Solving Equations with Variables on Both Sides11-3

Warm UpSolve.

1. 2x + 9x – 3x + 8 = 16

2. –4 = 6x + 22 – 4x

x = 1

x = –13

Solving Equations with Variables on Both Sides11-3

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

Solving Equations with Variables on Both Sides11-3

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.

Copy in Notes!

Move variables to the left!

Solving Equations with Variables on Both Sides11-3

Solve.

4x + 6 = x

Example 1: Solving Equations with Variables on Both Sides

4x + 6 = x– x – x

3x + 6 = 0

Subtract x from both sides.

Subtract 6 from both sides.

3x = -6

–6 –6

Divide both sides by 33 3

x = -2

Solving Equations with Variables on Both Sides11-3

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

Helpful Hint

Solving Equations with Variables on Both Sides11-3

Solve.

9b – 6 = 5b + 18

Example 2: 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.

Solving Equations with Variables on Both Sides11-3

Solve.

9w + 3 = 9w + 7

Example 3: 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.

Solving Equations with Variables on Both Sides11-3

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

Helpful Hint

Solving Equations with Variables on Both Sides11-3

Solve.

5x + 8 = x

Example 4

5x + 8 = x– x – x

4x +8 = 0

Subtract x from both sides.

Divide both sides by 4.

x= -2

4x 4

–84=

–8 -8 Subtract 8 from both sides.

Solving Equations with Variables on Both Sides11-3

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.

Example 5

Solving Equations with Variables on Both Sides11-3

Solve.

3w + 1 = 3w + 8

8 = 8

3w + 8 = 8 + 3w

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

All solutions. Any number you substitute will make the statement true.

Example 6

Solving Equations with Variables on Both Sides11-3

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.

Solving Equations with Variables on Both Sides11-3

Solve.

10z – 15 – 4z = 8 – 2z - 15

Example 7: 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=

Solving Equations with Variables on Both Sides11-3

Solve.

12z – 12 – 4z = 6 – 2z + 32

Example 8

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=

Solving Equations with Variables on Both Sides11-3

9) Solve3

81

4x

1

2x 3

4

Solving Equations with Variables on Both Sides11-3

Standard Lesson Quiz

Lesson Quizzes

Lesson Quiz for Student Response Systems

Solving Equations with Variables on Both Sides11-3

Lesson QuizSolve.

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

no solution

x = 3614

12

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

Solving Equations with Variables on Both Sides11-3

1. Combine like terms. 4p + 14 = 11p

A. p = 2

B. p = 7

C. p = 14

D. p = 15

Lesson Quiz for Student Response Systems

Solving Equations with Variables on Both Sides11-3

2. Combine like terms. 3g – 6 = 4g – 7

A. g = 1

B. g = 7

C. g = 13

D. g = –13

Lesson Quiz for Student Response Systems

Solving Equations with Variables on Both Sides11-3

3. Combine like terms. 3(g – 2) = 7g – 18

A. g = 2

B. g = 3

C. g = –3

D. g = –2

Lesson Quiz for Student Response Systems