Sole 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.

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 3:

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
    -11x + 45 = -14x
  • subtract 45 to both sides
    -11x + 45 - 45 = -14x -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

More references and links

on how to Solve Equations, Systems of Equations and Inequalities.