# 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.