Ch 1.7 (part 5)

Variable on Both Sides

Objective:

To solve equations where one variable exists on both sides of the equation.

GOAL: Isolate the variable on one side of the equation.

1) Use the Distributive Property.(then simplify by combining LIKE Terms)

2) Choose one of the variable expressions and use the Inverse Property of Addition

3) Apply the Inverse Property of Addition and/or the Inverse Property of Multiplication to the numbers.

Perform Inverse operations to both sides of the equation!

Rules

Special Cases

1) x = x

2) x + 1 = x

Plug in various numbers for x ……..

Plug in various numbers for x ……..

Solution: x = All Real Numbers

Solution: x = No solution

Every number makes a TRUE statement!

Every number makes a FALSE statement!

-x -x0 = 0

-x -x1 = 0

-3 - 3

2x + 4 = 5x - 17

2x + 4 = 5x - 17 2x + 4 = 5x - 17-2x -2x

4 = 3x - 17+17 +17

21 = 3x3 3

7 = x

-5x -5x-3x + 4 = -17

-4 -4

-3x = -21

x = 7

Example 1

Option 1: Subtract 2x from both sides

Option 2: Subtract 5x from both sides

4(x - 2) - 2x = 5(x - 4)4x - 8 - 2x = 5x - 20

2x - 8 = 5x - 20-2x -2x

-8 = 3x - 20+20 +20

12 = 3x3 3

€

x = 4

Example 2

Distributive Property

Combine LIKE Terms

Inverse Property of Addition for the variable

Inverse Property of Addition

Inverse Property of Multiplication

3x + 8 = 2(x + 4) + x3x + 2 = 2(x - 1) + x3x + 8 = 2x + 8 + x3x + 8 = 3x + 8

-3x -3x

8 = 8

3x + 2 = 2x - 2 + x3x + 2 = 3x - 2

-3x -3x

2 = -2

x = any real number

True !False !

No Solution

Example 3 Example 4

1) 3x - 5 = 2x + 12-2x -2x

x - 5 = 12+5 +5

x = 17

2) 5x - 3 = 13 – 3x+3x +3x8x - 3 = 13

+3 +38x = 168 8

x = 2

Classwork

3) 2b + 6 = 7b - 9-2b -2b

6 = 5b - 9+9 +915 = 5b 5 5

3 = b

4) -4c - 11 = 4c + 21+4c +4c

-11 = 8c + 21-21 -21-32 = 8c 8 8-4 = c

5) 3(x + 2) - (2x - 4) = - (4x + 5)

3x + 6 - 2x + 4 = - 4x - 5 x + 10 = - 4x - 5

+ 4x + 4x

5x + 10 = -5- 10 -105x = -155 5

x = -3

6) 4(y - 2) + 6y = 7(y - 8) - 3(10 - y)

4y - 8 + 6y = 7y - 56 - 30 + 3y

10y - 8 = 10y - 86

-10y -10y

-8 = -86 False

No Solution

7) 3(4 + k) - 2(3k + 4) = 5(k - 3) - (8k - 19)

12 + 3k - 6k - 8 = 5k - 15 - 8k + 19

-3k + 4 = -3k + 4

+3k +3k

4 = 4 True

Infinitely Many Solutions!

x = all real numbers

8) 5(m - 4) = 10 - 4[2(m - 5) - 5m]

5m - 20 = 10 - 4[2m - 10 - 5m]

5m - 20 = 10 - 4[-3m - 10]

5m - 20 = 10 + 12m + 40

5m - 20 = 12m + 50-5m -5m

-20 = 7m + 50-50 -50-70 = 7m

7 7

x = -10