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Simultaneous Equations

FoundationHigherAQAEdexcelOCR

Practise simultaneous equations with this free Foundation and Higher GCSE Maths worksheet from Virtus Academy. You'll work through solving simultaneous equations, building confidence for your exam, and every question comes with worked solutions in the mark scheme. Suitable for AQA, Edexcel and OCR. Match the coefficients of one variable, then add or subtract to eliminate it.

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These worksheets and mark schemes are original, written for Virtus Academy and checked against the current AQA, Edexcel and OCR specifications. Every worksheet comes with a full mark scheme.

Foundation

WorksheetFoundation · PDFMark SchemeFoundation · PDF

Higher

WorksheetHigher · PDFMark SchemeHigher · PDF

Topic overview

Simultaneous equations are two equations that share the same two unknowns, and you need both equations to pin down a single pair of values that satisfies each one.

The standard method is elimination. You scale one or both equations so that a variable has matching coefficients, then add or subtract the equations to remove that variable. That leaves one equation in one unknown, which you can solve directly.

The critical detail is the sign. If the matching coefficients have the same sign you subtract; if they have opposite signs you add. Getting this the wrong way round is the single most common source of lost marks, and it is easy to check because the variable you are eliminating should vanish entirely.

Revision notes

Making coefficients match

Look at one variable and multiply one or both equations so its coefficients are numerically equal.

For \(3x + 2y = 12\) and \(x + y = 5\), multiplying the second equation by \(2\) gives \(2x + 2y = 10\). Both now have \(2y\), so \(y\) can be eliminated. Multiply every term in the equation, including the number on the right.

Adding or subtracting

If the matching terms have the same sign, subtract one equation from the other. If they have opposite signs, add them.

Subtracting \(2x + 2y = 10\) from \(3x + 2y = 12\) gives \(x = 2\). The \(y\) terms cancel, which confirms the choice was right. If the variable does not disappear, you have added when you should have subtracted.

Finding the second unknown and checking

Substitute the value you found back into whichever original equation looks simpler.

Putting \(x = 2\) into \(x + y = 5\) gives \(y = 3\). Always check the pair in the equation you did not use for substitution: \(3(2) + 2(3) = 12\), which is correct, so the solution is confirmed.

Key points

  • You need two equations to find two unknowns.
  • Scale one or both equations so a variable has matching coefficients.
  • Same signs subtract, opposite signs add.
  • Multiply every term when scaling, including the right-hand side.
  • Substitute back to find the second unknown.
  • Check the pair in the other original equation.

Worked examples

Example 1

Solve \(3x + 2y = 12\) and \(x + y = 5\).

Working

\[2x + 2y = 10\]multiply the second equation by 2 so both have 2y
\[x = 2\]subtract, since both y terms are positive
\[2 + y = 5 \Rightarrow y = 3\]substitute x = 2 into the simpler equation

Example 2

Solve \(4x + 3y = 18\) and \(2x - 3y = 6\).

Working

\[6x = 24\]the y terms have opposite signs, so add the equations
\[x = 4\]divide both sides by 6
\[16 + 3y = 18 \Rightarrow y = \tfrac{2}{3}\]substitute x = 4 into the first equation

Example 3

Solve \(5x + 2y = 16\) and \(3x + 4y = 18\).

Working

\[10x + 4y = 32\]multiply the first equation by 2 to match the 4y
\[7x = 14 \Rightarrow x = 2\]subtract the second equation, since both y terms are positive
\[10 + 2y = 16 \Rightarrow y = 3\]substitute x = 2 back to find y

Common mistakes

  • Adding when you should subtract.

    If both matching terms are positive you must subtract. Check that the variable actually disappears — if it doubles instead, you added by mistake.

  • Forgetting to multiply the right-hand side.

    Doubling 2x + 2y = 10 must give 4x + 4y = 20. Leaving the 10 unchanged makes the equation false.

  • Sign errors when subtracting a negative term.

    Subtracting −3y is the same as adding 3y. Write the subtraction out in full rather than doing it mentally.

  • Only finding one variable.

    The answer is a pair of values. A solution with x but no y scores only part of the marks.

Exam tips

  • Number your equations 1 and 2 and label each new one, so the examiner can follow your method.
  • Decide which variable is easier to eliminate before you start multiplying.
  • Substitute into the simpler original equation to reduce arithmetic slips.
  • Always check both values in the equation you did not use for substitution.

Key terms

Simultaneous equations
Two or more equations that are true at the same time for the same unknowns.
Coefficient
The number in front of a variable, such as the 3 in \(3x\).
Elimination
Removing one variable by adding or subtracting the equations.
Substitution
Putting a known value back into an equation to find the remaining unknown.

Related topics

Written and reviewed against the current AQA, Edexcel and OCR specifications. Spotted an error? Let us know.