Algebraic Properties of Equality

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The equal sign that depicts the fact that both sides of it are equal is a very strange symbol with many properties. It tells you various traits of each side, and it allows you to manipulate each side in specific ways. Here are the different properties of that sign:

Property | Definition | Example

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Reflexive  | For all a, a=a
Symmetric  | If a = b, then b = a
Transitive  | If a = b & b = c, then a = c  | If 8 = (4)(2) and (4)(2) = (2)(4), then 8 = (2)(4)
Substitution  | If a = b, then one can replace a with b.  |  x − 6 = 14 {\displaystyle x-6=14\,}

x = 20 {\displaystyle x=20\,}

Subtraction  | You can subtract one number from both sides

x + 6 − 6 = 14 − 6 {\displaystyle x+6-6=14-6\,}

x = 8 {\displaystyle x=8\,}

Multiplication  | You can multiply both sides by a same number

x = 108 {\displaystyle x=108\,}

Division  | You can divide both sides by a same number different from zero

x = 3 {\displaystyle x=3\,}