Solving Linear Equations - Tutorial

Tutorials with detailed solutions and matched exercises on solving linear equations are presented. Detailed solutions and explanations ( in red) are provided.




Example 1: Solve the linear equation

-2(x + 3) = x + 6

Solution to Example 1

  • given
    -2(x + 3) = x + 6

  • multiply factors in left term
    -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

  • multiply factors in left term
    3x + 18 = 3x - 23

  • subtact 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

  • 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.







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Updated: 2 April 2013

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