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## Section3.1Solving Multistep Linear Equations

###### ObjectivesPCC Course Content and Outcome Guide

We have learned how to solve one-step equations in Section. In this section, we will learn how to solve multistep equations.

### Subsection3.1.1Solving Two-Step Equations

###### Example3.1.2

A water tank can hold $140$ gallons of water, but it has only $5$ gallons of water. A tap was turned on, pouring $15$ gallons of water into the tank every minute. After how many minutes will the tank be full? Let's find a pattern first.

 Minutes since Tap Amount of Water in Was Turned on the Tank (in Gallons) $0$ $5$ $1$ $15\cdot1+5=20$ $2$ $15\cdot2+5=35$ $3$ $15\cdot3+5=50$ $4$ $15\cdot4+5=65$ $\vdots$ $\vdots$ $m$ $15m+5$

We can see that after $m$ minutes, the tank has $15m+5$ gallons of water. This makes sense since the tap pours $15m$ gallons of water into the tank in $m$ minutes and it had $5$ gallons to start with. To find when the tank will be full (with $140$ gallons of water), we can write the equation

\begin{equation*} 15m+5=140 \end{equation*}

First, we need to isolate the variable term, $15m\text{,}$ in the equation. In other words, we need to remove $5$ from the left side of the equals sign. We can do this by subtracting $5$ from both sides of the equation. Once the variable term is isolated, we can eliminate the coefficient and solve for $m\text{.}$ The full process is:

\begin{align*} 15m+5\amp=140\\ 15m+5\subtractright{5}\amp=140\subtractright{5}\\ 15m\amp=135\\ \divideunder{15m}{15}\amp=\divideunder{135}{15}\\ m\amp=9 \end{align*}

Next, we need to substitute $m$ with $9$ in the equation $15m+5=140$ to check the solution:

\begin{align*} 15m+5\amp=140\\ 15(\substitute{9})+5\amp\stackrel{?}{=}140\\ 135+5\amp\stackrel{\checkmark}{=}140 \end{align*}

The solution $9$ is checked. In summary, the tank will be full after $9$ minutes.

In solving the two-step equation in Example 3.1.2, we first isolated the variable expression $15m$ and then eliminated the coefficient of $15$ by dividing each side of the equation by $15\text{.}$ These two steps will be at the heart of our approach to solving linear equations. For more complicated equations, we may need to simplify some of the expressions first. Below is a general approach to solving linear equations that we will use as we solve more and more complicated equations.

Let's look at some more examples.

###### Example3.1.5

Solve for $y$ in the equation $7-3y=-8\text{.}$

Explanation

To solve, we will first separate the variable terms and constant terms into different sides of the equation. Then we will eliminate the variable term's coefficient.

\begin{align*} 7-3y\amp=-8\\ 7-3y\subtractright{7}\amp=-8\subtractright{7}\\ -3y\amp=-15\\ \divideunder{-3y}{-3}\amp=\divideunder{-15}{-3}\\ y\amp=5 \end{align*}

Checking the solution $y=5\text{:}$

\begin{align*} 7-3y\amp=-8\\ 7-3(\substitute{5})\amp\stackrel{?}{=}-8\\ 7-15\amp\stackrel{\checkmark}{=}-8 \end{align*}

Therefore the solution to the equation $7-3y=-8$ is $5$ and the solution set is $\{5\}\text{.}$

### Subsection3.1.2Solving Multistep Linear Equations

###### Example3.1.6

Ahmed has saved $\2{,}500.00$ in his savings account and is going to start saving $\550.00$ per month. Julia has saved $\4{,}600.00$ in her savings account and is going to start saving $\250.00$ per month. If this situation continues, how many months later would Ahmed catch up with Julia in savings?

Ahmed saves $\550.00$ per month, so he can save $550m$ dollars in $m$ months. With the $\2{,}500.00$ he started with, after $m$ months he has $550m+2500$ dollars. Similarly, after $m$ months, Julia has $250m+4600$ dollars. To find when those two accounts will have the same amount of money, we write the equation

\begin{equation*} 550m+2500=250m+4600\text{.} \end{equation*}
\begin{align*} 550m+2500\amp=250m+4600\\ 550m+2500\subtractright{2500}\amp=250m+4600\subtractright{2500}\\ 550m\amp=250m+2100\\ 550m\subtractright{250m}\amp=250m+2100\subtractright{250m}\\ 300m\amp=2100\\ \divideunder{300m}{300}\amp=\divideunder{2100}{300}\\ m\amp=7 \end{align*}

Checking the solution $7\text{:}$

\begin{align*} 550m+2500\amp=250m+4600\\ 550(\substitute{7})+2500\amp\stackrel{?}{=}250(\substitute{7})+4600\\ 3850+2500\amp\stackrel{?}{=}1750+4600\\ 6350\amp\stackrel{\checkmark}{=}6350 \end{align*}

In summary, Ahmed will catch up with Julia's savings in $7$ months.

###### Example3.1.7

Solve for $x$ in $5-2x=5x-9\text{.}$

Explanation
\begin{align*} 5-2x\amp=5x-9\\ 5-2x\subtractright{5}\amp=5m-9\subtractright{5}\\ -2x\amp=5x-14\\ -2x\subtractright{5x}\amp=5x-14\subtractright{5x}\\ -7x\amp=-14\\ \divideunder{-7x}{-7}\amp=\divideunder{-14}{-7}\\ x\amp=2 \end{align*}

Checking the solution $2\text{:}$

\begin{align*} 5-2x\amp=5x-9\\ 5-2(\substitute{2})\amp\stackrel{?}{=}5(\substitute{2})-9\\ 5-4\amp\stackrel{?}{=}10-9\\ 1\amp\stackrel{\checkmark}{=}1 \end{align*}

Therefore the solution is $2$ and the solution set is $\{2\}\text{.}$

###### Remark3.1.8

In Example 3.1.7, we could have moved variable terms to the right side of the equals sign, and number terms to the left side. We chose not to. There's no reason we couldn't have moved variable terms to the right side though. Let's compare:

\begin{align*} 5-2x\amp=5x-9\\ 5-2x\addright{9}\amp=5x-9\addright{9}\\ 14-2x\amp=5x\\ 14-2x\addright{2x}\amp=5x\addright{2x}\\ 14\amp=7x\\ \divideunder{14}{7}\amp=\divideunder{7x}{7}\\ 2\amp=x \end{align*}

Lastly, we could save a step by moving variable terms and number terms in one step:

\begin{align*} 5-2x\amp=5x-9\\ 5-2x\addright{2x+9}\amp\phantom{=}5x-9\addright{2x+9}\\ 14\amp=7x\\ \divideunder{14}{7}\amp=\divideunder{7x}{7}\\ 2\amp=x \end{align*}

This textbook will move variable terms and number terms separately throughout this chapter. Check with your instructor for their expectations.

###### Checkpoint3.1.9

The next example requires combining like terms.

###### Example3.1.10

Solve for $n$ in $n-9+3n=n-3n\text{.}$

Explanation

To start solving this equation, we'll need to combine like terms. After this, we can put all terms containing $n$ on one side of the equation and finish solving for $n\text{.}$

\begin{align*} n-9+3n\amp=n-3n\\ 4n-9\amp=-2n\\ 4n-9\subtractright{4n}\amp=-2n\subtractright{4n}\\ -9\amp=-6n\\ \divideunder{-9}{-6}\amp=\divideunder{-6n}{-6}\\ n\amp=\frac{3}{2} \end{align*}

Checking the solution $\frac{3}{2}\text{:}$

\begin{align*} n-9+3n\amp=n-3n\\ \substitute{\frac{3}{2}}-9+3\left(\substitute{\frac{3}{2}}\right)\amp\stackrel{?}{=}\substitute{\frac{3}{2}}-3\left(\substitute{\frac{3}{2}}\right)\\ \frac{3}{2}-9+\frac{9}{2}\amp\stackrel{?}{=}\frac{3}{2}-\frac{9}{2}\\ \frac{12}{2}-9\amp\stackrel{?}{=}-\frac{6}{2}\\ 6-9\amp\stackrel{?}{=}-3\\ -3\amp\stackrel{?}{=}-3 \end{align*}

The solution to the equation $n-9+3n=n-3n$ is $\frac{3}{2}$ and the solution set is $\left\{\frac{3}{2}\right\}\text{.}$

###### Example3.1.12

Azul is designing a rectangular garden and they have $40$ meters of wood for the border. Their garden's length must be $4$ meters less than three times the width, and its perimeter must be $40$ meters. Find the garden's length and width.

Explanation

Reminder: A rectangle's perimeter formula is $P=2(L+W)\text{,}$ where $P$ stands for perimeter, $L$ stands for length and $W$ stands for width.

Let Azul's garden width be $W$ meters. We can then represent the length as $3W-4$ meters since we are told that it is $4$ meters less than three times the width. It's given that the perimeter is $40$ meters. Substituting those values into the formula, we have:

\begin{align*} P\amp=2(L+W)\\ 40\amp=2(3W-4+W)\\ 40\amp=2(4W-4)\amp\text{Like terms were combined.} \end{align*}

The next step to solve this equation is to remove the parentheses by distribution.

\begin{align*} 40\amp=2(4W-4)\\ 40\amp=8W-8\\ 40\addright{8}\amp=8W-8\addright{8}\\ 48\amp=8W\\ \divideunder{48}{8}\amp=\divideunder{8W}{8}\\ 6\amp=W\text{.} \end{align*}

Checking the solution $W=6\text{:}$

\begin{align*} 40\amp=2(4W-4)\\ 40\amp\stackrel{?}{=}2(4(\substitute{6})-4)\\ 40\amp\stackrel{\checkmark}{=}2(20)\text{.} \end{align*}

To determine the length, recall that this was represented by $3W-4\text{,}$ which is:

\begin{align*} 3W-4\amp=3(\substitute{6})-4\\ \amp=14\text{.} \end{align*}

Thus, the width of Azul's garden is $6$ meters and the length is $14$ meters.

###### Checkpoint3.1.13

We should be careful when we distribute a negative sign into the parentheses, like in the next example.

###### Example3.1.14

Solve for $a$ in $4-(3-a)=-2-2(2a+1)\text{.}$

Explanation

To solve this equation, we will simplify each side of the equation, manipulate it so that all variable terms are on one side and all constant terms are on the other, and then solve for $a\text{:}$

\begin{align*} 4-(3-a)\amp=-2-2(2a+1)\\ 4-3+a\amp=-2-4a-2\\ 1+a\amp=-4-4a\\ 1+a\addright{4a}\amp=-4-4a\addright{4a}\\ 1+5a\amp=-4\\ 1+5a\subtractright{1}\amp=-4\subtractright{1}\\ 5a\amp=-5\\ \divideunder{5a}{5}\amp=\divideunder{-5}{5}\\ a\amp=-1 \end{align*}

Checking the solution $-1\text{:}$

\begin{align*} 4-(3-a)\amp=-2-2(2a+1)\\ 4-(3-(\substitute{-1}))\amp\stackrel{?}{=}-2-2(2(\substitute{-1})+1)\\ 4-(4)\amp\stackrel{?}{=}-2-2(-1)\\ 0\amp\stackrel{\checkmark}{=}0 \end{align*}

Therefore the solution to the equation is $-1$ and the solution set is $\{-1\}\text{.}$

### Subsection3.1.3Differentiating between Simplifying Expressions, Evaluating Expressions and Solving Equations

Let's look at the following similar, yet different examples.

###### Example3.1.15

Simplify the expression $10-3(x+2)\text{.}$

Explanation
\begin{align*} 10-3(x+2)\amp=10-3x-6\\ \amp=-3x+4 \end{align*}

An equivalent result is $4-3x\text{.}$ Note that our final result is an expression.

###### Example3.1.16

Evaluate the expression $10-3(x+2)$ when $x=2$ and when $x=3\text{.}$

Explanation

We will substitute $x=2$ into the expression:

\begin{align*} 10-3(x+2)\amp=10-3(\substitute{2}+2)\\ \amp=10-3(4)\\ \amp=10-12\\ \amp=-2 \end{align*}

When $x=2\text{,}$ $10-3(x+2)=-2\text{.}$

Similarly, we will substitute $x=3$ into the expression:

\begin{align*} 10-3(x+2)\amp=10-3(\substitute{3}+2)\\ \amp=10-3(5)\\ \amp=10-15=-5 \end{align*}

When $x=3\text{,}$ $10-3(x+2)=-5\text{.}$

Note that the final results here are values of the original expression.

###### Example3.1.17

Solve the equation $10-3(x+2)=x-16\text{.}$

Explanation
\begin{align*} 10-3(x+2)\amp=x-16\\ 10-3x-6\amp=x-16\\ -3x+4\amp=x-16\\ -3x+4\subtractright{4}\amp=x-16\subtractright{4}\\ -3x\amp=x-20\\ -3x\subtractright{x}\amp=x-20\subtractright{x}\\ -4x\amp=-20\\ \divideunder{-4x}{-4}\amp=\divideunder{-20}{-4}\\ x\amp=5 \end{align*}

Checking the solution $x=5\text{:}$

\begin{align*} 10-3(x+2)\amp=x-16\\ 10-3(\substitute{5}+2)\amp\stackrel{?}{=}\substitute{5}-16\\ 10-3(7)\amp\stackrel{?}{=}-11\\ 10-21\amp\stackrel{\checkmark}{=}-11 \end{align*}

We have checked that $x=5$ is a solution of the equation $10-3(x+2)=x-16\text{.}$

Note that the final results here are solutions to the equations.

### Subsection3.1.4Exercises

###### 1

Solve the equation.

$\displaystyle{ {r+3}={-3} }$

###### 2

Solve the equation.

$\displaystyle{ {r+9}={6} }$

###### 3

Solve the equation.

$\displaystyle{ {t-6}={-2} }$

###### 4

Solve the equation.

$\displaystyle{ {t-2}={6} }$

###### 5

Solve the equation.

$\displaystyle{ {44}={-4x} }$

###### 6

Solve the equation.

$\displaystyle{ {42}={-7x} }$

###### 7

Solve the equation.

$\displaystyle{ {{\frac{10}{3}}a} = {2} }$

###### 8

Solve the equation.

$\displaystyle{ {{\frac{5}{9}}b} = {9} }$

###### 9

Solve the equation.

$\displaystyle{ {6A+4}={58} }$

###### 10

Solve the equation.

$\displaystyle{ {2B+2}={8} }$

###### 11

Solve the equation.

$\displaystyle{ {8m-6}={18} }$

###### 12

Solve the equation.

$\displaystyle{ {5n-5}={-50} }$

###### 13

Solve the equation.

$\displaystyle{ {-9} = {2q+3} }$

###### 14

Solve the equation.

$\displaystyle{ {-23} = {8y+1} }$

###### 15

Solve the equation.

$\displaystyle{ {-26} = {5r-6} }$

###### 16

Solve the equation.

$\displaystyle{ {6} = {2a-4} }$

###### 17

Solve the equation.

$\displaystyle{ {-5b+3}={48} }$

###### 18

Solve the equation.

$\displaystyle{ {-8A+1}={49} }$

###### 19

Solve the equation.

$\displaystyle{ {-2B-8}={-28} }$

###### 20

Solve the equation.

$\displaystyle{ {-5m-5}={5} }$

###### 21

Solve the equation.

$\displaystyle{ {17} = {-n+9} }$

###### 22

Solve the equation.

$\displaystyle{ {5} = {-q+3} }$

###### 23

Solve the equation.

$\displaystyle{ {7y+35}={0} }$

###### 24

Solve the equation.

$\displaystyle{ {4r+40}={0} }$

###### 25

A gym charges members ${\25}$ for a registration fee, and then ${\38}$ per month. You became a member some time ago, and now you have paid a total of ${\557}$ to the gym. How many months have passed since you joined the gym?

months have passed since you joined the gym.

###### 26

Your cell phone company charges a ${\18}$ monthly fee, plus ${\0.15}$ per minute of talk time. One month your cell phone bill was ${\90}\text{.}$ How many minutes did you spend talking on the phone that month?

You spent talking on the phone that month.

###### 27

A school purchased a batch of T-shirts from a company. The company charged ${\7}$ per T-shirt, and gave the school a ${\60}$ rebate. If the school had a net expense of ${\2{,}460}$ from the purchase, how many T-shirts did the school buy?

The school purchased T-shirts.

###### 28

Joshua hired a face-painter for a birthday party. The painter charged a flat fee of ${\80}\text{,}$ and then charged ${\5.50}$ per person. In the end, Joshua paid a total of ${\217.50}\text{.}$ How many people used the face-painter’s service?

people used the face-painter’s service.

###### 29

A certain country has $676.8$ million acres of forest. Every year, the country loses $7.52$ million acres of forest mainly due to deforestation for farming purposes. If this situation continues at this pace, how many years later will the country have only $368.48$ million acres of forest left? (Use an equation to solve this problem.)

After years, this country would have $368.48$ million acres of forest left.

###### 30

Heather has ${\87}$ in her piggy bank. She plans to purchase some Pokemon cards, which costs ${\1.55}$ each. She plans to save ${\62.20}$ to purchase another toy. At most how many Pokemon cards can he purchase?

Write an equation to solve this problem.

Heather can purchase at most Pokemon cards.

###### 31

Solve the equation.

${9q+10} = {q+50}$

###### 32

Solve the equation.

${8x+5} = {x+26}$

###### 33

Solve the equation.

$\displaystyle{ {-6r+9} = {-r-1} }$

###### 34

Solve the equation.

$\displaystyle{ {-8a+3} = {-a-39} }$

###### 35

Solve the equation.

$\displaystyle{ {5-7b} = {6b+96} }$

###### 36

Solve the equation.

$\displaystyle{ {2-2A} = {6A+82} }$

###### 37

Solve the equation.

$\displaystyle{ {5B+7}={9B+10} }$

###### 38

Solve the equation.

$\displaystyle{ {4m+4}={2m+3} }$

###### 39

Solve the equation.

1. $\displaystyle{ {7n+3} = {3n+39} }$

2. $\displaystyle{ {3x+3} = {7x-29} }$

###### 40

Solve the equation.

1. $\displaystyle{ {9q+10} = {3q+34} }$

2. $\displaystyle{ {3C+10} = {9C-44} }$

###### 41

Use a linear equation to solve the word problem.

Two trees are $6$ feet and $11.5$ feet tall. The shorter tree grows $2.5$ feet per year; the taller tree grows $2$ feet per year. How many years later would the shorter tree catch up with the taller tree?

It would take the shorter tree years to catch up with the taller tree.

###### 42

Use a linear equation to solve the word problem.

Massage Heaven and Massage You are competitors. Massage Heaven has $3400$ registered customers, and it gets approximately $900$ newly registered customers every month. Massage You has $10600$ registered customers, and it gets approximately $450$ newly registered customers every month. How many months would it take Massage Heaven to catch up with Massage You in the number of registered customers?

These two companies would have approximately the same number of registered customers months later.

###### 43

Use a linear equation to solve the word problem.

Two truck rental companies have different rates. V-Haul has a base charge of ${\60.00}\text{,}$ plus ${\0.60}$ per mile. W-Haul has a base charge of ${\52.60}\text{,}$ plus ${\0.65}$ per mile. For how many miles would these two companies charge the same amount?

If a driver drives miles, those two companies would charge the same amount of money.

###### 44

Use a linear equation to solve the word problem.

Massage Heaven and Massage You are competitors. Massage Heaven has $9200$ registered customers, but it is losing approximately $400$ registered customers every month. Massage You has $1200$ registered customers, and it gets approximately $400$ newly registered customers every month. How many months would it take Massage Heaven to catch up with Massage You in the number of registered customers?

These two companies would have approximately the same number of registered customers months later.

###### 45

Use a linear equation to solve the word problem.

Tammy has ${\85.00}$ in her piggy bank, and she spends ${\4.00}$ every day.

Laurie has ${\8.00}$ in her piggy bank, and she saves ${\1.50}$ every day.

If they continue to spend and save money this way, how many days later would they have the same amount of money in their piggy banks?

days later, Tammy and Laurie will have the same amount of money in their piggy banks.

###### 46

Use a linear equation to solve the word problem.

Lindsay has ${\95.00}$ in her piggy bank, and she spends ${\4.00}$ every day.

Derick has ${\18.00}$ in his piggy bank, and he saves ${\3.00}$ every day.

If they continue to spend and save money this way, how many days later would they have the same amount of money in their piggy banks?

days later, Lindsay and Derick will have the same amount of money in their piggy banks.

###### 47

Solve the equation.

$\displaystyle{ {4m+7m+2}={112} }$

###### 48

Solve the equation.

$\displaystyle{ {9n+2n+2}={90} }$

###### 49

Solve the equation.

$\displaystyle{ {6q+6+4}={40} }$

###### 50

Solve the equation.

$\displaystyle{ {3x+10+3}={22} }$

###### 51

Solve the equation.

$\displaystyle{ {-2+4}={-3r-r-30} }$

###### 52

Solve the equation.

$\displaystyle{ {-6+8}={-5t-t-46} }$

###### 53

Solve the equation.

$\displaystyle{ {2y+3-7y}={38} }$

###### 54

Solve the equation.

$\displaystyle{ {5r+7-7r}={27} }$

###### 55

Solve the equation.

$\displaystyle{ {-6r+10+r}={-20} }$

###### 56

Solve the equation.

$\displaystyle{ {-3r+5+r}={5} }$

###### 57

Solve the equation.

${45}={-8n-9-n}$

###### 58

Solve the equation.

${-58}={-5q-4-q}$

###### 59

Solve the equation.

$\displaystyle{ {2-x-x}={-7+3} }$

###### 60

Solve the equation.

$\displaystyle{ {8-r-r}={-2+14} }$

###### 61

Solve the equation.

$\displaystyle{ {3-2t-8}={-5} }$

###### 62

Solve the equation.

$\displaystyle{ {2-9b-10}={-8} }$

###### 63

Solve the equation.

$\displaystyle{ {A-8-5A} = {-6-8A+26} }$

###### 64

Solve the equation.

$\displaystyle{ {B-4-8B} = {-4-2B+25} }$

###### 65

Solve the equation.

$\displaystyle{ -10m+2m = 10-2m-40 }$

###### 66

Solve the equation.

$\displaystyle{ -9n+4n = 8-2n-11 }$

###### 67

Solve the equation.

${4q+10} = {-5q+10-2q}$

###### 68

Solve the equation.

${10x+5} = {-3x+5-2x}$

###### 69

Solve the equation.

$\displaystyle{ {-8+9}={7r-6-10r+4+2r} }$

###### 70

Solve the equation.

$\displaystyle{ {-2+\left(-1\right)}={4t-6-7t+2+2t} }$

###### 71

A $138$-meter rope is cut into two segments. The longer segment is $28$ meters longer than the shorter segment. Write and solve a linear equation to find the length of each segment. Include units.

The segments are and long.

###### 72

In a doctor’s office, the receptionist’s annual salary is ${\142{,}000}$ less than that of the doctor. Together, the doctor and the receptionist make ${\208{,}000}$ per year. Find each person’s annual income.

The receptionist’s annual income is . The doctor’s annual income is .

###### 73

Phil and Penelope went picking strawberries. Phil picked $116$ fewer strawberries than Penelope did. Together, they picked $214$ strawberries. How many strawberries did Penelope pick?

Penelope picked strawberries.

###### 74

Virginia and Ross collect stamps. Ross collected $27$ fewer than five times the number of Virginia’s stamps. Altogether, they collected $1005$ stamps. How many stamps did Virginia and Ross collect?

Virginia collected stamps. Ross collected stamps.

###### 75

Diane and Tracei sold girl scout cookies. Diane’s sales were ${\37}$ more than three times of Tracei’s. Altogether, their sales were ${\437}\text{.}$ How much did each girl sell?

Diane’s sales were . Tracei’s sales were .

###### 76

A hockey team played a total of $191$ games last season. The number of games they won was $11$ more than five times of the number of games they lost.

Write and solve an equation to answer the following questions.

The team lost games. The team won games.

###### 77

After a $55\%$ increase, a town has $155$ people. What was the population before the increase?

Before the increase, the town’s population was .

###### 78

After a $35\%$ increase, a town has $270$ people. What was the population before the increase?

Before the increase, the town’s population was .

###### 79

Solve the equation.

$\displaystyle{ {2\!\left(t+2\right)}={24} }$

###### 80

Solve the equation.

$\displaystyle{ {8\!\left(b+9\right)}={112} }$

###### 81

Solve the equation.

$\displaystyle{ {5\!\left(c-6\right)}={5} }$

###### 82

Solve the equation.

$\displaystyle{ {2\!\left(B-3\right)}={-18} }$

###### 83

Solve the equation.

$\displaystyle{ {24}={-8\!\left(m+7\right)} }$

###### 84

Solve the equation.

$\displaystyle{ {-30}={-5\!\left(n+1\right)} }$

###### 85

Solve the equation.

$\displaystyle{ {12}={-2\!\left(q-5\right)} }$

###### 86

Solve the equation.

$\displaystyle{ {128}={-8\!\left(x-9\right)} }$

###### 87

Solve the equation.

$\displaystyle{ {-\left(r-4\right)}={8} }$

###### 88

Solve the equation.

$\displaystyle{ {-\left(t-8\right)}={-2} }$

###### 89

Solve the equation.

$\displaystyle{ {-14}={-\left(7-b\right)} }$

###### 90

Solve the equation.

$\displaystyle{ {-2}={-\left(3-c\right)} }$

###### 91

Solve the equation.

$\displaystyle{ {10\!\left(10B-8\right)}={520} }$

###### 92

Solve the equation.

$\displaystyle{ {7\!\left(5C-8\right)}={-56} }$

###### 93

Solve the equation.

$\displaystyle{ {2}={-2\!\left(9-2n\right)} }$

###### 94

Solve the equation.

$\displaystyle{ {110}={-5\!\left(3-5q\right)} }$

###### 95

Solve the equation.

$\displaystyle{ {3+9\!\left(x+8\right)}={111} }$

###### 96

Solve the equation.

$\displaystyle{ {1+6\!\left(r+7\right)}={31} }$

###### 97

Solve the equation.

$\displaystyle{ {5-8\!\left(t+7\right)}={13} }$

###### 98

Solve the equation.

$\displaystyle{ {3-10\!\left(b+7\right)}={-137} }$

###### 99

Solve the equation.

$\displaystyle{ {22}={2-4\!\left(c-7\right)} }$

###### 100

Solve the equation.

$\displaystyle{ {97}={9-8\!\left(B-7\right)} }$

###### 101

Solve the equation.

$\displaystyle{ 3-6(C-7)=105 }$

###### 102

Solve the equation.

$\displaystyle{ 1-8(n-7)=17 }$

###### 103

Solve the equation.

$\displaystyle{ {3}={9-\left(5-q\right)} }$

###### 104

Solve the equation.

$\displaystyle{ {-1}={8-\left(3-x\right)} }$

###### 105

Solve the equation.

$\displaystyle{ {1-\left(r+10\right)}={-18} }$

###### 106

Solve the equation.

$\displaystyle{ {4-\left(t+7\right)}={-6} }$

###### 107

Solve the equation.

1. $\displaystyle{ {5+\left(b+4\right)}={12} }$

2. $\displaystyle{ {5-\left(b+4\right)}={12} }$

###### 108

Solve the equation.

1. $\displaystyle{ {2+\left(c+1\right)}={-6} }$

2. $\displaystyle{ {2-\left(c+1\right)}={-6} }$

###### 109

Solve the equation.

$\displaystyle{ {4\!\left(B+5\right)-10\!\left(B-7\right)}={90} }$

###### 110

Solve the equation.

$\displaystyle{ {3\!\left(C+10\right)-8\!\left(C-2\right)}={46} }$

###### 111

Solve the equation.

$\displaystyle{ {5+8\!\left(n-5\right)}={-28-\left(7-2n\right)} }$

###### 112

Solve the equation.

$\displaystyle{ {4+9\!\left(p-10\right)}={-84-\left(2-2p\right)} }$

###### 113

Solve the equation.

$\displaystyle{ {7\!\left(x-2\right)-x}={67-3\!\left(7+3x\right)} }$

###### 114

Solve the equation.

$\displaystyle{ {10\!\left(r-6\right)-r}={-90-3\!\left(2+3r\right)} }$

###### 115

Solve the equation.

$\displaystyle{ {7\!\left(-10t+10\right)}={14\!\left(-2-6t\right)} }$

###### 116

Solve the equation.

$\displaystyle{ {3\!\left(-10b+6\right)}={6\!\left(-9-6b\right)} }$

###### 117

Solve the equation.

$\displaystyle{ {23+6\!\left(6-4c\right)}={-4\!\left(c-13\right)+7} }$

###### 118

Solve the equation.

$\displaystyle{ {12+4\!\left(3-3B\right)}={-4\!\left(B-4\right)+8} }$

###### 119

A rectangle’s perimeter is ${78\ {\rm cm}}\text{.}$ Its base is ${27\ {\rm cm}}\text{.}$

Its height is .

###### 120

A rectangle’s perimeter is ${58\ {\rm m}}\text{.}$ Its width is ${12\ {\rm m}}\text{.}$ Use an equation to solve for the rectangle’s length.

Its length is .

###### 121

A rectangle’s perimeter is ${116\ {\rm in}}\text{.}$ Its length is ${8\ {\rm in}}$ longer than its width. Use an equation to find the rectangle’s length and width.

Its width is .

Its length is .

###### 122

A rectangle’s perimeter is ${120\ {\rm cm}}\text{.}$ Its length is $2$ times as long as its width. Use an equation to find the rectangle’s length and width.

It’s width is .

Its length is .

###### 123

A rectangle’s perimeter is ${106\ {\rm ft}}\text{.}$ Its length is ${2\ {\rm ft}}$ shorter than four times its width. Use an equation to find the rectangle’s length and width.

Its width is .

Its length is .

###### 124

A rectangle’s perimeter is ${184\ {\rm ft}}\text{.}$ Its length is ${4\ {\rm ft}}$ longer than three times its width. Use an equation to find the rectangle’s length and width.

Its width is .

Its length is .

###### 125

Solve the equation.

1. $\displaystyle{ {-b+7}={7} }$

2. $\displaystyle{ {-y+7}={-7} }$

3. $\displaystyle{ {-r-7}={7} }$

4. $\displaystyle{ {-a-7}={-7} }$

###### 126

Solve the equation.

1. $\displaystyle{ {-c+4}={4} }$

2. $\displaystyle{ {-m+4}={-4} }$

3. $\displaystyle{ {-B-4}={4} }$

4. $\displaystyle{ {-y-4}={-4} }$

###### 127
1. Solve the following linear equation:

$\displaystyle{ {r-2}={8} }$

2. Evaluate the following expression when $r=10\text{:}$

$\displaystyle{{r-2}=}$

###### 128
1. Solve the following linear equation:

$\displaystyle{ {r-8}={-3} }$

2. Evaluate the following expression when $r=5\text{:}$

$\displaystyle{{r-8}=}$

###### 129
1. Solve the following linear equation:

$\displaystyle{ {4\!\left(t+6\right)-4}={36} }$

2. Evaluate the following expression when $t=4\text{:}$

$\displaystyle{{4\!\left(t+6\right)-4}=}$

3. Simplify the following expression:

$\displaystyle{{4\!\left(t+6\right)-4}=}$

###### 130
1. Solve the following linear equation:

$\displaystyle{ {3\!\left(t-5\right)+9}={6} }$

2. Evaluate the following expression when $t=4\text{:}$

$\displaystyle{{3\!\left(t-5\right)+9}=}$

3. Simplify the following expression:

$\displaystyle{{3\!\left(t-5\right)+9}=}$

###### 131

Choose True or False for the following questions about the difference between expressions and equations.

1. $\text{We can evaluate }{-10x-10}\text{ when }x=1$

• True

• False

2. $-10x-10=-10x-10\text{ is an equation.}$

• True

• False

3. $\text{We can evaluate }{-10x-10}=-10x-10\text{ when }x=1$

• True

• False

4. $\text{We can check whether }x=1\text{ is a solution of }{-10x-10}.$

• True

• False

5. $-10x-10\text{ is an expression.}$

• True

• False

6. $\text{We can check whether }x=1\text{ is a solution of }{-10x-10}=-10x-10.$

• True

• False

7. $-10x-10=-10x-10\text{ is an expression.}$

• True

• False

8. $-10x-10\text{ is an equation.}$

• True

• False

###### 132

Choose True or False for the following questions about the difference between expressions and equations.

1. $-7x+4\text{ is an expression.}$

• True

• False

2. $4x-7\text{ is an equation.}$

• True

• False

3. $-7x+4=4x-7\text{ is an equation.}$

• True

• False

4. $\text{We can check whether }x=1\text{ is a solution of }{-7x+4}.$

• True

• False

5. $-7x+4=4x-7\text{ is an expression.}$

• True

• False

6. $\text{We can evaluate }{-7x+4}=4x-7\text{ when }x=1$

• True

• False

7. $\text{We can evaluate }{-7x+4}\text{ when }x=1$

• True

• False

8. $\text{We can check whether }x=1\text{ is a solution of }{-7x+4}=4x-7.$

• True

• False

###### 133

Think of a number. Add four to your number. Now double that. Then add six. Then halve it. Finally, subtract 7. What is the result? Do you always get the same result, regardless of what number you start with? How does this work? Explain using algebra.

###### 134

Write a linear equation whose solution is $x = -9\text{.}$

Note that you may not write an equation whose left side is just “$x$” or whose right side is just “$x\text{.}$”

There are infinitely many correct answers to this problem. Be creative. After finding an equation that works, see if you can come up with a different one that also works.