 (1693) = 940.55 This is smaller than 950°C Question 16. Your proﬁt for mowing lawns this week is$24. You are paid $8 per hour and you paid$40 for gas for the lawn mower. How many hours did you work this week?

You work for 8 hours this week.

Explanation:
Let x be the number of hours you work this week
8x – 40 = 24
8x = 24 + 40
8x = 64
x = 64/8
x = 8
You work for 8 hours this week.

### Equations Cumulative Practice

Question 1.
Which value of x makes the equation true?

A. 8
B. 28
C. 36
D. 128

A. 8

Explanation:
4x = 32
Divide each side by 4
4x/4 = 32/4
x = 8

Question 2.
A taxi ride costs $3 plus$2 for each mile driven. You spend $39 on a taxi. This can be modeled by the equation 2m + 3 = 39, where m represents the number of miles driven. How long was your taxi ride? F. 18 mi G. 21 mi H. 34 mi I. 72 mi Answer: F. 18 mi Explanation: To know the miles you traveled, we have to find the solution of the equation 2m + 3 = 39 2m = 39 – 3 2m = 36 m = 36/2 m = 18 So, you traveled 18 mi Question 3. Which of the following equations has exactly one solution? Answer: None of the equation has exactly one solution. Explanation: A. 2/3 (x + 6) = 2/3 x + 4 2/3x + 2/3 (6) = 2/3 x + 4 2/3 x + 4 = 2/3 x + 4 B. 3/7 y + 13 = 13 – 3/7 y 3/7 y + 3/7y = 13 – 13 6/7y = 0 C. 4/5(n + 1/3) = 4/5 n + 1/3 4/5n + 4/15 = 4/5n + 1/3 4/5n – 4/5n = 1/3 – 4/15 0 = 1/15 D. 7/8(2t + 1/8) = 7/4t 7t/4 + 7/64 = 7t/4 Question 4. The perimeter of the square is equal to the perimeter of the triangle. What are the side lengths of the square? Answer: The side length of the square is 12 units. Explanation: The perimeter of the square is equal to the perimeter of the triangle 4(3x + 3) = 2x + 4 + 7x – 2 + 7x – 2 12x + 12 = 16x 16x – 12x = 12 4x = 12 x = 12/4 x = 3 Side of square = 3x+ 3 = 3(3) + 3 = 9 + 3 = 12 The side length of the square is 12 units. Question 5. The formula d = rt relates distance, rate, and time. Solve the formula for t. Answer: G. t = d/r Explanation: Given that, d = rt Divide both sides by r d/r = rt/r d/r = t Question 6. What is a possible ﬁrst step to solve the equation 3x + 5 = 2(x + 7)? A. Combine 3x and 5. B. Multiply x by 2 and 7 by 2. C. Subtract x from 3x. D. Subtract 5 from 7. Answer: B. Multiply x by 2 and 7 by 2. Explanation: 3x + 5 = 2(x + 7) Multiply x by 2 and 7 by 2. 3x + 5 = 2x + 14 3x – 2x = 14 – 5 x = 9 Question 7. You work as a sales representative. You earn$400 per week plus 5% of your total sales for the week.

Part A
Last week, you had total sales of $5000. Find your total earnings. Show your work. Part B One week, you earned$1350. Let represent your total sales that week. Write an equation that you can use to ﬁnd s.
Part C
Using your equation from Part B, ﬁnd s. Show all steps clearly.

Part A
$650 Part B 400 + 0.05s = 1350 Part C s = 19,000 Explanation: Earnings = 400 + 0.05s a. Earnings = 400 + 0.05s Put s = 5000 400 + 0.05(5000) = 400 + 250 = 650 b. 400 + 0.05s = 1350 c. 400 + 0.05s = 1350 0.05s = 1350 – 400 0.05s = 950 s = 950/0.05 s = 19,000 Question 8. In 10 years, your aunt will be 39 years old. Let m represent your aunt’s age today. Which equation can you use to ﬁnd m? F. m = 39 + 10 G. m – 10 = 39 H. m + 10 = 39 I. 10m = 39 Answer: H. m + 10 = 39 Explanation: In 10 years, your aunt will be 39 years old. Let m represent your aunt’s age today. Equation is m = 39 – 10 m + 10 = 39 Question 9. Which value of y makes the equation 3y + 8 = 7y + 11 true? A. -4.75 B. -0.75 C. 0.75 D. 4.75 Answer: C. 0.75 Explanation: Given equation is 3y + 8 = 7y + 11 8 – 11 = 7y – 3y -3 = -4y y = 3/4 Question 10. What is the value of x? F. 23 G. 39 H. 58 I. 68 Answer: F. 23 Explanation: As shown in the figure 90 = 2x+ 12 + 32 90 = 2x + 44 90 – 44 = 2x 46 = 2x 46/2 = x 23 = x Question 11. You have already saved$35 for a new cell phone. You need $175 to buy the cell phone. You think you can save$10 per week. At this rate, how many more weeks will you need to save money before you can buy the new cell phone?

14 more weeks are required to save money to buy the new cell phone

Explanation:
The equation for the total amount saved
35 + 10x = 175
10x = 175 – 35
10x = 140
x = 140/10
x = 14
14 more weeks are required to save money to buy the new cell phone

Question 12.
What is the greatest angle measure in the triangle?

A. 26°
B. 78°
C. 108°
D. 138°

A. 26°

Explanation:
Sum of angle measures = 180°
3x + 2x + 50 = 180
5x + 50 = 180
5x = 180 – 50
5x = 130
x = 130/5
x = 26°

Question 13.
Which value x of makes the equation 6(x – 3) = 4x – 7 true?
F. -5.5
G. -2
H. 1.1
I. 5.5

I. 5.5

Explanation:
6(x – 3) = 4x – 7
6x – 18 = 4x – 7
6x – 4x = -7 + 18
2x = 11
x = 11/2
x = 5.5

Question 14.
The drawing below shows equal weights on two sides of a balance scale.

What can you conclude from the drawing?
A. A mug weighs one-third as much as a trophy.
B. A mug weighs one-half as much as a trophy.
C. A mug weighs twice as much as a trophy.
D. A mug weighs three times as much as a trophy.