Unit 2 · Lesson 2.1

Inverse operations & one-step equations

Every equation you'll ever solve is built from one move, repeated. Once you can solve a one-step equation, the longer ones ahead are just a few of these single moves in a row. So this is the piece worth getting solid.

Here's the idea, in something you can picture. Imagine a covered cup on a balance scale, sitting next to 5 coins, and the whole thing balancing 12 coins on the other pan. You want to know how much the cup weighs.

If you lift 5 coins off the left pan, the scale tips. To keep it level, you also lift 5 coins off the right. Do both, and it stays balanced: now the cup alone balances 7 coins. The cup weighs 7.

That's the same thing this equation says: $$x+5=12$$ The x is the cup. The +5 is sitting right next to it, and you want x by itself, alone on one side with a number on the other. (Getting x alone like this has a name: you're isolating the variable.)

To peel the +5 away, do the opposite of adding 5: subtract 5. That's the inverse operation, the move that undoes another. Adding and subtracting undo each other, and multiplying and dividing undo each other. And to keep the scale level, subtract 5 from the other side too: $$x+5=12 \;\xrightarrow{\,-5\text{ both sides}\,}\; x=7$$ When you subtract 5 from x + 5, the +5 goes to zero, and x is left standing alone. Notice what didn't happen: nothing got cancelled by magic. The +5 went to zero because you subtracted exactly 5.

One move left, and it's the one to build into a habit now: check it. Put your answer back into the original equation and see if both sides match. Here x = 7, so the left side is 7 + 5 = 12, which matches the right. So x = 7 is the solution, the value that makes the equation a true statement.

That check is your safety net for the whole book. If the two sides ever don't match, you haven't failed. Your check just did its job and caught something before it counted. That's exactly what it's for. Go back to your first step and re-run the arithmetic slowly. A mismatch is almost always one sign or one small slip, not the whole method.

The same move handles every one-step equation; only the operation you undo changes. If x has something subtracted, add it back. If x is multiplied, divide. If x is divided, multiply. The picture stays the balance scale: whatever you do, do it to both pans.