When you solve an equation, you use a process called deductive reasoning where you apply ideas that you know to be true. For example, you know that if you add the same number to both sides of an equation, the equation will not be changed. Many of the "ideas" that you use when solving are, in actuality, the mathematical properties (rules) that we saw in Real Numbers and Properties.

 
Using real number properties
(such as the commutative, associative, and distributive properties)
and the properties of equality
(such as adding, subtracting, multiplying and dividing by a non-zero)
judge
justify why each step in the process of solving a linear equation is legal!


Solve for x and justify each step with a reason: 3(x - 2) + 5x = 9x - 24
Steps:
Justification (Reasons):
3(x - 2) + 5x = 9x - 24
Given
3x - 6 + 5x = 9x - 24
Distributive Property
3x + 5x - 6 = 9x - 24
Commutative Property of Addition
8x - 6 = 9x - 24
Combine Like Terms
8x - 8x - 6 = 9x - 8x - 24
Subtraction Property of Equality
0 - 6 = x - 24
Additive Inverse Property (left)
Combine Like Terms (right)
-6 = x - 24
Additive Identity Property
-6 + 24 = x - 24 + 24
Addition Property of Equality
18 = x + 0
Addition (left)
Additive Inverse Property (right)
18 = x
Additive Identity Property

The justification method shown above is an example of one method.
There are other justification methods that are deemed acceptable.

Ask your teacher how you will be expected to write your justifications.

The justification below shows a more intense break-down of the concepts being used.

Steps:
Justification (Reasons):
3(x - 2) + 5x = 9x - 24
Given
3x - 6 + 5x = 9x - 24
Distributive Property
3x + 5x - 6 = 9x - 24
Commutative Property of Addition
x(3 + 5) - 6 = 9x - 24
Distributive Property in reverse
x(8) - 6 = 9x - 24
Addition
8x - 6 = 9x - 24
Commutative Property of Addition
8x - 8x - 6 = 9x - 8x - 24
Subtraction Property of Equality
x(8 - 8) - 6 = x(9 - 8) - 24
Distributive Property in reverse
x(0) - 6 = x(1) - 24
Additive Inverse Property (left side)
Subtraction (right side)
0 - 6 = x - 24
Zero Property of Multiplication (left side)
Multiplicative Identity (right side)
- 6 = x - 24
Additive Identity Property
-6 + 24 = x - 24 + 24
Addition Property of Equality
18 = x + 0
Addition (left side)
Additive Inverse Property (right side)
18 = x
Additive Identity Property



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