KS3 + GCSE mathematics

Algebra Hub

Learn the rules, follow worked examples, then test what you know. Start with the basics or jump to GCSE topics.

📘 Learn · Practise · Improve
KS3 foundations

Build confidence

Variables, expressions, substitution, simplifying and one-step or two-step equations.

GCSE algebra

Level up

Expanding brackets, factorising, sequences, inequalities, simultaneous equations and quadratics.

Check your progress

Quick quiz

Try questions from both levels. Get instant feedback and see your score.

What is algebra?

Algebra uses letters, called variables, to stand for numbers. It helps us describe patterns and solve problems even when a number is unknown.

3x + 2

If x = 4, then 3x + 2 = 3 × 4 + 2 = 14.

Remember: 3x means 3 multiplied by x. It does not mean 3 + x.

Maths Home Page      Sports Coach Home Page

KS3 · Expressions

1. Simplifying

Collect like terms. Only combine terms with the same letter and power.

3x + 2x − 4 = 5x − 4
  1. Find the x terms: 3x and 2x.
  2. Add their coefficients: 3 + 2 = 5.
  3. Keep the −4 because it is a constant.
KS3 · Substitution

2. Substituting

Replace each letter with its value, then calculate using the correct order of operations.

2a + 5, when a = 3
  1. Replace a with 3: 2 × 3 + 5.
  2. Multiply first: 6 + 5.
  3. Answer: 11.
KS3 · Equations

3. Solving equations

Keep the equation balanced: do the same operation to both sides.

x + 7 = 12
  1. Subtract 7 from both sides.
  2. x = 12 − 7.
  3. Answer: x = 5.
GCSE · Brackets

4. Expanding

Multiply the term outside the bracket by every term inside.

3(x + 4) = 3x + 12
  1. 3 × x = 3x.
  2. 3 × 4 = 12.
  3. Write the result: 3x + 12.
GCSE · Factorising

5. Factorising

Factorising reverses expanding. Find the highest common factor.

6x + 9 = 3(2x + 3)
  1. The highest common factor of 6 and 9 is 3.
  2. Divide each term by 3.
  3. Put the 3 outside the bracket.
GCSE · Quadratics

6. Quadratic equations

Some quadratics can be solved by factorising.

x² + 5x + 6 = 0
  1. Find two numbers that multiply to 6 and add to 5: 2 and 3.
  2. Factorise: (x + 2)(x + 3) = 0.
  3. So x = −2 or x = −3.
GCSE · Inequalities

7. Inequalities

Solve like an equation. If you multiply or divide by a negative number, reverse the inequality sign.

−2x > 8
  1. Divide both sides by −2.
  2. Reverse > to <.
  3. Answer: x < −4.
GCSE · Simultaneous

8. Simultaneous equations

Find values that satisfy both equations at the same time.

x + y = 7
x − y = 1
  1. Add the equations: 2x = 8.
  2. x = 4.
  3. Substitute: 4 + y = 7, so y = 3.
Mixed KS3 + GCSE

Practice quiz

Type your answer. For equations, enter just the value of the variable. The quiz has 5 questions.

Question 1 of 5
Simplify: 4x + 3x

Variable

A letter or symbol that represents a number, such as x.

Coefficient

The number multiplying a variable. In 5x, the coefficient is 5.

Constant

A number on its own, with no variable, such as 8 in 2x + 8.

Expression

A mathematical phrase without an equals sign, such as 3x + 1.

Equation

A statement that two expressions are equal, such as 2x + 1 = 9.

Factor

A number or expression that multiplies with another to make a product.