Forming an equation means turning the information in a word problem into an algebraic sentence that you can solve. Instead of guessing, you identify the unknown, give it a letter, write expressions for every other quantity, and connect them with an equals sign. This six-step strategy works across all KS3 algebra contexts.

What does "forming an equation" mean?

A word problem describes a mathematical situation in plain English. Forming an equation means extracting that information and writing it as algebra. The equals sign is placed wherever the problem uses words like "is", "equals", "gives", or "the result is". Once the equation is written, standard algebraic techniques — collecting like terms, expanding brackets, dividing both sides — lead to the answer.

The critical first habit is to define your variable clearly in writing before you set up any expression. Writing "Let n = the number I am thinking of" at the top of your working is not just good presentation — it ensures that every term you write afterwards refers to a clearly understood quantity, and it earns method marks if the arithmetic later goes wrong.

How do I choose what to call x?

Choose a letter to represent whichever unknown the question asks you to find. If the problem involves two unknowns, express the second in terms of the first. Sensible choices:

  • "Let n = the number" for "I think of a number" problems.
  • "Let w = the width in cm" for perimeter or area problems.
  • "Let e = Emma's number of books" for sharing problems.

If the problem mentions consecutive integers, let the smallest be n; then the next integers are n + 1, n + 2, and so on. This keeps the expressions tidy and avoids confusion between the integers and their sum.

What is the step-by-step method?

Follow these six steps on every word-problem question.

  1. Identify the unknown and write a clear definition: "Let n = …"
  2. Write expressions for any other quantities in terms of n.
  3. Form the equation using the total, sum, or condition stated in the problem.
  4. Solve the equation using algebraic steps — show each line of working.
  5. Interpret the solution — answer the question in a sentence, not just "n = 7".
  6. Check by substituting your answer back into the words of the original problem.

The table below shows how common English phrases translate into algebraic expressions.

English phrase Algebraic expression
Four more than x x + 4
Three times as many as y 3y
Twice a number decreased by 5 2n − 5
The sum of two consecutive integers n + (n + 1)
The product of n and m nm
Half of x x ÷ 2
Is / equals / gives / the result is =
x decreased by 7 x − 7
A number squared plus 3 n² + 3

How do I handle problems with two unknown quantities?

When two unknown quantities are linked by a relationship, express the second in terms of the first so that the equation contains only one variable.

Worked example 1 — "I think of a number"

I think of a number, multiply it by 3, and subtract 7. The result is 14. Find the number.

  1. Let n = the number.
  2. Expression for the result: 3n − 7.
  3. Form the equation: 3n − 7 = 14.
  4. Solve: 3n = 21, so n = 7.
  5. Check: 3 × 7 − 7 = 21 − 7 = 14 ✓ The number is 7.

Worked example 2 — rectangle perimeter

A rectangle has a length 4 cm more than its width. The perimeter is 40 cm. Find the dimensions.

  1. Let w = the width in cm.
  2. Length = w + 4 cm.
  3. Perimeter: 2(w + w + 4) = 40 → 2(2w + 4) = 40 → 4w + 8 = 40.
  4. Solve: 4w = 32, so w = 8 cm; length = 8 + 4 = 12 cm.
  5. Check: 2(8 + 12) = 2 × 20 = 40 ✓ Width = 8 cm, length = 12 cm.

Worked example 3 — sharing between two people

Emma and Liam have 63 books between them. Liam has twice as many as Emma. How many does each have?

  1. Let e = Emma's books.
  2. Liam's books = 2e.
  3. Form the equation: e + 2e = 63 → 3e = 63.
  4. Solve: e = 21; Liam = 2 × 21 = 42.
  5. Check: 21 + 42 = 63 ✓ Emma has 21 books, Liam has 42.

How do I check my answer?

Substitute your numerical answer back into the words of the original problem, not just the equation you formed. This step catches two types of error: arithmetic slips in the algebra, and interpreting the solution incorrectly (for example, solving correctly for e but forgetting to state Liam's total when the question asks for both).

Worked example 4 — three consecutive integers

Three consecutive integers sum to 48. Find the integers.

  1. Let n = the smallest integer; the others are n + 1 and n + 2.
  2. Form the equation: n + (n + 1) + (n + 2) = 48 → 3n + 3 = 48.
  3. Solve: 3n = 45, so n = 15.
  4. Integers: 15, 16, 17.
  5. Check: 15 + 16 + 17 = 48 ✓ The integers are 15, 16, and 17.

Worked example 5 — angles in a triangle

The angles in a triangle are x°, (2x + 10)°, and (x − 5)°. Find each angle.

  1. Angles in a triangle sum to 180°.
  2. Form the equation: x + (2x + 10) + (x − 5) = 180 → 4x + 5 = 180.
  3. Solve: 4x = 175, so x = 43.75.
  4. Angles: 43.75°, 2(43.75) + 10 = 97.5°, 43.75 − 5 = 38.75°.
  5. Check: 43.75 + 97.5 + 38.75 = 180° ✓

What are the most common mistakes when forming equations?

  • Not defining the variable clearly. Writing "3n − 7 = 14" without first stating "let n = the number" loses method marks even if the algebra is correct. Make the definition the first line of every solution.

  • Forgetting to answer the actual question. Solving for n when the question asks for the length (which is n + 4) is one of the most common sources of dropped marks. Read the question again before writing your final answer.

  • Skipping the check. A substitution check takes thirty seconds and catches arithmetic slips before you commit to a wrong answer. It also shows the examiner that you understand what the solution means.

  • Choosing the more complicated expression as x. Express the simpler unknown as n and write the more complex one in terms of n. For example, in Example 3, using Emma's total (not Liam's) keeps the equation 3e = 63 rather than the messier 3(l/2) = 63.

Frequently asked questions

Why does it matter which variable I choose?

The choice of variable does not affect the final answer, but it affects how cleanly you reach it. Choosing the simpler unknown — for example, Emma's books rather than Liam's, or the width rather than the length — usually produces tidier expressions and reduces the chance of a sign error when expanding or collecting terms.

What if the equation has a fraction in it?

Multiply every term on both sides by the denominator to clear the fraction before solving. For example, if forming the equation gives n/2 + 3 = 10, multiply through by 2 to get n + 6 = 20, then solve to give n = 14. This avoids fractions in subsequent steps.

My answer is a decimal — have I made a mistake?

Not necessarily. As Worked example 5 shows, angle problems in particular can produce non-integer answers. As long as the check works — the angles sum to 180° — the decimal answer is correct. If a check fails, look for an arithmetic slip in your equation formation or simplification step.

Can I use any letter instead of n or x?

Yes. The letter is simply a label for the unknown. Many students use a letter that reflects the quantity — w for width, t for time, h for height. Examiners accept any clearly defined letter. The definition line ("Let w = …") is what makes the algebra interpretable, not the choice of letter itself.

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