Algebra Basics — GCSE Maths

What is algebra?

• Algebra uses letters to represent unknown numbers.
• It lets us write general rules and solve problems we cannot work out by counting.
• At GCSE, you must: simplify expressions, expand brackets, factorise, and solve equations.
• Algebra is used across almost every topic in GCSE Maths.

Key Words

• Expression — A collection of terms with no equals sign (e.g. 3x + 2)
• Equation — Two expressions set equal to each other (e.g. 3x + 2 = 11)
• Like terms — Terms with the same letter(s) and power (e.g. 3x and 5x)
• Coefficient — The number in front of a letter (e.g. in 4x, the coefficient is 4)

Simplifying expressions

• Collect like terms — only add or subtract terms with the same letter.
• 3x + 5x = 8x. But 3x + 5y cannot be simplified.
• 4a + 2b − a + 3b = 3a + 5b.
• Numbers without letters are their own like terms: 4 + 7 = 11.
• Always write the simplified expression clearly.

Expanding brackets

• Multiply everything inside the bracket by the term outside.
• 3(x + 4) = 3x + 12.
• −2(3x − 5) = −6x + 10 (be careful with negatives).
• Double brackets: (x + 3)(x + 2) = x² + 2x + 3x + 6 = x² + 5x + 6.
• Use FOIL: First, Outer, Inner, Last for double brackets.

Solving equations

• Goal: get the unknown on its own on one side.
• Whatever you do to one side, do to the other.
• 3x + 7 = 22 → subtract 7: 3x = 15 → divide by 3: x = 5.
• 2(x − 3) = 14 → expand: 2x − 6 = 14 → 2x = 20 → x = 10.
• Always check your answer by substituting back.

Worked Examples

Practice Questions

• Simplify: 5x + 3y − 2x + y
  Answer: 3x + 4y
  5x − 2x = 3x. 3y + y = 4y. Answer: 3x + 4y.
• Expand: 4(3x − 2)
  Answer: 12x − 8
  4 × 3x = 12x. 4 × −2 = −8. Answer: 12x − 8.
• Solve: 4x + 5 = 21
  Answer: x = 4
  4x = 21 − 5 = 16. x = 16 ÷ 4 = 4.
• Expand: (x + 3)(x + 5)
  Answer: x² + 8x + 15
  FOIL: x²+5x+3x+15 = x² + 8x + 15.
• Solve: 2(x − 3) = 10
  Answer: x = 8
  2x − 6 = 10. 2x = 16. x = 8.