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.

Examples with Solutions

Example 1

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:

Example 2

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:

Example 3

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:

Example 4

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:

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