Who this is for: Cambridge IGCSE Mathematics (0580/0607) students who want simultaneous equations — linear pairs, one linear and one quadratic, and graphical solutions — to become a reliable source of marks instead of a topic they only half-remember.
What query it owns: how to understand and revise simultaneous equations in Cambridge IGCSE Mathematics.
Why this is safe: this page owns the simultaneous equations revision-guide angle, while Tutopiya’s Simultaneous Equations subtopic page owns the learning resource and the free simultaneous equations quiz owns the practice.

Simultaneous equations ask you to find values that satisfy two equations at the same time. In Cambridge IGCSE Mathematics (0580/0607) they appear as two linear equations, or one linear and one quadratic, and examiners reward a clear method — elimination, substitution or a careful sketch. This guide explains what the subtopic covers, how to read the command words on papers, and where to practise until the algebra is steady.

Key takeaways

• Simultaneous equations share the same unknowns; the solution is the (x, y) pair that satisfies both.
• For two linear equations, elimination is usually fastest; substitution suits when one equation is already rearranged.
• Linear + quadratic pairs often need substitution, giving a quadratic to solve — expect two solution pairs.
• Graphical questions want the point of intersection read from your sketch or plotted lines.

What are simultaneous equations in Cambridge IGCSE Maths?

Simultaneous equations are two (or more) equations involving the same variables, solved together so that every equation is true at the same time. In Cambridge IGCSE Mathematics the unknowns are usually x and y, and the solution is written as x = …, y = …. Extended papers also pair a linear equation with a quadratic, which typically yields two valid (x, y) pairs.

Work through the full notes on Tutopiya’s Simultaneous Equations subtopic page before attempting questions.

The core ideas you must master

Idea	What it means	How the exam uses it
Elimination	Add or subtract equations to remove one unknown	”Solve 3x + 2y = 12 and x − y = 1”
Substitution	Replace one unknown using the other equation	”Solve y = 2x + 1 and 3x + y = 11”
Linear + quadratic	Substitute linear into quadratic	”Solve y = x + 2 and x² + y² = 20”
Graphical solution	Intersection of two lines on a grid	”Solve by drawing graphs of y = 2x − 1 and y = −x + 5”
Word problems	Form two equations from context	”The sum of two numbers is 15 and their difference is 3”

How to solve two linear simultaneous equations — step by step

1. Label the equations (1) and (2). Check whether coefficients already match.
2. Elimination: multiply one or both equations so one unknown has the same coefficient (or opposite coefficients).
3. Add or subtract to eliminate that unknown. Solve the remaining equation.
4. Substitute back into either original equation to find the second unknown.
5. Check by substituting both values into the equation you did not use first.

Confirm the method with the free Simultaneous Equations quiz.

Linear and quadratic simultaneous equations

When one equation is quadratic, substitution is almost always the route:

1. Rearrange the linear equation to express y (or x) in terms of the other unknown.
2. Substitute into the quadratic equation to obtain a single quadratic in one unknown.
3. Solve the quadratic — factorise or use the formula.
4. Find the matching second value for each root. You should get two (x, y) pairs unless the line is a tangent.

Simultaneous equations in past-paper wording: command words that matter

Command word / phrase	What the question wants	Typical stem
Solve the simultaneous equations	Algebraic method with working	”Solve the simultaneous equations 2x + 3y = 13 and 4x − y = 5.”
Solve by substitution	Must use substitution — not elimination	”Solve by substitution: y = 3x − 2 and 2x + y = 12.”
Show your working clearly	Method marks even if the final answer is wrong	Any multi-mark simultaneous question
Solve by drawing graphs	Plot both lines; state intersection coordinates	”Use graphs to solve y = 2x + 1 and y = 7 − x.”
Form two equations and solve	Translate words into algebra first	”A cinema sells adult and child tickets…”

Worked exam-style stems (how to answer the wording)

1. “Solve the simultaneous equations 3x + 2y = 16 and x − y = 2.” From (2): x = y + 2. Substitute into (1): 3(y + 2) + 2y = 16 → y = 2, x = 4. Reward: substitution or elimination with both values correct.
2. “Solve y = x + 1 and x² + y² = 25.” Substitute: x² + (x + 1)² = 25 → 2x² + 2x − 24 = 0 → x = 3 or x = −4, giving (3, 4) and (−4, −3). Reward: correct quadratic, both pairs.
3. “Show that the lines y = 2x − 3 and 2y = 4x − 6 are the same line.” Rearrange the second: y = 2x − 3 — identical. Reward: algebraic proof that coefficients match.

Work the full set on the Algebra topical past-paper questions once the wording feels familiar.

How simultaneous equations connect to the wider Algebra unit

Simultaneous equations build on Linear Equations and Inequalities and feed into Quadratic Equations. Use the Cambridge IGCSE Maths resource hub to move between subtopics in one session.

Common mistakes students make

• Sign errors when subtracting equations — double-check whether you add or subtract.
• Finding x correctly but forgetting to find y (or vice versa).
• Using elimination when the question says “by substitution”.
• With linear + quadratic, finding x values but not pairing them with the correct y.
• Assuming one solution when the quadratic gives two roots.

When you need more support

If simultaneous equations keep costing marks — especially linear + quadratic pairs — revisit the Linear Equations quiz, then work through the Algebra topical past-paper questions. A Cambridge IGCSE Maths tutor can stabilise the method quickly.

Frequently asked questions

What is the easiest method for two linear equations? Elimination is usually fastest when coefficients are easy to match. Substitution is better when one equation is already y = … or x = ….

How many solutions can simultaneous equations have? Two non-parallel lines meet once. Parallel lines have no solution; identical lines have infinitely many. A line and a circle (or parabola) can meet twice, once or not at all.

Do I need to show working for simultaneous equations? Yes — method marks are common. Write each step clearly, especially when the question says “show your working”.

How do I revise simultaneous equations effectively? Practise elimination and substitution separately, then mixed pairs, then one linear + one quadratic. Use the simultaneous equations quiz to check each stage.

Ready to master Cambridge IGCSE Maths simultaneous equations?

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