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Non-linear Simultaneous Equations

HigherHigher tier onlyAQAEdexcelOCR

Master non-linear simultaneous equations for GCSE Maths with structured, exam-style practice. This Higher resource covers solving simultaneous equations and includes a complete mark scheme showing the steps examiners reward. Suitable for AQA, Edexcel and OCR. Free to download as a PDF. Substitute the linear equation into the quadratic, then solve.

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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.

This is a Higher tier only topic, so there's no Foundation paper — only the Higher worksheet and mark scheme below.

Higher

WorksheetHigher · PDFMark SchemeHigher · PDF

Topic overview

Non-linear simultaneous equations pair a straight line with a curve, usually a quadratic or a circle. Because a line can cut a curve twice, there are normally two pairs of solutions.

The reliable method is substitution rather than elimination. Rearrange the linear equation to make one variable the subject, then substitute that expression into the non-linear equation.

The result is a quadratic in one variable, which you solve as usual. Each solution then goes back into the linear equation to find its partner, and the answers must be given as pairs, since an \(x\) value only makes sense alongside its matching \(y\).

Revision notes

Substituting from the linear equation

Make \(y\) or \(x\) the subject of the straight line, then replace it in the curve's equation.

For \(y = x + 1\) and \(x^2 + y^2 = 25\): substitute to get \(x^2 + (x+1)^2 = 25\), which expands to \(2x^2 + 2x - 24 = 0\).

Solving the resulting quadratic

Simplify and solve by factorising or the formula.

Dividing by \(2\) gives \(x^2 + x - 12 = 0\), which factorises to \((x+4)(x-3)\), so \(x = -4\) or \(x = 3\).

Finding the partners and pairing up

Substitute each \(x\) back into the linear equation, which is simpler than the curve.

Here \(x = -4\) gives \(y = -3\), and \(x = 3\) gives \(y = 4\). The solutions are the pairs \((-4, -3)\) and \((3, 4)\), not four separate numbers.

Key points

  • Pairs a straight line with a curve.
  • There are usually two pairs of solutions.
  • Use substitution, not elimination.
  • Make a variable the subject of the linear equation.
  • Solve the resulting quadratic.
  • Give the answers as coordinate pairs.

Worked examples

Example 1

Solve \(y = x + 2\) and \(y = x^2\).

Working

\[x^2 = x + 2\]substitute the linear equation into the curve
\[x^2 - x - 2 = 0 \Rightarrow (x-2)(x+1)\]rearrange and factorise
\[(2, 4) \text{ and } (-1, 1)\]substitute each x back to find y

Example 2

Solve \(y = 2x\) and \(x^2 + y^2 = 20\).

Working

\[x^2 + 4x^2 = 20\]substitute y = 2x into the circle
\[5x^2 = 20 \Rightarrow x = \pm 2\]solve for x
\[(2, 4) \text{ and } (-2, -4)\]find the matching y values

Example 3

Solve \(y = x - 1\) and \(y = x^2 - 3x + 2\).

Working

\[x - 1 = x^2 - 3x + 2\]set the two expressions for y equal
\[x^2 - 4x + 3 = 0 \Rightarrow (x-1)(x-3)\]rearrange and factorise
\[(1, 0) \text{ and } (3, 2)\]substitute back to find each y

Common mistakes

  • Using elimination instead of substitution.

    Elimination works for two linear equations. A curve requires substitution.

  • Giving four separate numbers rather than two pairs.

    Each x belongs with a particular y, so the answers are coordinate pairs.

  • Substituting back into the quadratic.

    Use the linear equation — it is simpler and less error-prone.

  • Forgetting to square the whole bracket.

    (x + 1)² is x² + 2x + 1, not x² + 1.

Exam tips

  • Substitute into the non-linear equation, never the other way round.
  • Expand brackets carefully, especially squared ones.
  • Substitute back into the linear equation to find partners.
  • Present answers as pairs and check both in the original equations.

Key terms

Simultaneous equations
Equations true at the same time for the same unknowns.
Substitution
Replacing a variable with an equivalent expression.
Non-linear
Involving a curve rather than a straight line.
Coordinate pair
An x and y value belonging together.

Related topics

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