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Rule

# Properties of Equality

The Properties of Equality are rules that allow to manipulate an equation in a way that an equivalent equation is obtained. This permits to isolate the unknown variable and find the solutions to the original equation. Some of these properties are shown below.

Property of Equality Algebraic Representation
Reflexive $a=a$
Symmetric If $a=b,$ then $b=a$
Transitive If $a=b$ and $b=c,$ then $a=c.$

Even though some of the properties presented above may seem obvious or ineffective on their own, together they define what is called an equivalence relation, and they are needed to formally define the rest of properties. The remaining Properties of Equality are presented below.

Property of Equality Algebraic Representation
Addition If $a = b,$ then $a + c = b + c$
Subtraction If $a = b,$ then $a - c = b - c$
Multiplication If $a = b,$ then $a \cdot c = b \cdot c$
Division If $a = b$ and $c \neq 0,$ then $\dfrac{a}{c} = \dfrac{b}{c}$
Substitution If $a = b,$ then $a$ may be replaced by $b$ in any expression containing $a$ and vice versa.

All these properties can be used together with inverse operations to solve equations.