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7

1

2x

9x

8

16

4

x

2

7

7

Warm Up

Solve.

1.2x + 9x – 3x + 8 = 16

2. –4 = 6x + 22 – 4x

3. + = 5

4. – = 3

x = 1

x = –13

x = 34

x = 50

Standards

Extension of AF4.1 Solve two-step linear equations and inequalities in one variable over the rational numbers, interpret the solution or solutions in the context from which they arose, and verify the reasonableness of the results.

Also covered: AF1.1

–3x

=

–3

6

–3

Additional Example 1A: Solving Equations with Variables on Both Sides

Solve.

4x + 6 = x

4x + 6 = x

To collect the variable terms on one side, subtract 4x from both sides.

– 4x– 4x

6 = –3x

Since x is multiplied by -3, divide both sides by –3.

–2 = x

You can always check your solution by substituting the value back into the original equation.

4b

24

=

4

4

Additional Example 1B: Solving Equations with Variables on Both Sides

Solve.

9b – 6 = 5b + 18

9b – 6 = 5b + 18

To collect the variable terms on one side, subtract 5b from both sides.

– 5b– 5b

4b – 6 = 18

Since 6 is subtracted from 4b, add 6 to both sides.

+ 6+ 6

4b = 24

Since b is multiplied by 4, divide both sides by 4.

b = 6

9w + 3 = 9w + 7

– 9w– 9w

To collect the variable terms on one side, subtract 9w from both sides.

Additional Example 1C: Solving Equations with Variables on Both Sides

Solve.

9w + 3 = 9w + 7

3 ≠ 7

There is no solution. There is no number that can be substituted for the variable w to make the equation true.

if the variables in an equation are eliminated and the resulting statement is false, the equation has no solution.

–4x

=

–4

8

–4

Check It Out! Example 1A

Solve.

5x + 8 = x

5x + 8 = x

To collect the variable terms on one side, subtract 5x from both sides.

– 5x– 5x

8 = –4x

Since x is multiplied by –4, divide both sides by –4.

–2 = x

Check It Out! Example 1B

Solve.

3b – 2 = 2b + 12

3b – 2 = 2b + 12

To collect the variable terms on one side, subtract 2b from both sides.

– 2b– 2b

b – 2 = 12

Since 2 is subtracted from b, add 2 to both sides.

+ 2+ 2

b = 14

3w + 1 = 3w + 8

– 3w– 3w

To collect the variable terms on one side, subtract 3w from both sides.

Check It Out! Example 1C

Solve.

3w + 1 = 3w + 8

1 ≠ 8

No solution. There is no number that can be substituted for the variable w to make the equation true.

To solve more complicated equations, you may need to first simplify by combining like terms or clearing fractions. Then add or subtract to collect variable terms on one side of the equation. Finally, use properties of equality to isolate the variable.

8 simplify by combining like terms or clearing fractions. Then add or subtract to collect variable terms on one side of the equation. Finally, use properties of equality to isolate the variable.z8

=

8

8

Additional Example 2A: Solving Multi-Step Equations with Variables on Both Sides

Solve.

10z – 15 – 4z = 8 – 2z – 15

10z – 15 – 4z = 8 – 2z – 15

6z– 15 = –2z– 7

Combine like terms.

+ 2z+ 2z

Add 2z to both sides.

8z – 15 = – 7

+ 15+15

Add 15 to both sides.

8z = 8

Divide both sides by 8.

z = 1

7 simplify by combining like terms or clearing fractions. Then add or subtract to collect variable terms on one side of the equation. Finally, use properties of equality to isolate the variable.

10

7

10

7

10

7

10

3y

5

3y

5

3y

5

3y

5

y

5

y

5

y

5

y

5

3

4

3

4

3

4

3

4

+ – = y –

20( ) = 20( )

+ – y –

20() + 20( ) – 20( )= 20(y) – 20( )

Additional Example 2B: Solving Multi-Step Equations with Variables on Both Sides

+ – = y –

Multiply by the LCD, 20.

4y + 12y – 15 = 20y – 14

16y – 15 = 20y – 14

Combine like terms.

4 simplify by combining like terms or clearing fractions. Then add or subtract to collect variable terms on one side of the equation. Finally, use properties of equality to isolate the variable.y

4

–1

–1

4

= y

=

4

Additional Example 2B Continued

16y – 15 = 20y – 14

– 16y– 16y

Subtract 16y from both sides.

–15 = 4y – 14

+ 14+ 14

Add 14 to both sides.

–1 = 4y

Divide both sides by 4.

10 simplify by combining like terms or clearing fractions. Then add or subtract to collect variable terms on one side of the equation. Finally, use properties of equality to isolate the variable.z50

=

10

10

Check It Out! Example 2A

Solve.

12z – 12 – 4z = 6 – 2z + 32

12z – 12 – 4z = 6 – 2z + 32

8z– 12 = –2z+ 38

Combine like terms.

+ 2z+ 2z

Add 2z to both sides.

10z – 12 = 38

+ 12+12

Add 12 to both sides.

10z = 50

Divide both sides by 10.

z = 5

6 simplify by combining like terms or clearing fractions. Then add or subtract to collect variable terms on one side of the equation. Finally, use properties of equality to isolate the variable.

8

6

8

6

8

6

8

5y

6

5y

6

5y

6

5y

6

y

4

y

4

y

4

y

4

3

4

3

4

3

4

3

4

+ + = y –

24( ) = 24( )

+ + y –

24() + 24( )+ 24( )= 24(y) – 24( )

Check It Out! Example 2B

+ + = y –

Multiply by the LCD, 24.

6y + 20y + 18 = 24y – 18

26y + 18 = 24y – 18

Combine like terms.

2 simplify by combining like terms or clearing fractions. Then add or subtract to collect variable terms on one side of the equation. Finally, use properties of equality to isolate the variable.y

2

–36

2

=

Check It Out! Example 2B Continued

26y + 18 = 24y – 18

– 24y– 24y

Subtract 24y from both sides.

2y + 18 = – 18

– 18– 18

Subtract 18 from both sides.

2y = –36

Divide both sides by 2.

y = –18

Additional Example 3: simplify by combining like terms or clearing fractions. Then add or subtract to collect variable terms on one side of the equation. Finally, use properties of equality to isolate the variable.Business Application

Daisy’s Flowers sells a rose bouquet for $39.95 plus $2.95 for every rose. A competing florist sells a similar bouquet for $26.00 plus $4.50 for every rose. Find the number of roses that would make both florists' bouquets cost the same price. What is the price?

Write an equation for each service. Let c represent the total cost and r represent the number of roses.

total cost is flat fee plus cost for each rose

Daisy’s: c = 39.95 + 2.95 r

Other: c = 26.00 + 4.50 r

13.95 simplify by combining like terms or clearing fractions. Then add or subtract to collect variable terms on one side of the equation. Finally, use properties of equality to isolate the variable.

1.55r1.55

=

1.55

Additional Example 3 Continued

Now write an equation showing that the costs are equal.

39.95 + 2.95r = 26.00 + 4.50r

Subtract 2.95r from both sides.

– 2.95r– 2.95r

39.95 = 26.00 + 1.55r

Subtract 26.00 from both sides.

– 26.00– 26.00

13.95 = 1.55r

Divide both sides by 1.55.

9 = r

The two bouquets from either florist would cost the same when purchasing 9 roses.

Additional Example 3 Continued simplify by combining like terms or clearing fractions. Then add or subtract to collect variable terms on one side of the equation. Finally, use properties of equality to isolate the variable.

To find the cost, substitute 9 for r into either equation.

Daisy’s:

Other florist:

c = 39.95 + 2.95r

c = 26.00 + 4.50r

c = 39.95 + 2.95(9)

c = 26.00 + 4.50(9)

c = 39.95 + 26.55

c = 26.00 + 40.50

c = 66.5

c = 66.5

The cost for a bouquet with 9 roses at either florist is $66.50.

Check It Out! simplify by combining like terms or clearing fractions. Then add or subtract to collect variable terms on one side of the equation. Finally, use properties of equality to isolate the variable. Example 3

Marla’s Gift Baskets sells a muffin basket for $22.00 plus $2.25 for every balloon. A competing service sells a similar muffin basket for $16.00 plus $3.00 for every balloon. Find the number of balloons that would make both baskets cost the same price.

Write an equation for each service. Let c represent the total cost and b represent the number of balloons.

total cost is flat fee plus cost for each balloon

Marla’s: c = 22.00 + 2.25 b

Other: c = 16.00 + 3.00 b

6.00 simplify by combining like terms or clearing fractions. Then add or subtract to collect variable terms on one side of the equation. Finally, use properties of equality to isolate the variable.

0.75b0.75

=

0.75

Check It Out! Example 3 Continued

Now write an equation showing that the costs are equal.

22.00 + 2.25b = 16.00 + 3.00b

Subtract 2.25b from both sides.

– 2.25b– 2.25b

22.00 = 16.00 + 0.75b

Subtract 16.00 from both sides.

– 16.00– 16.00

6.00 = 0.75b

Divide both sides by 0.75.

8 = b

The two services would cost the same when purchasing a muffin basket with 8 balloons.

1 simplify by combining like terms or clearing fractions. Then add or subtract to collect variable terms on one side of the equation. Finally, use properties of equality to isolate the variable.

1

2

4

Lesson Quiz

Solve.

1. 4x + 16 = 2x

2. 8x – 3 = 15 + 5x

3. 2(3x + 11) = 6x + 4

4.x = x – 9

5. An apple has about 30 calories more than an orange. Five oranges have about as many calories as 3 apples. How many calories are in each?

x = –8

x = 6

no solution

x = 36

An orange has 45 calories. An apple has 75 calories.

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