Algebraic Proof & Justifying Each Step

The properties of equality

The operation properties justify the moves you make while solving. The Addition and Subtraction Properties of Equality let you add or subtract the same amount on both sides. The Multiplication and Division Properties let you multiply or divide both sides by the same nonzero number. The Distributive Property rewrites $a(b + c)$ as $ab + ac$.

The Substitution Property lets you replace a quantity with anything equal to it. It is the reason you can plug a known value into a later equation.

Reflexive, symmetric, and transitive

Three more properties describe equality itself. Reflexive: $a = a$ — anything equals itself. Symmetric: if $a = b$, then $b = a$ — an equation can be read in either direction. Transitive: if $a = b$ and $b = c$, then $a = c$ — equality chains through a shared middle quantity.

These three matter in geometry because segment lengths and angle measures are numbers. “If $AB = CD$ and $CD = EF$, then $AB = EF$” is the Transitive Property at work on segment lengths.

The table below sums up the three properties of equality itself.

Writing the proof

Set up two columns. The first line is the given equation, with reason “Given.” Each following line shows the equation after one move, and the reason names the property that justifies that move. The last line is the solved equation.

One move per line. If you subtract $7$ and divide by $3$ in the same line, the reader cannot check either move — and the reason column has no single property to point to.

Worked examples

Try one yourself

Common questions