Algebraic Manipulation Mastery

Intro to Proofs · Axiom Academy

Algebraic Manipulation Mastery Every algebraic step in a proof has to keep the equation true — and you have to be able to say why. An equation is a balance — keep it level You've solved equations for years. The reason all that algebra is allowed comes down to one idea: an equation is a balance scale, and a move is only legal if it leaves the scale balanced. Watch one legal move happen, then make some yourself. Here is x + 2 = 7 . Watch what "subtract 2 from both sides" does: the same weight lifts off each pan at the same time, so the beam never tips — and the equation lands on the answer, x = 5. Same operation, both sides — the balance is preserved, and the truth comes through untouched. Start from x + 2 = 7 again. Try each operation and watch the scale. If you change both sides the same way, the beam stays level — that move is legal. Change only one side and it tips: that move is off-limits in a proof. An operation is legal exactly when it's applied to both sides — that's the whole rule of equation solving. Strip away matching pieces from both sides Here's 2x + 3 = x + 5 as tiles — a blue tile is an x, a green tile is a 1. Removing the same tile from both sides keeps the picture balanced. Peel off one x, then 2 ones, and what's left reads off the answer. Each matched removal is the same legal move — subtracting equal amounts from both sides — done where you can watch it. From "do the algebra" to "justify every step"

This is the written version of the interactive lesson above. See the full Intro to Proofs course.