Learn to solve equations with variables on both sides of the equal sign.

Course 3

11-3 Solving Equations with Variables on Both Sides

Some problems produce equations that have variables on both sides of the equal sign.

Solving an equation with variables on both sides is similar to solving an equation with a variable on only one side. You can add or subtract a term containing a variable on both sides of an equation.

Course 3

11-3 Solving Equations with Variables on Both Sides

Solve.

4x + 6 = x

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

4x + 6 = x– 4x – 4x

6 = –3x

Subtract 4x from both sides.

Divide both sides by –3.

–2 = x

6–3

–3x–3=

Course 3

11-3 Solving Equations with Variables on Both Sides

Course 3

11-3 Solving Equations with Variables on Both Sides

Check your solution by substituting the value back into the original equation. For example, 4(2) + 6 = 2 or 2 = 2.

Helpful Hint

Solve.

9b – 6 = 5b + 18

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

9b – 6 = 5b + 18– 5b – 5b

4b – 6 = 18

4b 4

24 4 =

Subtract 5b from both sides.

Divide both sides by 4.

b = 6

+ 6 + 6

4b = 24Add 6 to both sides.

Course 3

11-3 Solving Equations with Variables on Both Sides

Solve.

9w + 3 = 9w + 7

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

3 ≠ 7

9w + 3 = 9w + 7

– 9w – 9w Subtract 9w from both sides.

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

Course 3

11-3 Solving Equations with Variables on Both Sides

Course 3

11-3 Solving Equations with Variables on Both Sides

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

Helpful Hint

Solve.

5x + 8 = x

Check It Out: Example 1A

5x + 8 = x– 5x – 5x

8 = –4x

Subtract 5x from both sides.

Divide both sides by –4.

–2 = x

8–4

–4x–4=

Course 3

11-3 Solving Equations with Variables on Both Sides

Solve.

3b – 2 = 2b + 123b – 2 = 2b + 12

– 2b – 2b

b – 2 = 12

Subtract 2b from both sides.

+ 2 + 2

b = 14Add 2 to both sides.

Check It Out: Example 1B

Course 3

11-3 Solving Equations with Variables on Both Sides

Solve.

3w + 1 = 3w + 8

1 ≠ 8

3w + 1 = 3w + 8

– 3w – 3w Subtract 3w from both sides.

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

Check It Out: Example 1C

Course 3

11-3 Solving Equations with Variables on Both Sides

To solve multi-step equations with variables on both sides, first combine like terms and clear fractions. Then add or subtract variable terms to both sides so that the variable occurs on only one side of the equation. Then use properties of equality to isolate the variable.

Course 3

11-3 Solving Equations with Variables on Both Sides

Solve.

10z – 15 – 4z = 8 – 2z - 15

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

10z – 15 – 4z = 8 – 2z – 15

+ 15 +15

6z – 15 = –2z – 7 Combine like terms.+ 2z + 2z Add 2z to both sides.

8z – 15 = – 7

8z = 8

z = 1

Add 15 to both sides.

Divide both sides by 8.8z 88 8=

Course 3

11-3 Solving Equations with Variables on Both Sides

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

Multiply by the LCD, 20.

4y + 12y – 15 = 20y – 14

16y – 15 = 20y – 14 Combine like terms.

y5

34

3y5

710

+ – = y –

y5

34

3y5

710

+ – = y –

20( ) = 20( )y5

34

3y5

710

+ – y –

20( ) + 20( ) – 20( )= 20(y) – 20( )y5

3y5

34

710

Course 3

11-3 Solving Equations with Variables on Both Sides

Additional Example 2B Continued

Add 14 to both sides.

–15 = 4y – 14

–1 = 4y

+ 14 + 14

–1 4

4y4 = Divide both sides by 4.

–14 = y

16y – 15 = 20y – 14

– 16y – 16y Subtract 16y from both sides.

Course 3

11-3 Solving Equations with Variables on Both Sides

Solve.

12z – 12 – 4z = 6 – 2z + 32

Check It Out: Example 2A

12z – 12 – 4z = 6 – 2z + 32

+ 12 +12

8z – 12 = –2z + 38 Combine like terms.+ 2z + 2z Add 2z to both sides.

10z – 12 = 38

10z = 50

z = 5

Add 12 to both sides.

Divide both sides by 10.10z 5010 10=

Course 3

11-3 Solving Equations with Variables on Both Sides

Multiply by the LCD, 24.

6y + 20y + 18 = 24y – 18

26y + 18 = 24y – 18 Combine like terms.

y4

34

5y6

68

+ + = y –

y4

34

5y6

68

+ + = y –

24( ) = 24( )y4

34

5y6

68

+ + y –

24( ) + 24( )+ 24( )= 24(y) – 24( )y4

5y6

34

68

Check It Out: Example 2B

Course 3

11-3 Solving Equations with Variables on Both Sides

Subtract 18 from both sides.

2y + 18 = – 18

2y = –36

– 18 – 18

–36 2

2y2 = Divide both sides by 2.

y = –18

26y + 18 = 24y – 18

– 24y – 24y Subtract 24y from both sides.

Check It Out: Example 2B Continued

Course 3

11-3 Solving Equations with Variables on Both Sides

Additional Example 4 Continued

Now find the amount of money Jamie spends each morning.

1.25 + 2d Choose one of the original expressions.

Jamie spends $1.75 each morning.

1.25 + 2(0.25) = 1.75

0.25n0.25

1.75 0.25 =

Let n represent the number of doughnuts.

Find the number of doughnuts Jamie buys on Tuesday.

0.25n = 1.75

n = 7; Jamie bought 7 doughnuts on Tuesday.

Divide both sides by 0.25.

Course 3

11-3 Solving Equations with Variables on Both Sides

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 = 6

x = –8

Insert Lesson Title Here

no solution

x = 3614

12

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

Course 3

11-3 Solving Equations with Variables on Both Sides