What Is a Two-Column Proof?

A proof is an argument where nothing is taken on faith. Every statement you write comes with a reason that justifies it, and every line follows from the ones already above it. The two-column format keeps you honest: statements go on the left, and the reason for each goes on the right.

You have actually been writing proofs for years without calling them that. Every time you solve an equation, each step is justified by a rule you are allowed to use — a Property of Equality. Writing it in two columns just makes those reasons explicit.

Solving an equation is a proof

Take $2x + 5 = 13$. To find $x$ you subtract $5$ from both sides, then divide both sides by $2$. Each move is legal for a reason:

The Given is the starting point — the equation you were handed. The Subtraction Property of Equality says you may subtract the same amount from both sides; the Division Property says you may divide both sides by the same nonzero number. Line by line, the reasons force the conclusion $x = 4$ — you are not just showing the answer, you are proving it must be so.

Build the proof

Read the Given and Prove at the top, then, for each statement, choose the reason that justifies it. A line only locks into place when it genuinely follows from what is already proved above it — so the argument stays airtight from top to bottom.