Expressions 2 That Play Together - Carnegie Learning

Expressions 2

That Play Together . . .

Solving Equations on a Double Number Line

WARM UP

Explain whether or not Expression B is equivalent to Expression A. If the expressions are not equivalent, determine an expression equivalent to Expression A.

1. A: 2(x 2 5)

B: 2x 2 5

2. A: 8 2 2(n 1 3) B: 6(n 1 3)

3. A: 2(x 2 4)

B: 2x 1 4

LEARNING GOALS

? Identify relationships between expressions.

? Decompose an equation to isolate the unknown.

? Model and solve equations using a double number line.

? Construct equations to solve problems by reasoning about the quantities.

Previously, you have identified expressions that are equivalent. In this lesson, you will model situations as equal expressions on a double number line. How can you maintain equality to determine the unknown quantities in linear equations?

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A double number line diagram is a model used to show equivalent relationships.

Getting Started

. . . Stay Together

Consider this double number line. The expressions 12 and 3x have the same location, so they have the same value.

0

12

0x

3x

1. Write an equation to show that 3x and 12 have the same value.

2. Extend each number line in both directions by identifying and labeling additional equivalent relationships. Explain the reasoning you used to place each relationship.

3. Consider this double number line.

2x

0x

?14

0

a. Write an equation to show that 214 and 2x have the same value.

b. Extend each number line in both directions by identifying and labeling additional equivalent relationships. Explain the reasoning you used to place each relationship.

M3-66 ? TOPIC 2: Two-Step Equations and Inequalities

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4. Consider this double number line.

0

17

0x

x + 4

a. Write an equation to show that x 1 4 and 17 have the same value.

b. Extend each number line in both directions by identifying and labeling additional equivalent relationships. Explain the reasoning you used to place each relationship.

5. Consider this double number line.

0

21

0x

3 4

x

a. Write an equation to show that _34_x and 21 have the same value.

b. Extend each number line in both directions by identifying and labeling additional equivalent relationships. Explain the reasoning you used to place each relationship.

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LESSON 2: Expressions That Play Together . . . ? M3-67

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Think about how to transform the equation to isolate the variable.

AC TIVIT Y

2.1 Solving a Two-Step Equation

In the previous lesson, you modeled a problem in which Fido and Jet are two small dogs. Fido weighs exactly 10 pounds more than Jet. Together, they weigh exactly 46 pounds.

This situation can be represented by the equation 2j 1 10 5 46, where j represents Jet's weight. You can also represent this situation and solve the equation using a double number line.

WORKED EXAMPLE

Solve the equation 2j 1 10 5 46.

First, draw a model to set up the equation.

0

46

0j

2j + 10

Next, start decomposing the variable expression. Place 2j in relationship to 2j 1 10. The expression 2j is 10 to the left of 2j 1 10. To maintain equality, place a number that is 10 to the left of 46. So, 2j 5 36.

0

36

46

0j

2j 2j + 10

The expression 1j, or j, is halfway between 0j and 2j. And 18 is halfway between 0 and 36. So, j 5 18.

0

18

36

46

0j

1j

2j 2j + 10

M3-68 ? TOPIC 2: Two-Step Equations and Inequalities

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1. How can you check to see if j 5 18 is the solution to the original equation.

2. What does the solution j 5 18 represent in terms of this problem situation?

3. What operation is used in each step to move toward the solution?

Is there another way to use the double number line to determine the value of j?

4. Use the double number line shown to solve the equation 21 5 2d 1 3. Describe the steps you use, including the operations represented at each step.

0

21

0d

2d + 3

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ACTIVITY Practice Solving Two-Step

2.2 Equations

Don't forget to place 0 and 0x on your double number line model.

In this activity, you will use double number lines to solve equations.

1. Model each equation on the double number line given. Then use the model to solve the equation. Describe the steps and operations you used and explain your reasoning.

a. _12_x 1 5 5 15

b. 52.5 5 t 2 3.1

M3-70 ? TOPIC 2: Two-Step Equations and Inequalities

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c. 4(b 1 1) 5 20 d. m 1 4.5 5 210

Are you checking your answers?

2. Brent showed how he started to solve the equation _12_x 1 5 5 15. Describe his method. Then complete his process to solve the equation.

0

15

30

0x

1 2

x

+

5

x + 10

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ACTIVITY Reasoning with Negatives

2.3 to Solve Equations

The double number line shows one way to start to solve the equation 22(3x 1 4) 5 10. A through D represent the order in which the steps were completed.

D

B

CA

?9

0

5

9 10

3x

0x

?3x ? 4

?3x ?2(3x + 4)

1. Describe each step used to solve the equation. List the operation used at each step.

From Step A to Step B:

From Step B to Step C:

From Step C to Step D:

2. Between which two steps is there a reflection across 0 on the number line? What operation is used to accomplish this reflection?

3. What is the solution to the equation?

M3-72 ? TOPIC 2: Two-Step Equations and Inequalities

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