# Chapter 3: Linear Equations

**Basic**Created by: CK-12

## Introduction

Aside from simplifying algebraic expressions and graphing functions, solving equations is one of the most important concepts in mathematics. To successfully manipulate an equation, you must understand and be able to apply the rules of mathematics.

Mathematical equations are used in many different career fields. Medical researchers use equations to determine the length of time it takes for a drug to circulate throughout the body, botanists use equations to determine the amount of time it takes a Sequoia tree to reach a particular height, and environmental scientists can use equations to approximate the number of years it will take to repopulate the bison species.

In this chapter, you will learn how to manipulate linear equations to solve for a particular variable. You already have some experience solving equations. This chapter is designed to help formalize the mental math you use to answer questions in daily life.

- 3.1.
## One-Step Equations and Inverse Operations

- 3.2.
## Applications of One-Step Equations

- 3.3.
## Two-Step Equations and Properties of Equality

- 3.4.
## Multi-Step Equations with Like Terms

- 3.5.
## Distributive Property for Multi-Step Equations

- 3.6.
## Equations with Variables on Both Sides

- 3.7.
## Equations with Ratios and Proportions

- 3.8.
## Scale and Indirect Measurement Applications

- 3.9.
## Conversion of Decimals, Fractions, and Percent

- 3.10.
## Percent Equations

- 3.11.
## Percent of Change

- 3.12.
## Formulas for Problem Solving

### Chapter Summary

## Summary

This chapter begins by talking about one-step equations and provides some real-world problems to solve by using these types of equations. It then moves on to two-step equations and multi-step equations, covering both the combining of like terms and the Distributive Property. Next, solving equations with variables on both sides and with ratios and proportions is discussed, as well as solving problems using scales and indirect measurement. The chapter then gives instruction on percents, including the percent equation and the percent of change. Finally, the using a formula problem-solving strategy is highlighted.

### Linear Equations Review

Solve for the variable.

- \begin{align*}a + 11.2 = 7.3\end{align*}
a+11.2=7.3 - \begin{align*}9.045+j=27\end{align*}
9.045+j=27 - \begin{align*}11 = b + \frac{5}{7}\end{align*}
11=b+57 - \begin{align*}-22 = -3 + k\end{align*}
−22=−3+k - \begin{align*}-9 = n-6\end{align*}
−9=n−6 - \begin{align*}-6+l=-27\end{align*}
−6+l=−27 - \begin{align*}\frac{s}{2} = -18\end{align*}
s2=−18 - \begin{align*}29= \frac{e}{27}\end{align*}
29=e27 - \begin{align*}u \div -66 = 11\end{align*}
u÷−66=11 - \begin{align*}-5f = -110\end{align*}
−5f=−110 - \begin{align*}76=-19p\end{align*}
76=−19p - \begin{align*}-h=-9\end{align*}
−h=−9 - \begin{align*}\frac{q+1}{11} = -2\end{align*}
q+111=−2 - \begin{align*}-2-2m=-22\end{align*}
−2−2m=−22 - \begin{align*}-5+ \frac{d}{6} = -7\end{align*}
−5+d6=−7 - \begin{align*}32=2b-3b+5b\end{align*}
32=2b−3b+5b - \begin{align*}9=4h+14h\end{align*}
9=4h+14h - \begin{align*}u-3u-2u=144\end{align*}
u−3u−2u=144 - \begin{align*}2i+5-7i=15\end{align*}
2i+5−7i=15 - \begin{align*}-10=t+15-4t\end{align*}
−10=t+15−4t - \begin{align*}\frac{1}{2} k-16+2 \frac{1}{2} k=0\end{align*}
12k−16+212k=0 - \begin{align*}\frac{-1543}{120} = \frac{3}{5} x + \frac{11}{4} \left (- \frac{11}{5} x + \frac{8}{5} \right )\end{align*}
−1543120=35x+114(−115x+85) - \begin{align*}- 5.44x + 5.11 (7.3x + 2) = - 37.3997 + 6.8x\end{align*}
−5.44x+5.11(7.3x+2)=−37.3997+6.8x - \begin{align*}-5(5r+7) = 25+5r\end{align*}
−5(5r+7)=25+5r - \begin{align*}-7p + 37 = 2(-6p + 1)\end{align*}
−7p+37=2(−6p+1) - \begin{align*}3(-5y-4)=-6y-39\end{align*}
3(−5y−4)=−6y−39 - \begin{align*}5(a-7)+2(a-3(a-5) )=0\end{align*}
5(a−7)+2(a−3(a−5))=0

Write the following comparisons as ratios. Simplify when appropriate.

- 10 boys to 25 students
- 96 apples to 42 pears
- $600 to $900
- 45 miles to 3 hours

Write the following as a unit rate.

- $4.99 for 16 ounces of turkey burger
- 40 computers to 460 students
- 18 teachers to 98 students
- 48 minutes to 15 appointments

Solve the proportion.

- Solve for \begin{align*}n: \ - \frac{6}{n - 7} = - \frac{2}{n + 1}\end{align*}
n: −6n−7=−2n+1 . - Solve for \begin{align*}x: \ - \frac{9}{5} = \frac{x - 7}{x + 10}\end{align*}
x: −95=x−7x+10 . - Solve for \begin{align*}b: \ \frac{5b}{12} = \frac{3}{11}\end{align*}
b: 5b12=311 . - Solve for \begin{align*}n: \ - \frac{12}{n} = \frac{5}{2n + 6}\end{align*}
n: −12n=52n+6 .

Write each decimal as a percent.

- 0.4567
- 2.01
- 0.005
- 0.043

Write each percent as a decimal.

- 23.5%
- 0.08%
- 0.025%
- 125.4%

Write each percent as a fraction.

- 78%
- 11.2%
- 10.5%
- \begin{align*}33.\bar{3}\%\end{align*}
33.3¯%

Solve using the percent equation.

- 32.4 is 45% of what number?
- 58.7 is what percent of 1,000?
- What is 12% of 78?
- The original price is $44 and the mark-up is 20%. What is the new price?
- An item originally priced at $240 has a 15% discount. What is the new price?
- A pair of shoes originally priced at $89.99 is discounted to $74.99. What is the percent of mark-down?
- A salon’s haircut rose in price from $10 to $14. What is the percent of mark-up?
- The price of an item costing \begin{align*}c\end{align*}
c dollars decreased by $48, resulting in a 30% mark-down. What was the original price? - The width of a rectangle is 15 units less than its length. The perimeter of the rectangle is 98 units. What is the rectangle’s length?
- George took a cab from home to a job interview. The cab fare was $4.00 plus $0.25 per mile. His total fare was $16.75. How many miles did he travel?
- The sum of twice a number and 38 is 124. What is the number?
- The perimeter of a square parking lot is 260 yards. What are the dimensions of the parking lot?
- A restaurant charges $3.79 for \begin{align*}\frac{1}{8}\end{align*}
18 of a pie. At this rate, how much does the restaurant charge for the entire pie? - A 60-watt light bulb consumes 0.06
*kilowatts/hour*of energy. How long was the bulb left on if it consumed 5.56*kilowatts/hour*of energy? - A \begin{align*}6 \frac{1}{2}\end{align*}
612 -foot-tall car casts a 33.2-foot shadow. Next to the car is an elephant casting a 51.5-foot shadow. How tall is the elephant? - Two cities are 87 miles apart. How far apart would they be on a map with a scale of 5
*inches*: 32*miles?*

### Linear Equations Test

- School lunch rose from $1.60 to $2.35. What was the percent of mark-up?
- Solve for \begin{align*}c: \ \frac{3c}{8} = 11\end{align*}
c: 3c8=11 . - Write 6.35 as a percent.
- Write the following as a simplified ratio: 85 tomatoes to 6 plants.
- Yvonne made 12 more cupcakes than she did yesterday. She made a total of 68 cupcakes over the two days. How many cupcakes did she make the second day?
- A swing set 8 feet tall casts a 4-foot-long shadow. How long is the shadow of a lawn gnome 4 feet tall?
- Solve the proportion: \begin{align*}\frac{v}{v-2} = - \frac{9}{5}\end{align*}
vv−2=−95 . - Find the distance between Owosso and Perry if they are 16 cm apart on a map with a scale of 21
*km*: 4*cm*. - Solve for \begin{align*}j: \ - \frac{13}{4} - \frac{3}{2} \left (\frac{3}{4} j - \frac{4}{5} \right )= - \frac{14}{5}\end{align*}
j: −134−32(34j−45)=−145 . - Solve for \begin{align*}m: \ 2m(2-4)+5m(-8)=9\end{align*}
m: 2m(2−4)+5m(−8)=9 . - Job A pays $15 plus $2.00 per hour. Job B pays $3.75 per hour. When will the two jobs pay exactly the same?
- Solve for \begin{align*}k: \ 9.0604 + 2.062k = 0.3(2.2k + 5.9)\end{align*}
k: 9.0604+2.062k=0.3(2.2k+5.9) . - Solve for \begin{align*}a: \ -9-a = 15\end{align*}
a: −9−a=15 . - 46 tons is 11% of what?
- 17% of what is 473 meters?
- Find the percent of change from 73 to 309.
- JoAnn wants to adjust a bread recipe by tripling its ingredients. If the recipe calls for \begin{align*}4 \frac{1}{3}\end{align*}
413 cups of pastry flour, how much should she use? - A sweater originally marked $80.00 went on sale for $45. What was the percent of change?

#### Texas Instruments Resources

*In the CK-12 Texas Instruments Algebra I FlexBook® resource, there are graphing calculator activities designed to supplement the objectives for some of the lessons in this chapter. See http://www.ck12.org/flexr/chapter/9613.*

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