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9.2      Solving One-Step Equations Using Addition and Subtraction      •      615

You’ve certainly seen parallel lines before. Railroad tracks look like parallel 

lines. The opposite sides of a straight street form parallel lines. Even a very 

important symbol in mathematics looks like parallel lines: the equals sign (). 

Did you know there is a reason for why an equals sign looks the way it does?  

In 1557, mathematician Robert Recorde first used parallel line segments to 

represent equality because he didn’t want to keep writing the phrase “is equal to” 

and, as he explained, “no two things can be more equal” than parallel lines.

What does equality mean in mathematics? How can you determine whether two or 

more things are equal?

Key Terms  one-step equation

  Properties of Equality for Addition 

and Subtraction

  solution

  inverse operations

Learning GoalsIn this lesson, you will:

  Use inverse operations to solve 

one-step equations.

  Use models to represent one-step 

equations.

OppositesAttracttoMaintainaBalanceSolving One-Step Equations Using Addition and Subtraction

616      •      Chapter 9      Inequalities and Equations

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Problem 1 Maintaining Balance

Each representation shows a balance. Determine what will

balance 1 rectangle in each. Adjustments can be made in

each pan as long as the balance is maintained. Then,

describe your strategies.

1.

a. Strategies:

b. What will balance one rectangle?

You might want to get your algebra

tiles out.

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9.2      Solving One-Step Equations Using Addition and Subtraction      •      617

2.

a. Strategies:

b. What will balance one rectangle?

3. Describe the general strategy you used to maintain balance in Questions 1 and 2.

4. Generalize the strategies for maintaining balance by completing each sentence.

a. To maintain balance when you subtract a quantity from one side, you must

b. To maintain balance when you add a quantity to one side, you must

618      •      Chapter 9      Inequalities and Equations

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Problem 2 One Step at a Time

1. Write an equation that represents each pan balance. These are the same pan

balances that you analyzed for Question 1 and Question 2 in Problem 1. Use the

variable x to represent , and count the units to determine the number

they represent together. Then, describe how the strategies you used to determine what

balanced one rectangle can apply to an equation. In other words, what balances x?

a.

b.

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9.2      Solving One-Step Equations Using Addition and Subtraction      •      619

You just wrote and solved one-stepequations. Previously, you wrote an equation by

setting two expressions equal to each other. You solve an equation by determining what

value will replace the variable to make the equation true. If you can solve an equation

using only one operation, this equation is called a one-stepequation. To determine if

your value is correct, substitute the value for the variable in the original equation. If the

equation is true, or remains balanced, then you correctly solved the equation.

2. Check each of your solutions to Question 1, part (a) and part (b), by substituting your

value for x into the original equation you wrote. Show your work.

You just determined solutions to your equations. A solution to an equation is any value for

a variable that makes the equation true.

3. State the operations in each equation you wrote for Question 1, and the operation

you used to determine the value of x. Describe how they relate to each other.

To solve an equation, you must isolate the variable by performing inverseoperations.

Inverseoperations are pairs of operations that undo each other.

4. State the inverse operation for each stated operation.

a. addition

b. subtraction

To isolate the variable

means to get the variable by itself

on one side of the equation.

620      •      Chapter 9      Inequalities and Equations

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Problem 3 Solving Equations

1. Analyze each example and the different methods used to solve each equation.

a. Describe the difference in strategy between Method 1 and Method 2 for Example 1.

b. The final step in each method shows the variable isolated. What is the coefficient

of each variable?

Example1 Example2

a 1 7 5 9 12 5 b 2 8

Method1: Method1:

a 1 7 2 7 5 9 2 7

a 5 2

12 1 8 5 b 2 8 1 8

20 5 b

Method2: Method2:

a 1 7 5 9

27 5 27

a 5 2

12 5 b 2 8

18 5 1 8

20 5 b

The answers are the same. What is different about the two methods?

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9.2      Solving One-Step Equations Using Addition and Subtraction      •      621

2. Consider the equations shown. State the inverse operation needed to isolate the

variable. Then, solve the equation. Make sure you show your work. Finally, check to

see if the value of your solution maintains balance in the original equation.

a. m 1 7 11

b. 5 x 2 8

c. b 1 5.67 12.89

d. 5 3 __ 4 5 x 2 4 1 __

2

622      •      Chapter 9      Inequalities and Equations

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e. 23.563 5 a 2 345.91

f. 7 ___ 12

5 y 1 1 __ 4

g. w 1 3.14 5 27

h. 13 7 __ 8

5 c 1 9 3 __ 4

Don't forget to check your

answers!

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9.2      Solving One-Step Equations Using Addition and Subtraction      •      623

3. Determine if each solution is true. Explain your reasoning.

a. Is x 5 25 a solution to the equation x 1 17 5 8?

b. Is x 5 16 a solution to the equation x 2 12 5 4?

c. Is x 5 2 1 __ 3

a solution to the equation 17 2 __ 3

5 x 1 15 1 __ 3

?

d. Is x 5 4.567 a solution to the equation x 1 19.34 5 23.897?

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Talk the Talk

The PropertiesofEquality allow you to balance and solve equations involving

any number.

Properties of Equality For all numbers a, b, and c,…

Addition Property of Equality If a 5 b, then a 1 c 5 b 1 c.

Subtraction Property of Equality If a 5 b, then a 2 c 5 b 2 c.

1. Describe in your own words what the Properties of Equality represent.

2. What does it mean to solve a one-step equation?

3. Describe how to solve any one-step equation.

4. How do you check to see if a value is the solution to an equation?

5. Given the solution x 12, write two different equations using the

Properties of Equality.

Be prepared to share your solutions and methods.