Definition

The balance method uses the idea of a balance scale to solve equations: whatever you do to one side of the equation, you must do the exact same thing to the other side to keep it balanced. Also called the properties of equality, it applies the addition, subtraction, multiplication, and division properties of equality systematically, each step justified by a specific property, which trains mathematical reasoning and proof-writing skills. Formally, the balance method rests on the substitution property of equality (if $a = b$, then $a$ may replace $b$ in any expression) and on equality being an equivalence relation: reflexive ($a = a$), symmetric (if $a = b$ then $b = a$), and transitive (if $a = b$ and $b = c$ then $a = c$); solving an equation this way is a constructive proof that the solution set is non-empty, and each step is a biconditional, meaning the final solution set equals the original one with no extraneous or lost solutions.