Forming and Solving Equations

Forming equations from word problems is the skill that bridges arithmetic and algebra. Exam boards love it because it tests whether you can translate English into mathematics — and then solve what you have built. This topic appears on both Foundation and Higher tiers for AQA, Edexcel, and OCR, often within geometry, perimeter, and angle questions. On this page you will learn a reliable strategy for turning worded problems into algebraic equations, see how to solve those equations, and practise with realistic exam-style questions. Once you have this skill, many seemingly hard problems become straightforward.

What Is Forming an Equation?

Forming an equation means reading a problem, choosing a letter to represent the unknown quantity, writing a mathematical statement that captures the relationships described, and then solving it.

The key phrase to look for is any statement that two things are equal, or that a total has a given value. Words like "is," "equals," "the same as," "total," and "altogether" signal that you can write an equation.

These angle facts are the most common context for forming equations at GCSE.

Step-by-Step Method

1. Read the problem twice. Underline key information and the quantity you need to find.
2. Choose a variable. Let x represent the unknown quantity. If there are two unknowns, try to express the second in terms of x.
3. Write expressions for each part of the problem using x.
4. Form the equation by connecting the expressions with an equals sign, using the given relationship (e.g., perimeter = 34, angles sum to 180°).
5. Solve the equation using the methods covered in Solving Linear Equations.
6. Answer the question. The problem might ask for x, or it might ask for something calculated from x, such as the length of a side. Re-read the question to check what is required.
7. Check your answer makes sense in context (e.g., a length cannot be negative).

Worked Example 1 — Foundation Level

Question: The three angles of a triangle are x°, (2x + 10)°, and (x + 30)°. Find the value of x and state the size of each angle.

Working:

Step 1: Angles in a triangle add to 180°.

x + (2x + 10) + (x + 30) = 180

Step 2: Simplify the left side.

4x + 40 = 180

Step 3: Subtract 40.

4x = 140

Step 4: Divide by 4.

x = 35

Step 5: Find each angle.

• First angle: 35°
• Second angle: 2(35) + 10 = 80°
• Third angle: 35 + 30 = 65°

Check: 35 + 80 + 65 = 180 ✓

Answer: x = 35; the angles are 35°, 80°, and 65°.

Worked Example 2 — Higher Level

Question: A rectangle has length (3x + 2) cm and width (x + 5) cm. The perimeter of the rectangle is 54 cm. Find the area of the rectangle.

Working:

Step 1: Perimeter of a rectangle = 2(length + width).

2[(3x + 2) + (x + 5)] = 54

Step 2: Simplify inside the bracket.

2(4x + 7) = 54

Step 3: Divide both sides by 2.

4x + 7 = 27

Step 4: Subtract 7.

4x = 20

Step 5: Divide by 4.

x = 5

Step 6: Find length and width.

• Length = 3(5) + 2 = 17 cm
• Width = 5 + 5 = 10 cm

Step 7: Area = length × width = 17 × 10 = 170 cm²

Check: Perimeter = 2(17 + 10) = 2 × 27 = 54 ✓

Answer: The area is 170 cm².

Common Mistakes

• Not reading the question carefully enough. Students find x but forget the question asks for the length or the area. Always re-read what is being asked.
• Setting up the wrong equation. Confusing perimeter with area, or using the wrong angle fact. Double-check which formula applies.
• Forgetting brackets in expressions. If the width is (x + 5), make sure you keep the brackets when substituting into a formula, especially with multiplication.
• Accepting impossible answers. If x comes out negative and represents a length, something has gone wrong. Go back and check your equation.
• Using two variables when one is enough. In most GCSE questions, you can express everything in terms of a single unknown. This keeps the equation simple and solvable.

Exam Tips

1. Write "Let x = ..." at the start. This tells the examiner what your variable represents and often earns the first mark.
2. Show the equation clearly before solving. Examiners award a mark for the correct equation, separate from the solving marks. Write it on its own line.
3. Context questions often have a follow-up. After finding x, you may need to calculate an area, a missing angle, or a cost. Do not stop at x.
4. On geometry forming questions, sketch a diagram if one is not given. Labelling sides or angles with your expressions helps you see the equation more easily.

Practice Questions

Q1 (Foundation): The angles of a quadrilateral are x°, (x + 20)°, (2x − 10)°, and (x + 50)°. Find x.

Q2 (Foundation/Higher): A triangle has sides of length x cm, (x + 3) cm, and (2x − 1) cm. The perimeter is 30 cm. Find the length of each side.

Q3 (Higher): Amy is x years old. Ben is twice as old as Amy. Chloe is 5 years younger than Ben. The sum of their ages is 75. Form an equation and find each person's age.

Want to practise forming equations with instant hints? Start revising with GCSEMathsAI — our AI tutor breaks down word problems step by step and helps you set up the algebra.

Related Topics

• Solving Linear Equations
• Solving Inequalities
• Simultaneous Equations — Substitution
• Factorising Expressions

For more on the formulas you need, see GCSE Maths Formulas You Must Know.

Summary

• Forming an equation means translating a word problem into algebra.
• Define your variable clearly — write "Let x = ..." at the start.
• Use known facts (angle sums, perimeters, totals) to build the equation.
• Solve using inverse operations, showing every step.
• Always answer the question asked — it may not be just "find x."
• Check that your answer is sensible in context (no negative lengths, angles that sum correctly, etc.).
• This skill appears in geometry, number, and real-life contexts across all exam boards.