Unit 1 · Lesson 1.1

Variables and the meaning of equations

You already know how to work with numbers. Algebra adds just one new idea: sometimes a number is hidden, and we use a letter to stand for it until we track it down.

Start with the equals sign, because almost everything ahead leans on reading it the right way. The equals sign means the same value: whatever is on the left is worth exactly what's on the right.

Here's a picture for it. Imagine an old balance scale with two pans. If the two pans rest level, the weights on them are equal; that's all "=" claims, that the two sides weigh the same.

So read x + 3 = 7 as that scale resting level: the left pan holds x and 3, the right pan holds 7, and they balance. Here is the one rule to remember: whatever you do to one pan, you do to the other, or it tips.

Reading "=" this way turns a string of symbols into a sentence. On its own, "4 + 3" is just a phrase, the name of a number, the way "the cost of two coffees" names an amount; it doesn't say anything yet. Drop an "=" in and you get a full sentence, with "=" acting as the verb: "4 + 3 = 7" claims the left side and the right side have the same value. A claim like that can be true, false, or, when a piece is still hidden, neither yet. Try reading each of these as a balance and saying which it is:

• 7 = 4 + 3. The right pan is 7, the left pan is 7. They match, so it's true. The single number sits on the left, which can look "backwards" until you remember = just means same as.
• 8 = 8. Nothing to work out, both pans already weigh 8. Still true.
• 5 + 2 = 9. The left pan is 7, the right is 9. They don't match, so it's false.
• □ + 3 = 5 + 7. The right pan is 12, and the left pan won't balance it until the box holds 9. Until you fill the box, it's neither true nor false. It's open.

That box is where algebra begins. Instead of an empty box we usually write a letter, say x, and call it a variable: a closed box with a number hidden inside that we don't know yet.

So "3 + x = 7" asks the balance question again: what number in the box keeps the two sides equal?

There's one way to answer it that keeps working no matter how unfriendly the numbers get, and it's the move the rest of the course is built on: undo what was done to x. The 3 is added onto x, so undo it by subtracting 3. To keep the scale level you subtract from both pans, and when you take 3 away from x + 3, the +3 goes to zero, leaving x by itself.

$$\begin{aligned} &x + 3 = 7 \\ &\xrightarrow{\;-3 \text{ from both sides}\;}\; x = 4 \end{aligned}$$

With small numbers you could also just ask the question out loud, "what plus 3 is 7?", and land on four. That's a fine quick check, and here it agrees. But undoing is the method to lean on, because it still works when the numbers don't let you guess.

Now do the one move worth building into a habit today: check it. Put your answer back into the original and see if both sides match. Here x = 4, so the left side is 4 + 3 = 7, which matches the right. So x = 4 is correct.

If a check ever doesn'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.

Notice one small thing in passing. The rule "+3" took an input and turned it into an output: feed in 4, out comes 7. An equation pins down the output and asks you for the input. That input-to-output idea gets its own name later in the course; for now just notice it's there.