What actually is a linear equation?
Every time you figure out how many months until two phone plans cost the same, or how long until two runners are at the same point on a track, you're already thinking in linear equations , you just haven't written them down yet.
A linear equation is any equation where the variable appears only to the first power and never gets multiplied by itself or another variable. So , , , and all qualify. The moment you see or , you've left linear territory.
The equals sign is the whole point , it tells you the two sides are perfectly balanced, like a set of scales. A solution is whatever value of the variable keeps that balance true when you substitute it in. Two equations are called equivalent equations if they share exactly the same solution , they might look different, but they're describing the same balance point.
Here's the counterintuitive bit: you can rewrite an equation in dozens of different-looking ways and still be talking about the exact same solution. The art of solving is choosing rewrites that make the answer obvious.
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