Mathematics 0580 · syllabus C2.5, E2.5
Cambridge IGCSE Mathematics 0580 — Equations topical past papers
Linear, simultaneous and quadratic equations, forming equations from problems, and changing the subject of a formula, with the harder cases on the Extended route.
- Questions
- 572
- Years
- 2009–2026
- With mark scheme
- 91%
- Questions 2022–2026
- 208
Anyone can browse these pages. Opening the questions and their mark schemes takes a free Momentum account.
What Equations covers
Solving an equation means finding the value that makes it true, by doing the same thing to both sides until the unknown stands alone. Core covers linear equations with one unknown, including brackets and unknowns on both sides, such as 7 − 3x = 2(x + 1), and pairs of simultaneous linear equations, solved by elimination or substitution. You also construct equations from information, for example from the angles of a shape or the cost of two kinds of ticket, and change the subject of a simple formula, where the subject appears once and has no power or root.
Extended adds fractional equations, with numbers or linear expressions in the denominators, quadratic equations solved by factorising, by completing the square or with the quadratic formula, which is on the formula list, and simultaneous equations in which one is linear and the other is not, such as a line and a curve. Some answers have to be left in surd form, others given to a stated accuracy. Changing the subject on the Extended route includes formulas where the subject appears twice, so it has to be collected and factorised out, or sits under a power or a root.
Linear and simultaneous linear equations and simple rearrangements are Core content, so they can be set on all four papers; the rest is Extended, examined only on Papers 2 and 4. Many problems end in an equation: a question about angles, areas, ages or prices is turned into one and solved. Expanding, factorising and algebraic fractions are the tools, and graphs of functions show the same solutions as the points where a line meets a curve.
In the syllabus
The headings this topic has in the Cambridge International 2025–2027 syllabus, as the board prints them.
- C2.5 Equations
- E2.5 Equations
What you should be able to do
- Solve linear equations with brackets and unknowns on both sides, and with fractions on the Extended route
- Form an equation from a problem in words or a diagram, and interpret the solution
- Solve simultaneous linear equations by elimination or substitution
- Change the subject of a formula, including Extended cases where the subject appears twice or under a power or root
- Solve a quadratic equation by factorising, by completing the square or with the formula (Extended)
- Solve simultaneous equations where one is linear and the other quadratic (Extended)
Key ideas
- Whatever you do to one side of an equation, you do to the other
- Elimination: make the coefficients of one unknown the same size, then add or subtract the equations
- If ab = 0, then a = 0 or b = 0, which is why a factorised quadratic gives two solutions
- x = (−b ± √(b² − 4ac)) / (2a) solves ax² + bx + c = 0 (Extended)
- A linear and a quadratic equation can share two solutions, one or none, just as a line can cross a curve twice, touch it or miss it
How Equations has been examined
From 2022 to 2026, Equations was on Paper 4 · Calculator (Extended) in 14 of 14 exam series, worth about 15% of that paper’s marks.
| Paper | Questions | Share of marks | Series |
|---|---|---|---|
| Paper 1 · Non-calculator (Core) | 48 | 8% | 13 of 14 |
| Paper 2 · Non-calculator (Extended) | 61 | 9% | 14 of 14 |
| Paper 3 · Calculator (Core) | 41 | 10% | 13 of 14 |
| Paper 4 · Calculator (Extended) | 58 | 15% | 14 of 14 |
Momentum’s own count, over the questions in its bank for these years: a series is one exam session, every zone included, and a question filed under two chapters counts in both. Median marks per question: 4.
Filed in Momentum under: Linear Equation, Simultaneous Equations, Quadratic Equations, Changing the Subject.
A worked example
Solve the simultaneous equations y = x² − 2x − 1 and y = x + 3.
- Both expressions equal y, so set them equal: x² − 2x − 1 = x + 3.
- Rearrange to equal zero: x² − 3x − 4 = 0.
- Factorise: (x − 4)(x + 1) = 0, so x = 4 or x = −1.
- Substitute each into the linear equation: x = 4 gives y = 7, and x = −1 gives y = 2.
- Check in the quadratic: 4² − 2 × 4 − 1 = 7 and (−1)² − 2 × (−1) − 1 = 2.
Answer: x = 4, y = 7 or x = −1, y = 2
Written by Momentum for this page, with its own numbers; it is not taken from a past paper.
Mistakes that cost marks
- Losing a sign when expanding: 5 − 2(x − 3) is 5 − 2x + 6, not 5 − 2x − 6.
- Subtracting the equations in elimination when the signs need adding: if the y-terms are +3y and −3y, add the equations to remove y.
- Dividing a quadratic by x and losing the solution x = 0: x² = 5x gives x = 0 or x = 5, so factorise x(x − 5) = 0 instead.
- Rounding the square root part-way through the quadratic formula, which can move the final answer away from the accuracy asked for.
- Changing the subject by doing an operation to only one term, such as taking the square root of one term of a sum.
- Stopping at one value in simultaneous equations: both x and y are needed, and with a line and a curve each x must be paired with its own y.
Exam technique
- Write each step on a new line and do one operation at a time; method marks are given for correct steps even if an arithmetic slip follows.
- Check solutions by substituting them back into the original equations, especially simultaneous ones.
- For a quadratic, rearrange to = 0 first; try factorising, and use the formula when the question asks for decimals or surds.
Practising in Momentum
- Every question in a chapter as one list, from every paper and year in the bank, newest first or shuffled.
- Each question turns over to its own part of the mark scheme, where the board published one.
- Narrow by paper, year, session, zone and marks, and star the questions worth keeping.
- Pick chapters, a number of questions and a time, and the exam builder draws a mock to sit and mark.
Past-paper questions and mark schemes are the copyright of Cambridge International. Momentum shows them to signed-in learners for revision and does not publish them on these pages.