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Algebra practice problems GCSE Maths, Topic 2

Written by STEM-Tutor's tutors · Updated October 2026 · 8 questions and 1 exam-style question

Solving equations, sequences and the nth term, factorising, simultaneous equations, straight-line graphs and quadratics. Try each question before you look: use the hint if you are stuck, then reveal the worked solution one step at a time.

On this page
  1. Key equations
  2. Warm-up questions
  3. Exam-style questions
  4. Exam question practice
  5. Common mistakes
  6. Keep going

Key equations for algebra

y = mx + cstraight line: gradient m, y-intercept c
m = (y2 − y1)/(x2 − x1)gradient between two points
x = (−b ± √(b2 − 4ac))/(2a)quadratic formula (on the formulae sheet)

A formulae sheet is provided in the 2025, 2026 and 2027 GCSE Maths exams. Answers within 2.5% are marked correct, and you can type fractions such as 23/20. Every topic, Foundation and Higher, is in our GCSE Maths topic list.

Warm-up questions

Quick checks on the core ideas. If these feel easy, go straight to the exam-style questions.

Q1Warm-up
Solve 3x + 7 = 22.

Hint

Subtract 7 from both sides, then divide by 3.

Worked solution
  1. 3x = 15.
  2. x = 5.
Q2Warm-up
A sequence has nth term 4n − 1. Find the 10th term.

Hint

Substitute n = 10.

Worked solution
  1. 4 × 10 − 1 = 39.
Q3Warm-up
Factorise x2 + 5x + 6.

Hint

Find two numbers that multiply to 6 and add to 5.

Worked solution
  1. 2 × 3 = 6 and 2 + 3 = 5.
  2. So x2 + 5x + 6 = (x + 2)(x + 3): B.

Exam-style questions

Questions at the level of the GCSE Maths exam. The ones marked Challenge combine more than one idea.

Q4Exam-style
Solve 5(x − 2) = 2x + 11.

Hint

Expand the bracket first.

Worked solution
  1. 5x − 10 = 2x + 11.
  2. 3x = 21.
  3. x = 7.
Q5Exam-style
Solve the simultaneous equations 2x + y = 11 and x − y = 1. What is x?

Hint

Add the equations to eliminate y.

Worked solution
  1. Adding: 3x = 12, so x = 4.
  2. Then y = 3. So x = 4.
Q6Exam-style
Find the gradient of the straight line through (1, 3) and (4, 12).

Hint

Change in y divided by change in x.

Worked solution
  1. (12 − 3)/(4 − 1) = 9/3.
  2. Gradient = 3.
Q7Exam-styleHigher
Solve x2 − 4x − 7 = 0, giving the positive solution to 2 decimal places.

Hint

It does not factorise. Use the quadratic formula with a = 1, b = −4, c = −7.

Worked solution
  1. x = (4 ± √(16 + 28))/2 = (4 ± √44)/2.
  2. x = (4 + 6.633)/2 = 5.32 (the other solution is −1.32).
Q8ChallengeHigher
Solve (x + 3)/2 + (x − 1)/3 = 4.

Hint

Multiply every term by 6 to clear the fractions.

Worked solution
  1. 3(x + 3) + 2(x − 1) = 24.
  2. 3x + 9 + 2x − 2 = 24, so 5x = 17.
  3. x = 3.4.

Exam question practice

One original exam-style question in the style of a GCSE paper. Answer it in full, with working, before you open the mark scheme: most marks are for method, so show every step.

Exam-style question style · 5 points · about 6 minutes

A rectangle has length (2x − 1) cm and width (x + 3) cm. Its perimeter is 34 cm.

  1. (a) Show that 6x + 4 = 34.
  2. (b) Solve the equation to find x.
  3. (c) Work out the area of the rectangle.

Write your full answer on paper first, then open the scoring guide and tick each point you earned.

Scoring guide and model answer

Part (a)

Perimeter = 2(2x − 1) + 2(x + 3) = 4x − 2 + 2x + 6 = 6x + 4, so 6x + 4 = 34.

Part (b)

6x = 30, so x = 5.

Part (c)

Length 2(5) − 1 = 9 cm, width 5 + 3 = 8 cm. Area = 72 cm2.

Your mark: 0 of 5

Common mistakes in algebra

Keep going

Revise the ideas behind these questions:

Then try real exam questions: GCSE Maths past papers, with official scoring guidelines. · All GCSE Maths units