PDF 1.3 Solving Equations with Variables on Both Sides

1.3

Solving Equations with Variables

on Both Sides

How can you solve an equation that has

variables on both sides?

1

ACTIVITY: Perimeter and Area

Work with a partner.

��

Each figure has the unusual property that the value of its perimeter

(in feet) is equal to the value of its area (in square feet). Write an

equation for each figure.

��

Solve each equation for x.

��

Use the value of x to find the perimeter and the area of each figure.

��

Describe how you can check your solution.

b.

a.

3

x

x

c.

4

d.

x

5

2

18

x

e.

f.

1

4

2

2

3

COMMON

CORE

Solving Equations

In this lesson, you will

�� solve equations with

variables on both sides.

�� determine whether

equations have no solution

or infinitely many solutions.

Learning Standards

8.EE.7a

8.EE.7b

x��1

x

g.

3x

x

1

3

2

1

x

18

Chapter 1

Equations

x

Laurie��s Notes

Introduction

Standards for Mathematical Practice

?

MP6 Attend to Precision: Mathematically proficient students try to

communicate precisely with others. In today��s activities, students will work

with vocabulary related to measurement.

Motivate

��What balances with the cylinder? Explain.��

Common Core State Standards

8.EE.7a Give examples of linear

equations in one variable with one

solution, infinitely many solutions, or

no solutions. Show which of these

possibilities is the case by successively

transforming the given equation into

simpler forms, until an equivalent

equation of the form x = a, a = a,

or a = b results (where a and b are

different numbers).

8.EE.7b Solve linear equations with

rational number coefficients, including

equations whose solutions require

expanding expressions using the

distributive property and collecting

like terms.

Previous Learning

?

?

2 cubes; Remove one cube and one cylinder from each side. 2 cylinders

balance with 4 cubes, so 1 cylinder would balance with 2 cubes.

The balance problem is equivalent to x + 5 = 3x + 1, where x is a cylinder

and the whole numbers represent cubes. This is an example of an equation

with variables on both sides, the type students will solve today. Return to

this equation at the end of class.

MP4 Model with Mathematics: If students are familiar with algebra tiles,

you can model the problem using the tiles. The cylinder is replaced with an

x-tile, and the cubes are replaced with unit tiles.

Students should know common formulas

for perimeter, area, surface area,

and volume.

Technology

ogy for the

Teacher

Lesson Plans

Complete Materials List

Activity Notes

Activity 1

?

?

?

Discuss with students the general concept of what it means to measure

the attributes of a two-dimensional figure. In other words, what is the

difference between a rectangle��s perimeter and a rectangle��s area? What

type of units are used to measure each? linear units for perimeter and

square units for area

MP6: Be sure to make it clear that the directions are saying that perimeter

and area are not the same, but their values are equal. For example, a

square that measures 4 centimeters on each edge has a perimeter of

16 centimeters and an area of 16 square centimeters. The value (16) is the

same, but the units of measure are not.

Before students begin, ask a few review questions.

? ��How do you find the perimeter P and the area A of a rectangle with

length ? and width w ?�� P = 2? + 2w and A = ?w

��How

do you find the perimeter and the area of a composite

?

figure?�� Listen for students�� understanding that perimeter is the sum

of all of the sides. The area is found in parts and then added together.

Have a few groups share their work at the board, particularly for part (d),

fractions, and part (g), algebraic expressions.

1.3 Record and Practice Journal

Essential Question How can you solve an equation that has variables

on both sides?

1

ACTIVITY: Perimeter and Area

Work with a partner.

?

Each figure has the unusual property that the value of its perimeter

(in feet) is equal to the value of its area (in square feet).Write an

equation for each figure.

?

Solve each equation for x.

?

Use the value of x to find the perimeter and the area of each figure.

?

Describe how you can check your solution.

a.

b.

3

x

x

4

2x + 6 = 3x

x=6

P = 18 ft

A = 18 ft2

c. x

2x + 8 = 4x

x=4

P = 16 ft

A = 16 ft2

d.

5

2

18

2x + 36 = 18x

1

x = 2��

4

P = 40.5 ft

A = 40.5 ft2

x

5

2

2x + 5 = ��x

x = 10

P = 25 ft

A = 25 ft2

T-18

Laurie��s Notes

Differentiated Instruction

Auditory

Point out to students that skills used to

solve equations in this lesson are the

same skills they have used before. The

goal is to isolate the variable on one

side of the equation. Just as they used

the Addition Property of Equality to

remove a constant term from one side

of the equation, they will use the same

property to remove the variable term

from one side of the equation.

Activity 2

?

?

Activity 3

?

?

1.3 Record and Practice Journal

e.

f.

1

4

2

2

3

This activity is similar to Activity 1.

��How do you find the surface area S and volume V of a rectangular prism

with length ?, width w, and height h?�� S = 2?w + 2?h + 2wh and

V = ?wh

Students may guess that part (a) is a cube, suggesting x = 6. Ask students

to verify their guesses.

��The larger triangle is a scale drawing of the smaller triangle. How can

you find the missing dimensions?�� Sample answer: Write a proportion to

solve for the side labeled x in the smaller triangle, then use this value to

find the side labeled 2x in the larger triangle.

The above described method is a good preview of similar figures in the

next chapter.

Another Way: There are other approaches students may take in solving

this problem.

? One method is to solve the equation: 150% of the smaller triangle��s

perimeter is equal to the perimeter of the larger triangle.

x+1

150% of (18 + x) = 24 + 2x

1.5(18 + x) = 24 + 2x

x

2x + 8 = 3x + 2

x=6

P = 20 ft

A = 20 ft2

g.

x

1

2

2x + 16 = 2x + 4(x + 1)

x=3

P = 22 ft

A = 22 ft2

3x

?

3

1

x

2

x

6x + 10 = 9x + x + x

x=2

P = 22 ft

A = 22 ft2

?

ACTIVITY: Surface Area and Volume

This method reviews decimal multiplication.

Another method is to find the scale factor, and then use it to determine

the missing dimensions.

Neighbor Check: Have students work independently and then have their

neighbors check their work. Have students discuss any discrepancies.

Work with a partner.

?

Each solid on the next page has the unusual property that the value

of its surface area (in square inches) is equal to the value of its volume

(in cubic inches).Write an equation for each solid.

?

Solve each equation for x.

?

Use the value of x to find the surface area and the volume of each solid.

?

Describe how you can check your solution.

Closure

?

a.

b.

x

4

8

6

6

x

8x + 16x + 64 = 32x

x=8

S = 256 in.2; V = 256 in.3

12x + 72 + 12x = 36x

x=6

S = 216 in.2; V = 216 in.3

3

ACTIVITY: Puzzle

Work with a partner. The perimeter of the larger triangle is 150% of the

perimeter of the smaller triangle. Find the dimensions of each triangle.

10

x

15

9

8

x=6

6, 8, 10

2x

2x = 12

9, 12, 15

What Is Your Answer?

4. IN YOUR OWN WORDS How can you solve an equation that has variables

on both sides? How do you move a variable term from one side of the

equation to the other?

Add or subtract variable terms and constant terms to

collect the variable terms on one side and constant

terms on the other side.

5. Write an equation that has variables on both sides. Solve the equation.

See Additional Answers.

T-19

Describe how to solve x + 5 = 3x + 1. Sample answer: Subtract x from

both sides, subtract 1 from both sides, and then divide both sides by 2.

2

ACTIVITY: Surface Area and Volume

Work with a partner.

Math

Practice

��

Each solid has the unusual property that the value of its surface area

(in square inches) is equal to the value of its volume (in cubic inches).

Write an equation for each solid.

��

Solve each equation for x.

��

Use the value of x to find the surface area and the volume of each solid.

��

Describe how you can check your solution.

Use Operations

What properties

of operations do

you need to use in

order to find the

value of x?

a.

b.

x

4

8

6

x

6

3

ACTIVITY: Puzzle

Work with a partner. The perimeter of the larger triangle is 150% of the

perimeter of the smaller triangle. Find the dimensions of each triangle.

10

x

15

9

8

2x

4. IN YOUR OWN WORDS How can you solve an equation that has variables

on both sides? How do you move a variable term from one side of the

equation to the other?

5. Write an equation that has variables on both sides. Solve the equation.

Use what you learned about solving equations with variables on

both sides to complete Exercises 3 �C 5 on page 23.

Section 1.3

Solving Equations with Variables on Both Sides

19

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