Introduction

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Use What You Know

Lesson 14 Solutions of Linear Equations

Lesson 14Solutions of Linear Equations

124

You’ve learned how to solve linear equations and how to check your solution. In this lesson, you’ll learn that not every linear equation has just one solution. Take a look at this problem.

Jason and his friend Amy are arguing. Jason says that a linear equation always has just one solution. Amy says that some linear equations have more than one solution. Who’s right? Amy asked Jason to explore solutions to the following equation.

2x 1 1 1 x 5 3(x 2 2) 1 7

Use the math you already know to solve this problem.

Remember that a solution to an equation is a number that makes the equation true. To check to see if a number is a solution to this equation, replace x with its value.

a. Is 6 a solution to the equation? Show your work.

b. Is 22 a solution to the equation? Show your work.

c. Is 0 a solution to the equation? Show your work.

d. Can an equation have more than one solution? Explain. Who is right—Jason or Amy?

M.8.9a

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Lesson 14 Solutions of Linear Equations 125

Look at how you could solve Amy’s equation.

2x 1 1 1 x 5 3(x 2 2) 1 7

First, simplify each side:

Use the distributive property. 2x 1 1 1 x 5 3x 2 6 1 7

Combine like terms. 3x 1 1 5 3x 1 1

Once both sides of an equation are in simplest form, you can say a lot about the solution without actually solving the equation. You can just look at the structure.

Think about how you might solve this equation with pictures. Look at the pan balance below. The left pan represents 3x 1 1. So does the right pan.

xxx xxx

If you take away 3x from both sides, you end up with 1 5 1, a true statement. If you take away 1 from both sides, you end up with 3x 5 3x, a true statement. You can replace x with any number and you will always get a true statement. The pan will remain balanced. This equation has infinitely many solutions.

You’ve seen that a linear equation can have only one solution or, in a case like this, infinitely many solutions. You will also see that a linear equation can have no solution.

Reflect1 Once both sides of an equation are in simplest form, how can you tell if it has infinitely

many solutions?

Modeled and Guided Instruction

Learn About

Lesson 14

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Determining the Number of Solutions of an Equation

Read the problem below. Then explore how to identify when an equation has only one solution, infinitely many solutions, or no solution.

Yari and her friends, Alyssa and David, were each given an equation to solve.

Yari: 2(x 1 10) 2 17 5 5 1 2x 2 2

Alyssa: 5x 1 3 2 3x 5 2(x 1 3) 2 5

David: 2(x 2 3) 1 9 5 5 1 x 2 1

Whose equation has only one solution? Infinitely many solutions? No solution?

Model It You can use properties of operations to simplify each side of Yari’s equation.

2(x 1 10) 2 17 5 5 1 2x 2 2

2x 1 20 2 17 5 3 1 2x

2x 1 3 5 3 1 2x

Model It You can use properties of operations to simplify each side of Alyssa’s equation.

5x 1 3 2 3x 5 2(x 1 3) 2 5

2x 1 3 5 2x 1 6 2 5

2x 1 3 5 2x 1 1

Model It You can use properties of operations to simplify each side of David’s equation.

2(x 2 3) 1 9 5 5 1 x 2 1

2x 2 6 1 9 5 4 1 x

2x 1 3 5 4 1 x

xx xx

xx xx

xx x

The variable terms and the constants are the same on both sides of the equation. No matter what value you choose for x, the equation will always be true.

The variable terms are the same on both sides of the equation but the constants are different. There is no value for x that will make the equation true.

The variable terms are different. There is only one value for x that will make the equation true.

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Connect It Now you will use the models to solve this problem.

2 Look at Model It for Yari’s equation. What do you notice about both sides of the equation? What equation do you get if you subtract 2x from both sides of the equation?

3 Look at Model It for Alyssa’s equation. How is it different from Yari’s equation? How is

it similar?

4 Look at the pan balance for Alyssa’s equation. Is there any way to balance the pans? Explain. What equation do you get if you subtract 2x from both sides of the equation?

5 Look at Model It for David’s equation. Are the variable terms on each side of the

equation the same or different? Solve David’s equation.

6 Explain how you know when an equation has only one solution, no solution, or infinitely

many solutions.

Try It Use what you just learned about equations with only one solution, no solution, or infinitely many solutions. Show your work on a separate sheet of paper.

Replace c and d in the equation cx 1 d 5 8x 1 12 with the given values. Tell whether the equation has only one solution, no solution, or infinitely many solutions.

7 c 5 6 and d 5 34 8 c 5 8 and d 5 6

Guided Practice

Practice

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Lesson 14

Lesson 14 Solutions of Linear Equations

Determining the Number of Solutions of an Equation

Study the example below. Then solve problems 9–11.

Example

Michelle looks at the equation 26x 2 30 5 6x 2 30 and says there is no solution. Is she correct? Explain.

Look at how you can show your work.

Solution

9 Draw lines to match each linear equation to its correct number of solutions.

Show your work.

5(4 2 x) 5 25x 1 20 no solution

25(4 2 x) 5 25x 1 20 infinitely many solutions

5(5 2 x) 5 25x 1 20 only one solution

26x 2 30 5 6x 2 30

26x 2 30 2 6x 5 6x 2 30 2 6x

212x 2 30 5 230

212x 2 30 1 30 5 230 1 30

212x 5 0

212x ····· 212 5 0 ···· 212

x 5 0

No; There is one solution to this equation, x 5 0.

How can you tell when a linear equation has no solution?

Pair/ShareHow could you convince Michelle that there is a solution to this equation?

Pair/ShareWhen the variable terms on both sides of an equation are the same, what does that tell you about the solution(s) to the equation?

The student solved the equation to find that 0 is a solution.

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10 Write a number in the box so that the equation will have the type of solution(s) shown.

no solution

1 ·· 3 x 1 5 5 1 ·· 3 x 1

infinitely many solutions

1 ·· 3 x 1 5 5 1 ·· 3 x 1

only one solution

1 ·· 3 x 1 5 5 x 1 5

11 What is the solution to the equation 3(x 2 4) 5 2(x 2 6)?

A x 5 0

B x 5 1

C There are infinitely many solutions.

D There is no solution.

Brian chose D as the correct answer. How did he get that answer?

What is the difference between an equation with no solution and an equation with infinitely many solutions?

How can you simplify this equation to justify the correct answer?

Pair/ShareWhat do you notice about the solution to equations that have the same variable term on each side of the equation?

Pair/ShareHow could you explain the correct response to Brian?

Independent Practice

Practice

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Lesson 14

Lesson 14 Solutions of Linear Equations

Determining the Number of Solutions of an Equation

Solve the problems.

1 How many solutions does the equation 2(2x 2 10) 2 8 5 22(14 2 3x) have?

A exactly one solution

B exactly two solutions

C no solution

D infinitely many solutions

2 How many solutions does the equation 10 2 3x 1 10x 2 7 5 5x 2 5 1 2x 1 8 have?

A exactly one solution

B exactly two solutions

C no solution

D infinitely many solutions

3 For each linear equation in the table, shade in the appropriate box to indicate whether the equation has no solution, only one solution, or infinitely many solutions.

Equation No SolutionnOnly

One SolutionInfinitely Many

Solutions

8x 1 16 5 8x 2 16

23x 2 17 5 2(17 1 3x)

9x 1 27 5 27

2x 2 6 5 6 1 2x

4 Which equation has an infinite number of solutions? Select all that apply.

A 3x 2 2(x 1 10) 5 x 2 20 D 5 ·· 2 x 2 2 5 9 ·· 2 x 2 2(x 1 1)

B 5x 1 2(x 2 3) 5 5x 1 2(3 2 x) E 7 ·· 2 x 1 x 5 x 1 7 ·· 4

C x ·· 2 1 1 5 3x ·· 10 1 3

Self Check

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Go back and see what you can check off on the Self Check on page 99.

Lesson 14 Solutions of Linear Equations

5 Consider the equation 2(5x 2 4) 5 ax 1 b.

Part A Find a value for a and a value for b so that the equation has only one solution. Explain your reasoning.

Show your work.

a 5

b 5

Part B Find a value for a and a value for b so that the equation has no solution. Explain your reasoning.

Show your work.

a 5

b 5

Part C Find a value for a and a value for b so that the equation has infinitely many solutions. Explain your reasoning.

Show your work.

a 5

b 5