Academic Learning Center
Math Handout
Addition and Multiplication Properties of Equality
Addition Property of Equality – To move a term from one side of an equation to the other side, add the opposite
sign to both sides of the equation.
x + 4 = 11
-4 -4
x
= 7
To move the 4 from the left side of the equation to the right side
subtract 4 from both sides of the equation.
This leaves you with x on the left side and 7 on the right side.
In other words, x = 7.
Also, when you are solving an equation always remember to check your answer in the original equation to see if it
is the correct answer.
Check:
x + 4 = 11
(7) + 4 = 11
11 = 11
Replace x with (7) and simplify.
This is correct so the answer checks out.
Here are some more examples using the Addition Property of Equality:
a. x – 3 = 9
+3 +3
x
= 12
check:
b. 3x = 2x – 10
- 2x - 2x
x=
-10
check: 3 x = 2x - 10
3(-10) = 2(-10) -10
-30 = -20 - 10
-30 =
-30 Correct.
c.
check: -5 x = -6 x + 4
-5(4) = -6(4) + 4
-20 = -24 + 4
-20 =
-20 Correct.
–5x = -6x + 4
+ 6x + 6x
x=
4
x–3= 9
(12) – 3 = 9
9 = 9 Correct.
Academic Learning Center ~ Central Piedmont Community College ~ Central High Room 103 ~ 704-330-6474
Multiplication Property of Equality:
Case 1: To eliminate a number (coefficient) in front of a variable, divide all terms on both sides of the equation by
the number.
3x = 12
3x = 12
3
3
x = 4
To get the x by itself on the left side of the equation
divide both sides of the equation by 3.
This leaves you with x on the left side and 4 on the right side.
Check
3 x = 12
3(4) = 12
12 = 12 Correct.
Case 2: If a variable is already divided by a number, then multiply all terms on both sides of the equation by that
number.
x
=4
5
⎛ x⎞
(5)⎜ ⎟ = 4(5)
⎝5⎠
x = 20
The x is already divided by 5
So multiply both sides of the equation by 5.
The 5’s cancel on the left leaving x on the left and 20 on
the right.
Check:
x
=4
5
(20)
=4
5
4
= 4 Correct.
Case 3: If a variable has a fraction in front of it (fractional coefficient), then multiply all terms on both sides of the
equation by the reciprocal of that fraction.
3
x = 15
2
⎛ 2⎞
⎛ 2 ⎞⎛ 3 ⎞
⎜ ⎟⎜ ⎟ x = 15⎜ ⎟
⎝ 3⎠
⎝ 3 ⎠⎝ 2 ⎠
x = 10
The x has a fraction 3/2 in front of it.
So multiply both sides of the equation by the reciprocal
of 3/2 which is 2/3.
The 2/3 and 3/2 cancel each other leaving x on the left and
10 on the right.
Check:
3
x = 15
2
3
(10) = 15
2
15 = 15 Correct.
Academic Learning Center ~ Central Piedmont Community College ~ Central High Room 103 ~ 704-330-6474
Here are some examples using the Multiplication Property of Equality:
a. 7x = 21
7x = 21
7
7
x= 3
check: 7 x = 21
7(3) = 21
21 = 21 Correct.
b. 6x = 30
6x = 30
6
6
x=5
check: 6 x = 30
6(5) = 30
30 = 30 Correct.
c.
x
=3
8
⎛ x⎞
(8)⎜ ⎟ = 3(8)
⎝8⎠
check:
x = 24
d.
x
= 12
2
⎛ x⎞
(2)⎜ ⎟ = 12(2)
⎝ 2⎠
x = 24
x
=3
8
(24)
=3
8
3 = 3 Correct.
check:
x
= 12
2
(24)
= 12
2
12 = 12 Correct.
Academic Learning Center ~ Central Piedmont Community College ~ Central High Room 103 ~ 704-330-6474