What an Equation Is

You have spent this year writing expressions like 3x + 2 and simplifying them. An expression is a recipe: it tells you what to do with x, but it never asks a question. Put an equals sign and another value on the end and everything changes. 3x + 2 = 14 is a claim — it says these two things are worth the same. That claim is only true for one particular value of x, and finding it is what solving means.

The best picture for an equation is a set of balance scales. x + 3 = 8 means the left pan and the right pan weigh exactly the same. That picture gives you the single rule that governs all of this: whatever you do to one side, you must do to the other. Take 3 grams off the left pan only and it tips — the two sides are no longer equal, and your equation has become false. Do it to both and the balance holds.

Solving an equation means getting x by itself on one side, because once it stands alone the other side simply tells you what it is worth. In x + 3 = 8 the x has a 3 stuck to it, so take 3 away — from both pans. The left becomes just x, the right becomes 5, and you have your answer: x = 5. Notice you did not guess. You peeled away what surrounded x until it stood alone.

How do you peel? With the inverse — the operation that undoes the one you can see. If something is stuck to x by adding, subtract it. If by subtracting, add it. If x has been multiplied, divide; if divided, multiply. These are the same inverse pairs you have used since primary school, now doing a job in algebra. Look at what is being done to x, then do the opposite to both sides.

Here is the habit worth building from the very first equation you solve: put your answer back in and check. You found x = 5 for x + 3 = 8, so substitute: 5 + 3 is 8, and the right side is 8. Both sides match, so you are right. This is exactly the substitution you learned earlier this year, and it means you never have to wonder whether an answer is correct — you can prove it yourself in five seconds.