IGCSE Mathematics: Simultaneous Equations Practice Questions
Simultaneous equations are two equations that are both true for the same pair of values. Solve a linear pair by elimination or substitution, and solve a linear and non-linear pair by substituting the linear equation into the non-linear one.
Simultaneous equations appear on every Cambridge IGCSE Mathematics 0580 paper, usually once as a pure algebra question and once hidden inside a word problem. Extended candidates also face a linear pair combined with a quadratic or a circle. The marks are almost entirely method marks, so showing every line matters more than reaching the answer quickly.
What you need to know for Simultaneous Equations
- EliminationMultiply one or both equations so that the coefficients of one variable match, then add or subtract the equations to remove that variable.
- SubstitutionRearrange one equation to make a variable the subject, then substitute that expression into the other equation.
- Choosing the methodUse elimination when both equations are in the form ax + by = c. Use substitution when one equation already gives y in terms of x, or when one equation is non-linear.
- Linear and non-linear pairsSubstituting a linear equation into a quadratic gives a quadratic in one variable, so there are usually two solution pairs. Always give both x values with their matching y values.
- Checking your answerSubstitute both values back into the equation you did not use to find the second variable. This catches almost every sign error.
- Word problemsDefine your variables in words first, form two equations from the two pieces of information given, then solve. The definition line often carries a mark.
IGCSE Mathematics Simultaneous Equations questions and answers
4 exam-style questions written to the 0580 syllabus. Try each one on paper first, then open the worked answer to check your method against the marks.
Solve the simultaneous equations 4x + 3y = 25 and 2x - y = 5.
Show the worked answer
- Multiply the second equation by 3 so the y coefficients match: 6x - 3y = 15.
- Add this to the first equation to eliminate y: (4x + 3y) + (6x - 3y) = 25 + 15, giving 10x = 40.
- So x = 4.
- Substitute into 2x - y = 5: 8 - y = 5, so y = 3. Check in the first equation: 16 + 9 = 25, correct.
Solve the simultaneous equations y = x + 1 and x2 + y2 = 25.
Show the worked answer
- Substitute y = x + 1 into the second equation: x2 + (x + 1)2 = 25.
- Expand: x2 + x2 + 2x + 1 = 25, so 2x2 + 2x - 24 = 0.
- Divide through by 2: x2 + x - 12 = 0, which factorises to (x + 4)(x - 3) = 0.
- So x = -4 or x = 3. Substitute each back into y = x + 1 to get y = -3 and y = 4 respectively.
Three pens and two notebooks cost RM26. Five pens and four notebooks cost RM46. Find the cost of one pen and the cost of one notebook.
Show the worked answer
- Let p be the cost of a pen in ringgit and n be the cost of a notebook in ringgit.
- Form the equations: 3p + 2n = 26 and 5p + 4n = 46.
- Multiply the first equation by 2: 6p + 4n = 52. Subtract the second: 6p - 5p = 52 - 46, so p = 6.
- Substitute p = 6 into 3p + 2n = 26: 18 + 2n = 26, so 2n = 8 and n = 4.
The line y = 3x - 2 meets the curve y = x2 at two points. Find the coordinates of both points.
Show the worked answer
- At the points of intersection the y values are equal, so x2 = 3x - 2.
- Rearrange to x2 - 3x + 2 = 0.
- Factorise: (x - 1)(x - 2) = 0, so x = 1 or x = 2.
- Substitute into y = x2: when x = 1, y = 1, and when x = 2, y = 4. The points are (1, 1) and (2, 4).
Common mistakes in this topic
- Multiplying only one term of an equation instead of every term when scaling it.
- Sign errors when subtracting equations. Write the subtraction out fully rather than doing it in your head.
- Forgetting to find the second variable after solving for the first.
- Pairing x and y values incorrectly in non-linear problems.
- In word problems, forgetting to answer in context with units such as ringgit or centimetres.
Exam tips
- Number your equations (1) and (2) and label your working, for example (1) x 2. Markers award method marks for clearly labelled steps.
- If the coefficients of one variable are already equal or opposite, eliminate that variable immediately rather than rearranging.
- Always substitute your answers back into the equation you did not use. It costs ten seconds and catches most errors.
- For a linear and non-linear pair, expect two solution pairs. One pair usually means an error, unless the line is a tangent.
- Write the final answer as a clear statement, for example x = 4, y = 3, rather than leaving it buried in the working.
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Simultaneous Equations FAQs
When should I use elimination instead of substitution?
Use elimination when both equations are in the form ax + by = c, because matching one coefficient and adding or subtracting is quicker and less error prone. Use substitution when one equation already expresses one variable in terms of the other, or when one of the equations is not linear.
How do I solve simultaneous equations when one is a quadratic?
Rearrange the linear equation to make one variable the subject, substitute that expression into the non-linear equation, and simplify to a quadratic in one variable. Solve it by factorising or with the quadratic formula, then substitute each root back into the linear equation to find the matching value of the other variable.
How many marks do I lose for not showing working?
Usually most of them. Simultaneous equations questions are marked with method marks for the elimination or substitution step and the rearrangement. A correct final answer with no working typically scores far below full marks, and an answer with an arithmetic slip but clear method often scores nearly all of them.
What if the two equations have no solution?
If eliminating a variable leaves a false statement such as 0 = 7, the equations represent parallel lines and there is no solution. If it leaves a true statement such as 0 = 0, the equations describe the same line and there are infinitely many solutions.
This page covers solving two equations together. For a single linear equation or an inequality, see Linear Equations and Inequalities.
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Written to the published Cambridge IGCSE Mathematics (0580) syllabus. Check your school entry code and syllabus year, because Core and Extended candidates are assessed on different content. Last reviewed 2026-08-12.