# Free Mathematics Tutorials # Solve Linear Equations with Parentheses Using the Distributive Law

Tutorials with detailed solutions and matched exercises on solving linear equations with parentheses are presented. Detailed solutions and explanations ( in red) are provided. Parentheses are eliminated from the equation by using the distributive law in algebra used as follows.

a ( b + c) = a � b + a � c
where a, b and c are real numbers or variables.

## Examples with Solutions

### Example 1

Solve the linear equation
- 2(x + 3) = x + 6

#### Solution to Example 1

• given
- 2(x + 3) = x + 6
• Use the distributive law to multiply factors in left term to eliminate parentheses.
- 2(x) - 2(3) = x + 6
• Simplify
- 2x - 6 = x + 6
• add 6 to both sides
- 2x - 6 + 6 = x + 6 + 6
• group like terms
-2x = x + 12
• subtract x to both sides
-2x - x = x + 12 -x
• group like terms
-3x = 12
• multiply both sides by -1/3
x = -4
• Check the solution
left side:-2(-4 +3) = 2
right side:-4 + 6 = 2
• Conclusion
x = -4 is the solution to the given equation

#### Matched Exercise 1:

Solve the linear equation
- 3(-x +3) = x - 7

### Example 2

Solve the linear equation
-3(-x - 6) = 3x - 23

#### Solution to Example 2

• given
-3(-x - 6) = 3x - 23
• The distributive law is used to multiply factors in left term to eliminate parentheses
- 3 (- x) - 3 (-6) = 3x - 23
• Simplify
3 x + 18 = 3 x - 23
• subtract 18 to both sides
3x + 18 - 18 = 3x - 23 - 18
• group like terms
3x = 3x - 41
• subtract 3x to both sides
3x - 3x = 3x - 41 -3x
• group like terms
0x = - 41
• As you can see no real value for x can satisfy the above equation, the above equation has no solutions.

#### Matched Exercise 2:

Solve linear the equation
4(-x +3) = -4x - 7

### Example 3

Solve the linear equation
-7(x - 6) - 3x - 3 = 3(x + 5) - 2x

#### Solution to Example 3

• given
-7(x - 6) - 3x - 3 = 3(x + 5) - 2x
• The distributive law is used to multiply all factors to eliminate parentheses multiply factors
-7x + 42 - 3x - 3 = 3x + 15 - 2x
• group like terms
-10x + 39 = x + 15
• subtract 39 to both sides
-10x + 39 - 39 = x + 15 -39
• group like terms
-10x = x - 24
• subtract x to both sides
-10x - x = x - 24 -x
• group like terms
-11x = - 24
• multiply both sides by -1/11
x = 24/11
• Check the solution
left side:-7(24/11 - 6) - 3(24/11) - 3 = 189/11
right side:3(24/11 + 5) - 2(24/11) = 189/11
• Conclusion
x = 24/11 is the solution to the given equation

#### Matched Exercise 3:

Solve the equation
-5(x - 4) - x + 23 = 5(x - 5) - x

### Example 4

Solve the linear equation
-2(x - 6) / 7 - (x - 3) / 2 = - x

#### Solution to Example 4

• It can be noted that this equation has rational expressions. The first step is to eliminate the denominators by multiplying by the LCD
-2(x - 6) / 7 - (x - 3) / 2 = - x
• The LCD is equal to 7*2 = 14. Multiply both sides of the equation by the LCD.
14 * [-2(x - 6) / 7 - (x - 3) / 2] = 14* [ - x ]
• Simplify to eliminate the denominator.
-4(x - 6) - 7(x - 3) = -14x
• Multiply factors and group like terms
- 11 x + 45 = - 14 x
• subtract 45 to both sides
- 11 x + 45 - 45 = - 14 x - 45
• group like terms
-11x = -14x - 45
• add 14x to both sides
-11x + 14x = -14x - 45 +14x
• group like terms
3x = -45
• multiply both sides by 1/3
x = -15
• Check the solution
left side:-2(-15 - 6) / 7 - (-15 - 3) / 2 = 15
right side:-(-15) = 15
• Conclusion
x = -15 is the solution to the given equation

#### Matched Exercise 4:

Solve the equation
-3(x + 4)/4 - x - 2 = (x - 4)/3 - x