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.
 Examples with SolutionsExample 1
Solve the linear equation
 2(x + 3) = x + 6Solution 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  7Example 2
Solve the linear equation
3(x  6) = 3x  23Solution 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  7Example 3
Solve the linear equation
7(x  6)  3x  3 = 3(x + 5)  2xSolution 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)  xExample 4
Solve the linear equation
2(x  6) / 7  (x  3) / 2 =  xSolution 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.
