Unit 2

Solving Linear Equations

One- and two-step equations, like terms, distributive property, variables on both sides, fractions in equations

This unit is about finding a hidden number when an equation tells you what it has to equal. It's the everyday workhorse of algebra, and once you have it, a lot of what comes later is just this same skill in longer form. It helps to have your work with negative numbers fresh, since a few answers here land below zero.

This unit turns the question "what's the hidden number?" into a steady, repeatable procedure. It rests on one picture: an equals sign means the two sides balance, like a level scale. Every move you'll make is really the same move. You do the same thing to both sides so the scale stays level.

A word that names what these equations have in common. A linear equation has its variable only to the first power. No x², no x in the bottom of a fraction, no two variables multiplied together. That's why they almost always have a single answer. A couple of them turn out to have no answer, or to be true for every number, but those are the rare cases.

Two habits run through every example here, and they're worth adopting from the first page. The first is to check by substituting back. Once you find x, put it into the original equation and see that both sides match. It's your safety net, and it's a habit you'll carry into every later unit.

The second is a way of talking about the moves that keeps them honest. When a +5 leaves, it doesn't "cancel". It goes to zero, because you subtracted 5. When a coefficient leaves, it goes to one, because you divided. Saying it that way keeps the symbols from feeling like they vanish by magic.

Before you start a new lesson, redo two or three problems from a lesson or two back from memory. That short warm-up keeps the older moves sharp while you add new ones.