Solving equations:
keep the scale
balanced.

What an equation actually is.

Let's name the parts, because once you can name them, the rest gets easy. An equation is a statement that two things are equal — it always has an equals sign with a left side and a right side, and it promises they're worth the same amount. So x + 5 = 12 reads: "some mystery number, plus 5, is worth exactly 12."

That mystery number has a name too: the unknown. It's the value we don't know yet, and we give it a letter — usually x — so we can talk about it before we've found it. The whole job of solving is to discover what number x must be for the equation to be true. There's only one number that fits x + 5 = 12, and the balance is going to help us hunt it down.

One thing worth clearing up straight away, because it trips people up for years: the letter x is not a mysterious code or a secret you're supposed to already know. It's simply a placeholder — a box waiting to be filled with a number. When a recipe says "add the same number of eggs as cups of flour," it's using a placeholder too; algebra just gives the placeholder a short name so we can do arithmetic with it before we know its value. There's nothing magic about the letter, either. People write x out of habit, but n, t, or a little drawing of a star would work exactly the same. The letter is a nickname for "the number we're hunting."

A quick test of whether you've understood: in the equation 3x = 12, what's the left side, and what's the unknown? (The left side is 3x, meaning "three lots of x"; the unknown is x; and the right side is 12. And notice there's no plus or times sign written between the 3 and the x — in algebra, a number written right against a letter always means "multiply.") Hold that vocabulary loosely in your head — left side, right side, equals, unknown — and let's go make a scale tip.

Find the value that makes it balance.

Here's the equation x + 3 = 8 drawn as a real scale. On the left pan sits the unknown x together with 3 little weights; on the right sit 8 weights. Drag the slider to try different values for x. Too small and the left pan floats up; too big and it crashes down. The answer is the value that makes the beam sit perfectly flat — because that's the only number that makes both sides truly equal.

But the tipping scale isn't just a clumsy first attempt — it's quietly teaching you what "solving" even means. The solution to an equation is the single value of the unknown that makes the two sides genuinely equal: the one number that lets the beam rest flat. Every other value tips it. So when we say "solve x + 3 = 8," we're really asking "which value of x balances this scale?" Keep that picture in your mind for the next demo, because we're about to find that value without any guessing at all — by rearranging the scale itself.

Every operation has an "undo".

Think of the operations around x as knots tied around the unknown. To set x free, you untie each knot — and untying is just doing the opposite operation. Maths calls the opposite an inverse operation, but "undo" is exactly the right word for it.

They come in pairs, and the pairs are friendly and obvious once you say them out loud. Tap each one to see its undo.

So if x has had 5 added to it, you subtract 5. If x has been multiplied by 3, you divide by 3. If x has had 5 taken away — as in x − 5 = 7 — you add 5 back to both sides, and out pops x = 12. Each undo strips one layer off the unknown. Get the undos right and solving becomes almost mechanical.

Why does subtracting 5 from both sides leave x perfectly alone on its side? Because adding 5 and then subtracting 5 cancel out completely — they take you on a round trip back to where you started, like walking five steps forward and five steps back. The "+ 5" and the "− 5" undo each other and vanish, leaving just x behind. That's the engine under the whole method: an operation and its inverse, side by side, simply disappear, and whatever's left is your answer. The skill you're building isn't really arithmetic — it's reading an equation backwards, spotting what was done to x and calmly undoing it.

Solve it — and watch the scale stay dead level.

This is the heart of the page. Pick an equation, then choose a move. Each move does the same undo to both pans at once, so the scale never tips — yet every move strips away one more thing standing between you and x. Keep going until the unknown is alone on the left and its value is sitting on the right. The big blue tile is x; the small tiles are single units.

Run x + 4 = 11 first: one move (subtract 4 from both sides) and you're done — that's a one-step equation, because a single undo frees x. Then try 3x = 12: here x has been multiplied, so you divide both sides into 3 equal groups and keep one group. Finally tackle 2x + 3 = 11 — that one needs two undos, and the order matters, which is exactly where we're headed next.

Pay special attention to the divide move on 3x = 12, because it's the one people find slipperiest. On the left pan you have three identical x-tiles; on the right, twelve units. The equation says those two piles weigh the same, so each single x must be worth one-third of the right pile. "Dividing both sides by 3" is exactly that: share the right pan fairly into three equal groups, and keep one group to stand beside one x. Twelve shared into three is four, so x = 4. You didn't shrink x — you just discovered how much one of them was worth all along. And because you split both sides into three, the scale stayed honest while you did it.

Two-step equations: untie the knots in a smart order.

A two-step equation like 2x + 3 = 11 has two things wrapped around x: it's been multiplied by 2 and had 3 added. To free x you need two undos — and the neat trick is to peel them off in reverse, like taking off your shoes before your socks. The "+ 3" is the outermost layer, so you undo it first: subtract 3 from both sides to get 2x = 8. Then undo the "× 2": divide both sides by 2 to get x = 4.

Why reverse order? Think about how the equation was built in the first place. To make 2x + 3, someone started with x, multiplied by 2, and then added 3 — the adding happened last, so it's sitting on the outside, like the wrapping paper on a present. To unwrap, you peel the outermost layer first. That's why you subtract the 3 before you touch the multiplication: you're undoing the steps in the opposite order they were done, the same way you take off your coat before your jumper even though you put the jumper on first.

Does the order really matter? Try it both ways below. Here's the honest truth: both orders are completely legal — the scale stays balanced either way — but one of them keeps you in tidy whole numbers, and the other drags you through fractions. Tap each button to see.

Always check: put your answer back in.

Here's a superpower that almost nobody uses but everybody should: once you think you've found x, substitute it back into the original equation — that just means swapping the letter for your number — and see whether both sides really do come out equal. If they balance, you're right. If they don't, you've caught your own mistake before anyone else does.

This is the one habit that separates people who are nervous about algebra from people who are quietly confident. When you can check your own work, you never have to wonder whether you got it right — you can simply prove it to yourself in a few seconds. Slide the answer in below and watch the two sides settle. When they land on the same number, the original scale would sit perfectly level, and that's your guarantee. Pick an equation and try both a right answer and a wrong one, so you can feel the difference.

The trap: "just move it over and flip the sign."

You may have heard a shortcut: "to solve x + 5 = 12, move the 5 to the other side and flip its sign, so x = 12 − 5 = 7." And here's the thing — the answer is right. But if that's all you know, you're one slip away from disaster, because you're chanting a spell instead of understanding a balance. The moment an equation looks a little different, the spell misfires.

See the connection? "Move it over and flip the sign" is just a shorthand for "do the same thing to both sides" — it's the same maths, with the reasoning hidden. The danger is using the shorthand without the picture. With 2x = 12, a sign-flipper might write x = 12 − 2 = 10 (wrong — the 2 was multiplying, so you divide, giving x = 6). Someone who pictures the balance never makes that mistake, because they ask "what was done to x, and what's the undo?" Learn the balance first; let the shortcut be the fast version you've earned.

From a story to an equation.

Equations aren't just rows of symbols in a textbook — they're what you get when you translate a real question into maths. The trick to the translation is small but it changes everything: let x stand for "the thing the story is asking about," and then write down, in symbols, exactly what the story tells you. "A friend gives you 7 more" becomes "+ 7." "Three identical stacks" becomes "3 ×". The words turn into operations, and the operations turn into a scale you already know how to balance.

Read each riddle, decide what x stands for, then tap to reveal the equation it hides and its answer. Try to set up the equation yourself before you peek — that setup is the genuinely hard part, and the part worth practising.

That's the real reason this matters. Once a messy situation becomes a clean equation, you don't have to be clever or lucky — you just balance the scale and let the method hand you the answer. Marbles, money, distances, recipes, the speed of a falling ball: behind an astonishing amount of the world sits a quiet little equation, waiting for someone who knows how to keep it level.