Two-Step Equations

Picture an old set of kitchen scales, perfectly level. On the left pan sit three identical mystery boxes and five loose marbles; on the right pan sit twenty marbles. The pans balance, so the two sides weigh exactly the same. Written down, that balance is an equation:

3x + 5 = 20

Each mystery box holds the same secret number of marbles — call it x. The whole game of algebra is to work out that number without ever tipping the scales. The golden rule is simple: whatever you do to one side, you must do to the other. Take three marbles off the left and the pan drops — so you take three off the right too, and it stays level.

Undo the operations — but backwards

Look at how x got buried in 3x + 5 = 20. Starting from the box, the machine did two things: first it multiplied by 3, then it added 5. To dig x back out you run the machine in reverse — and reverse means last thing first:

Undo the +5 first — subtract 5 from both sides.
Then undo the ×3 — divide both sides by 3.

It is exactly like getting undressed: shoes went on last, so shoes come off first. This is nothing more than a one-step equation done twice in a row — and the order is order of operations played backwards.

See it solved

Watch 2x + 3 = 11 come apart one line at a time. Whatever we do to the left, we do to the right too — so the scales stay balanced. Step through it.

Worked example 1 — solve the scales puzzle

Back to our balancing pans, 3x + 5 = 20. Undo in reverse:

3x + 5 = 20

3x + 5 - 5 = 20 - 5 \quad\Rightarrow\quad 3x = 15

\frac{3x}{3} = \frac{15}{3} \quad\Rightarrow\quad x = 5

Each mystery box holds 5 marbles. Now the most important habit in all of algebra: check it. Put x = 5 back into the original equation and make sure both sides come out equal:

3(5) + 5 = 15 + 5 = 20 \;\checkmark

Both sides make 20, so the scales really do balance. The answer is right.

A story hiding an equation

Two-step equations aren't just rows of symbols — they are everyday situations in disguise. A taxi charges a £3 flag-fall to get in, then £2 for every mile. Your ride cost £17. How many miles was it?

Let x be the number of miles. The cost is "£2 a mile, plus the £3 to start", which is exactly a two-step equation:

2x + 3 = 17

Undo the +3 (subtract 3 → 2x = 14), then undo the ×2 (divide by 2 → x = 7). Seven miles. The exact same balancing move that cracks a bare equation also answers a real question — that is why this skill is worth owning.

Worked example 2 — subtraction, then division

The two steps do not have to be "add" and "multiply". Here the box was multiplied by 5 and then 7 was taken away:

5x - 7 = 8

Undo the −7 first (so add 7), then undo the ×5 (so divide by 5):

5x - 7 + 7 = 8 + 7 \quad\Rightarrow\quad 5x = 15

\frac{5x}{5} = \frac{15}{5} \quad\Rightarrow\quad x = 3

Check: 5(3) - 7 = 15 - 7 = 8 — matches, so x = 3.

Worked example 3 — the answer can be negative

Nothing says x has to be a friendly positive number. Solve:

2x + 11 = 3

2x + 11 - 11 = 3 - 11 \quad\Rightarrow\quad 2x = -8

\frac{2x}{2} = \frac{-8}{2} \quad\Rightarrow\quad x = -4

A negative answer is perfectly fine — the balancing rules never changed. Check it the same way: 2(-4) + 11 = -8 + 11 = 3. It works, so x = -4.

See it explained

Sal Khan solves two-step equations by doing the same thing to both sides.

Two mistakes trip up almost everyone with two-step equations:

Wrong order. In 3x + 5 = 20 it is tempting to "divide by 3" first. But the 3 is multiplied only into the x, not into the +5 — divide the whole side by 3 and you get an ugly x + \tfrac{5}{3} = \tfrac{20}{3}. Undo the +5 first, then the ×3. Reverse order of operations: BIDMAS backwards.
Only doing it to one side. If you subtract 5 from the left but forget the right, the scales tip and the equation is no longer true. Every move happens to both sides — that is the one rule you can never break.

From a book. Around the year 820, the Persian scholar Muhammad al-Khwarizmi wrote a maths masterpiece whose title included the word al-jabr — meaning "the restoring" or "the balancing", the act of adding the same thing to both sides to put an equation back in order. That one word travelled across languages and became algebra. So every time you keep the scales level by doing the same move to both sides, you are performing the very act the whole subject was named after. (His name, worn down over centuries, also gave us the word algorithm.)

Because an equation literally is a see-saw. The = sign is the pivot in the middle, and the two sides must always weigh the same. Every trick you will ever learn — from GCSE right up to the equations engineers solve to land a spacecraft on Mars — is just a clever way of keeping the see-saw level while you shuffle things around until the unknown sits by itself.

That is why "do the same thing to both sides" is not a random rule someone invented to annoy you. It is the whole point. Master this two-step undoing and you have the foundation of all equation-solving — the same balancing move, used over and over, forever.