Solve Linear Equations with Parentheses Using the Distributive Law

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

More references and links

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