solve two-step equations solve equations with variables on both sides

1. Solve two-step equations

2. Solve equations with variables on both sides

3. Solve equations containing grouping symbols.

ObjectivesThe student will be able to:

Adapted from original designed by Skip Tyler, Varina High School

Solve: 8x + 5 = 61

Step 1

Examine the problem

Figure out the order that you would DO to solve if you knew the value of the variable.

Use PEMDAS!

Step 2

Completely UNDO those steps

Working in reverse order using inverse operations!

Solve: 8x + 5 = 61 – 5 – 5

8x = 56

x = 7

8(7) + 5 = 61

56 + 5 = 61

1. Draw “the river”2. Subtract 5 from

both sides3. Simplify4. Divide both sides

by 85. Simplify

6. Check your answer

8 8

1. 245

2. 5

3. 9

4. 315

Find the value of z if 7z + 5 = 40

Answer Now

Solve:

+ 9 + 9

= 40

a = −160

40 − 9 = 31

1. Draw “the river”2. Add 9 to both

sides3. Simplify4. Multiply both

sides by -45. Simplify

6. Check your answer

3194

a

4a

3194

160

−4 −4

Solve:

− 8 − 8

= − 20

y = −80

−20 + 8 = −12

1. Draw “the river”2. Subtract 8 from

both sides3. Simplify4. Multiply both

sides by 45. Simplify

6. Check your answer

1284

y

4

y

1284

80

4 4

1. 30

2. 4

3. 100

4. 1.2

Find the value of z if

Answer Now

1375

m

Solve:

= (4)3

m – 8 = 12 + 8 = + 8

m = 20

12/3 = 4

1. Draw “the river”2. Multiply both

sides by 33. Simplify4. Add 8 to both

sides 5. Simplify

6. Check your answer

43

8

m

3

83

m

43

820

1. 186

2. 51

3. 39

4. 174

Find the value of z if

Answer Now

902

6

t

Solve: 3x + 2 = 4x – 1

You need to get the variables on one side of the equation.

It does not matter which variable you move.

Try to move the one that will keep your variable positive.

Solve: 3x + 2 = 4x – 1

- 3x - 3x

2 = x – 1

+ 1 + 1

3 = x

3(3) + 2 = 4(3) – 1

9 + 2 = 12 – 1

1. Draw “the river”2. Subtract 3x from

both sides3. Simplify4. Add 1 to both

sides5. Simplify

6. Check your answer

Solve: 8y – 9 = –3y + 2 + 3y + 3y

11y – 9 = 2

+ 9 + 9

11y = 11

11 11

y = 1

8(1) – 9 = –3(1) + 2

1. Draw “the river”2. Add 3y to both sides3. Simplify4. Add 9 to both sides5. Simplify6. Divide both sides by

117. Simplify

8. Check your answer

1. – 3

2.

3.

4. 3

What is the value of x if 3 – 4x = 18 + x?

1

3

1

3

Answer Now

Solve: 4 = 7x – 3x 4 = 4x

4 4

1 = x

4 = 7(1) – 3(1)

1. Draw “the river” – Notice the variables

are on the same side!2. Combine like terms3. Divide both sides by 44. Simplify5. Check your answer

Solve: – 7(x – 3) = – 7

– 7x + 21 = – 7

– 21 – 21

– 7x = – 28

– 7 – 7

x = 4

– 7(4 – 3) = – 7

– 7(1) = – 7

1. Draw “the river” 2. Distribute3. Subtract 21 from both

sides4. Simplify5. Divide both sides by -76. Simplify

7. Check your answer

What is the value of x if 3(x + 4) = 2(x – 1)?

Answer Now

1. -14

2. -13

3. 13

4. 14

3 - 2x = 4x – 6

+ 2x +2x

3 = 6x – 6

+ 6 + 6

9 = 6x

6 6

or 1.5 = x

Solve:

1. Draw “the river”2. Clear the fraction – multiply

each term by the LCD3. Simplify4. Add 2x to both sides5. Simplify6. Add 6 to both sides7. Simplify8. Divide both sides by 69. Simplify

10. Check your answer

3

8

1

4x

1

2x

3

4

(8)3

8 (8)

1

4x (8)

1

2x (8)

3

4

1.53 1 1 3

8 4 2 41.5

3

2

Special Case #1 6) 2x + 5 = 2x – 3

– 2x – 2x

5 = – 3

1. Draw “the river” 2. Subtract 2x from both

sides3. Simplify

This is never true!

No solutions

Special Case #2 3(x + 1) – 5 = 3x – 2

3x + 3 – 5 = 3x – 2

3x – 2 = 3x – 2

– 3x – 3x

– 2 = – 2

1. Draw “the river” 2. Distribute3. Combine like terms4. Subtract 3x from both

sides5. Simplify

This is always true!

Infinite solutions or identity

What is the value of x if – 3 + 12x = 12x – 3?

1. 0

2. 4

3. No solutions

4. Infinite solutions

Answer Now

1. -8

2. -2

3. 2

4. 8

Challenge! What is the value of x if – 8(x + 1) + 3(x – 2) = – 3x + 2?

Answer Now