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