Forming Equations

Here is a secret about algebra that school rarely says out loud: the hard part was never solving the equation. Once you have 2x + 7 = 19 in front of you, the moves are mechanical — you already know them. The real skill, the one that pays off for the rest of your life, is turning a sentence into that equation in the first place.

Try this riddle: "I think of a number, double it, add 7, and get 19." Read it slowly, one phrase at a time. "A number" — call it x. "Double it" is 2x. "Add 7" makes 2x + 7. "And get 19" means the whole thing equals 19:

2x + 7 = 19

That leap — from muddled English to one clean line of maths — is called forming an equation, and it is the most creative, most useful thing algebra can teach you. Solve it (subtract 7, then halve) and you get x = 6. The riddle is cracked.

The five-step method

Every "wordy" problem yields to the same routine. Learn it once and word problems stop being scary.

Name the unknown. Pick a letter and say exactly what it stands for — including its units. "Let d be the distance in miles."
Translate each phrase into maths, in the order the sentence gives them.
Build the equation — the word that means "equals" (is, gives, costs, totals) is the = sign.
Solve it with the moves you already know.
Interpret the answer back in the real situation — and check it makes sense.

Reading the story phrase by phrase is really just turning words into algebra, and the equals sign is the word that finishes the sentence. Once it is an equation, you already know how to undo each step to find the answer.

See it built

Watch the words appear, then the equation form underneath them one phrase at a time — and finally get solved.

Worked example 1 — the taxi fare

A taxi charges a £3 flag fee just for getting in, then £2 for every mile. Your fare came to £15. How far did you travel?

Name it: let m be the distance in miles.
Translate: the £3 is fixed; the mileage part is 2m pounds; "came to £15" is the equals sign.
Build: 3 + 2m = 15.
Solve: subtract 3 → 2m = 12; divide by 2 → m = 6.
Interpret: the journey was 6 miles. Check: 3 + 2\times 6 = 15. It fits.

Worked example 2 — a rectangle's sides

A rectangle is 3 cm longer than it is wide. Its perimeter is 26 cm. Find its width.

Name it: let the width be w cm. Then the length, "3 cm longer", is w + 3.
Translate: perimeter is two widths plus two lengths: 2w + 2(w+3).
Build & tidy: 2w + 2(w+3) = 26, which opens to 4w + 6 = 26.
Solve: subtract 6 → 4w = 20; divide by 4 → w = 5.
Interpret: width 5 cm, length 8 cm. Check: 5 + 8 + 5 + 8 = 26.

Worked example 3 — sharing out

Three friends share £90. Bina gets twice as much as Ali, and Cara gets £10 more than Ali. How much does Ali get?

Name it: let Ali's share be a pounds. Then Bina has 2a and Cara has a + 10.
Build: the three shares total £90: a + 2a + (a + 10) = 90.
Solve: 4a + 10 = 90; subtract 10 → 4a = 80; divide by 4 → a = 20.
Interpret: Ali £20, Bina £40, Cara £30 — and 20 + 40 + 30 = 90.

Word order in English does not always march in step with the maths, so read slowly. Three traps catch almost everyone:

Translate the structure, not the word sequence. "5 more than double a number" is 2x + 5 — you double first, then add 5. It is not 2(x+5) (that doubles the sum) and not 5x + 2 (that swaps the numbers). When you mean "double the result of adding 5", that is when brackets appear.
Say what your letter is — with units. "Let x = the number of children" is fine; a bare "let x = children" leaves you unsure later whether x is a count, a cost, or a length.
Sanity-check the answer. If solving gives -4 people, or a cake shared into 0.5 guests, something is wrong — a negative number of people is nonsense. Re-read the sentence and re-translate.

Because forming equations is exactly what scientists, engineers and economists do all day. A physicist watching a rocket, an economist watching prices, a game programmer watching a bouncing ball — each starts with a messy real situation and turns it into clean symbols they can actually solve. That translation step is the job.

So the dreaded "word problem" is really the most authentic maths there is: pure equation-solving with the numbers handed to you is the artificial version. Being able to look at a tangled real situation and write down the one equation that captures it is a genuine superpower — and it is the entire point of algebra.

See it explained

Sal Khan turns word problems into equations and then solves them.