Two friends set off on a long bike ride. Amara starts 9 km ahead but pedals steadily; Ben starts at the gate but rides faster and eats into her lead every hour. The natural question — when does Ben catch up? — is really a question about two things being equal. Write the distance each has travelled after x hours and set them equal, and the unknown x lands on both sides of the equals sign at once:

That is the whole story of this page. Whenever you ask "when are two changing things the same?" — two phone tariffs, two taxis, a head start being reeled in — you land on an equation with the unknown on both sides. This page teaches the one reliable trick that cracks every one of them.

Sometimes the unknown x turns up on both sides of the equals sign:

We can't divide straight away, because there's an x term left and right. The trick is to get every x onto one side first. Take away the smaller x term — here 3x — from both sides, so the equation stays balanced and no x goes negative:

On the left, 5x - 3x = 2x; on the right the 3x cancels and we're left with 10. (That tidying-up is just collecting like terms.) Now the unknown lives on one side only:

From here it's an ordinary one-sided equation — exactly like the kind we met after expanding brackets. Subtract 2 from both sides, then divide by 2:

Always strip the smaller x term from both sides to gather the unknown on one side; then solve as usual.

The recipe, in three moves

Every one of these equations yields to the same three-step routine — and each move is just a balance rule you already know:

Then check: put your answer back into the original equation and make sure both sides really do come out equal.

Worked example 1 — collect, then solve

Solve 5x + 3 = 2x + 12.

The smaller x term is 2x, so subtract 2x from both sides to gather the x on the left:

Now move the number across — subtract 3 from both sides — and divide:

Check: left side 5(3) + 3 = 18; right side 2(3) + 12 = 18. Both are 18 — it works.

Worked example 2 — a negative answer is fine

Solve 4x + 11 = 7x + 2.

Here the smaller x term is 4x, so subtract 4x from both sides — this keeps the x coefficient positive on the right:

Subtract 2 from both sides, then divide by 3:

Nothing negative there — but answers are allowed to be negative. Solve 2x + 9 = 5x + 15: gather with -2x to get 9 = 3x + 15, then -6 = 3x, so x = -2. A negative solution is not a mistake — it just means the answer sits to the left of zero.

Check the second one: 2(-2) + 9 = 5 and 5(-2) + 15 = 5. Equal — good.

Worked example 3 — brackets first

If either side hides a bracket, expand it before you collect. Solve 3(x + 2) = x + 10.

Multiply the bracket out: 3 \times x = 3x and 3 \times 2 = 6:

Now it's an ordinary both-sides equation. The smaller x term is x, so subtract x from both sides:

Check: 3(2 + 2) = 12 and 2 + 10 = 12. Equal — solved.

See it balance

Think of the equation as a balanced scale: whatever you do to one pan you must do to the other. Step through removing the smaller x term from both pans, then the number, until a single x is left. Each Refresh gives a new example.

See it explained

Sal Khan works through an equation with the unknown on both sides, gathering the variable onto one side before solving.