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Edexcel IGCSE Maths (4MA1) · Unit 3: Equations & Sequences

Edexcel IGCSE 4MA1 Simultaneous Equations: exam questions and answers, notes and worked examples

Updated October 2026 · By Pete Bromfield

Simultaneous equations are two equations that are true at the same time; the solution is the pair of values that fits both, which is where their graphs meet. For two linear equations, eliminate one unknown by adding or subtracting the equations (after scaling so a coefficient matches), or substitute. For one linear and one quadratic equation, rearrange the linear one and substitute it into the quadratic: you get up to two pairs of solutions.

Unit 3: Equations & Sequences
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Notes
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Worked examples
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FAQ

Simultaneous Equations notes

Elimination

Make the coefficients of one unknown the same size, then add the equations if the signs are different and subtract them if the signs are the same. For $3 x + 2 y = 16$ and $5 x - 2 y = 16$ , add: $8 x = 32$ , so $x = 4$ and $y = 2$ . If no coefficient matches, multiply first: $2 x + 3 y = 13$ and $3 x + 2 y = 12$ become $6 x + 9 y = 39$ and $6 x + 4 y = 24$ ; subtracting gives $5 y = 15$ , so $y = 3$ and $x = 2$ .

Substitution

If one equation already gives $y$ (or $x$ ) on its own, substitute it into the other. From $y = 2 x - 1$ and $3 x + 2 y = 19$ : $3 x + 2 (2 x - 1) = 19$ , so $7 x = 21$ , $x = 3$ and $y = 5$ .

Word problems

Choose a letter for each unknown and write one equation for each fact. Two adult tickets and three child tickets cost £31; one adult and two child tickets cost £18. So $2 a + 3 c = 31$ and $a + 2 c = 18$ , giving $c = 5$ and $a = 8$ : an adult ticket is £8 and a child ticket £5.

One linear and one quadratic equation

Rearrange the linear equation to $y = \dots$ or $x = \dots$ , substitute into the quadratic, rearrange to $= 0$ and solve. From $y = x + 3$ and $y = x^{2} + 1$ : $x^{2} - x - 2 = 0$ , so $(x - 2) (x + 1) = 0$ , giving $x = 2, y = 5$ or $x = - 1, y = 2$ . Find each $y$ from the linear equation and give the answers as pairs.

What the solutions mean on a graph

Each solution pair is a point where the two graphs cross. A line can cross a curve twice, touch it once or miss it altogether, so a linear and a quadratic equation have two, one or no pairs of real solutions. The graph shows worked example 2: the line $y = x - 1$ crosses the circle $x^{2} + y^{2} = 13$ at its two solutions.

Graph of the line y = x − 1 and the circle x² + y² = 13, crossing at (3, 2) and (−2, −3), the two solutions of the simultaneous equations — Worked example 2 drawn with our Grapher: the line meets the circle at the two solution pairs.

Key formulas

Quadratic formula, $a x^{2} + b x + c = 0$ On the formula sheet: $x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$

On the formula sheet = printed on the formulae sheet in the 4MA1 Higher papers, so know where to find it. The whole 4MA1 formula sheet (free PDF) → · 4MA1 formulas to learn

Simultaneous Equations worked examples

Worked example 1: Elimination

Solve $2 x + 3 y = 12$ and $4 x - 3 y = 6$ .

Solution

Add the equations (the $y$ terms cancel): $6 x = 18$ , so $x = 3$ .
Substitute: $6 + 3 y = 12$ , so $y = 2$ .

3 steps · 1 to fill in

Tip: Same signs subtract, different signs add.

Worked example 2: Line and circle

Solve $y = x - 1$ and $x^{2} + y^{2} = 13$ .

Solution

Substitute: $x^{2} + (x - 1)^{2} = 13$ .
$2 x^{2} - 2 x - 12 = 0$ , so $x^{2} - x - 6 = 0$ .
$(x - 3) (x + 2) = 0$ : $x = 3, y = 2$ or $x = - 2, y = - 3$ .

4 steps

Tip: Pair each x with its own y.

Worked example 3: Forming and solving a pair of equations · 4 marks

At a climbing centre, 3 adult sessions and 5 junior sessions cost £65.75 in total. 4 adult sessions and 2 junior sessions cost £58.50 in total. Work out the cost of one adult session and the cost of one junior session.

Mark scheme

M1 3a + 5j = 65.75 and 4a + 2j = 58.50
M1 matches coefficients and eliminates, e.g. 12a + 20j = 263 and 12a + 6j = 175.50
A1 j = 6.25 or a = 11.50
A1 adult £11.50, junior £6.25

Answer: adult £11.50, junior £6.25

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Simultaneous Equations exam-style questions

Try each question before you open the answer (one has its full worked solution open). The mark schemes use Edexcel-style marks: M for method, A for accuracy (after the method mark), B for an independent result.

Question 1 · easy · 3 marks

Solve the simultaneous equations $2 x + 5 y = 1$ and $4 x - 5 y = 17$ .

Show the answer and mark scheme

M1 adds to eliminate y: 6x = 18
M1 substitutes their x = 3 into either equation
A1 x = 3, y = −1

Answer: $x = 3$ , $y = - 1$

Question 2 · easy · 3 marks

Solve the simultaneous equations $5 x + 4 y = 7$ and $5 x - 2 y = 19$ .

Show the answer and mark scheme

M1 subtracts to eliminate x: 6y = −12
M1 substitutes their y = −2 into either equation
A1 x = 3, y = −2

Answer: $x = 3$ , $y = - 2$

Question 3 · medium · 3 marks

Solve the simultaneous equations $4 x + 3 y = 25$ and $3 x + 2 y = 18$ .

Full worked solution Free, no sign-in

M1 multiplies to match coefficients, e.g. 8x + 6y = 50 and 9x + 6y = 54
M1 eliminates a variable and substitutes back
A1 x = 4, y = 3

Answer: $x = 4$ , $y = 3$

Every question here has a full worked solution. See them all →

Question 4 · medium · 3 marks

Solve the simultaneous equations $6 x + 4 y = 7$ and $4 x - 6 y = 9$ .

Show the answer and mark scheme

M1 e.g. 18x + 12y = 21 and 8x − 12y = 18
M1 26x = 39 and substitutes their x
A1 x = 1.5, y = −0.5

Answer: $x = 1.5$ , $y = - 0.5$

Question 5 · hard · 5 marks

The line $a x + b y = 14$ passes through the points $(2, 3)$ and $(- 1, 9)$ . (a) Find the values of $a$ and $b$ . (b) The line meets the line $3 x - 2 y = 5$ at the point P. Find the coordinates of P.

Show the answer and mark scheme

M1 2a + 3b = 14 and −a + 9b = 14
M1 eliminates a, e.g. 2a + 3b = 14 and −2a + 18b = 28 so 21b = 42
A1 a = 4, b = 2
M1 4x + 2y = 14 and 3x − 2y = 5 added: 7x = 19
A1 (19/7, 11/7)

Answer: (a) $a = 4$ , $b = 2$ (b) $(\frac{19}{7}, \frac{11}{7})$

Question 6 · hard · 3 marks

Solve the simultaneous equations $0.4 x + 1.5 y = 3.4$ and $1.2 x - 0.5 y = 4.2$ .

Show the answer and mark scheme

M1 e.g. multiplies the first by 3: 1.2x + 4.5y = 10.2
M1 subtracts: 5y = 6, and substitutes their y
A1 x = 4, y = 1.2

Answer: $x = 4$ , $y = 1.2$

Question 7 · hard · 5 marks

Solve the simultaneous equations $y = 2 x - 1$ and $x^{2} + y^{2} = 34$ .

Show the answer and mark scheme

M1 x² + (2x − 1)² = 34
A1 5x² − 4x − 33 = 0
M1 (x − 3)(5x + 11) = 0 or correct use of the formula
A1 x = 3 and x = −2.2
A1 y = 5 and y = −5.4, correctly paired

Answer: $x = 3, y = 5$ or $x = - 2.2, y = - 5.4$

Common simultaneous equations mistakes

Adding when you should subtract. If the matching terms have the same sign, such as $+ 6 x$ and $+ 6 x$ , subtract the equations; if the signs differ, add them.
Stopping after one unknown. Substitute your first value back into one of the original equations to find the other, and give both values.
Unpaired answers. In a linear and quadratic pair, each $x$ has its own $y$ . Write $x = 2, y = 5$ and $x = - 1, y = 2$ , not $x = 2, - 1$ and $y = 5, 2$ .
Finding y from the quadratic. Find $y$ from the linear equation. Putting $x = 3$ into $x^{2} + y^{2} = 13$ gives $y = \pm 2$ , but only $y = 2$ is on the line $y = x - 1$ .

Where students lose marks across 4MA1, from the examiner reports →

Simultaneous Equations FAQ

Should I use elimination or substitution?

Use elimination when both equations are in the form ax + by = c. Use substitution when one equation already gives x or y on its own, and always for one linear and one quadratic equation.

How many solutions can a linear and a quadratic equation have?

Up to two pairs. The line can cross the curve twice, touch it once (one repeated solution) or miss it (no real solutions). Each pair is a point where the graphs meet.

Can I use a calculator for simultaneous equations in 4MA1?

Yes. A calculator may be used in all four 4MA1 papers, so both Higher papers (1H and 2H) allow one. Most marks are for the method you write down, so show your working.

How long are the 4MA1 Higher papers?

Papers 1H and 2H are 2 hours each and 100 marks each, worth 50% each. Each tier has its own formulae sheet in the exam.

Next steps

Simultaneous Equations revision notes · 4MA1 formula sheet · Taking Cambridge 0580 instead? Cambridge 0580 simultaneous equations questions

Every question and worked example here is original, written for IGCSE Math Revision rather than copied from Pearson papers, and every result was re-solved independently before publishing.

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Edexcel IGCSE 4MA1 Simultaneous Equations: exam questions and answers, notes and worked examples

Try it first

Simultaneous Equations notes

Elimination

Substitution

Word problems

One linear and one quadratic equation

What the solutions mean on a graph

Key formulas

Simultaneous Equations worked examples

Worked example 1: Elimination

Solution

Worked example 2: Line and circle

Solution

Worked example 3: Forming and solving a pair of equations · 4 marks

Mark scheme

Simultaneous Equations exam-style questions

Question 1 · easy · 3 marks

Question 2 · easy · 3 marks

Question 3 · medium · 3 marks

Full worked solution Free, no sign-in

Question 4 · medium · 3 marks

Question 5 · hard · 5 marks

Question 6 · hard · 3 marks

Question 7 · hard · 5 marks

Common simultaneous equations mistakes

Simultaneous Equations FAQ

Next steps

Practise simultaneous equations with your progress saved

More Unit 3 topics: Equations & Sequences