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Parallel Graphs

FoundationChallengeAQAEdexcelOCR

Parallel Graphs is a key algebra topic at GCSE Maths. This Foundation worksheet gives you exam-style questions on recognising parallel lines from their equations, with a full mark scheme so you can check every method mark. Suitable for AQA, Edexcel and OCR. Download the free PDF and answers below. Parallel lines share the same gradient but have different intercepts.

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Topic overview

Parallel lines never meet, and on a graph that happens precisely when they have the same gradient. Their equations therefore share the same value of \(m\).

So \(y = 3x + 1\) and \(y = 3x - 4\) are parallel. Both climb three units for every one across, so the gap between them stays constant however far the lines are extended.

The \(y\)-intercepts must differ, though. If both the gradient and the intercept matched, the two equations would describe the same line rather than two parallel ones.

Revision notes

Recognising parallel lines

Compare the coefficients of \(x\) once both equations are in the form \(y = mx + c\). Equal gradients mean parallel.

So \(y = 2x + 5\) and \(2y = 4x - 6\) are parallel, because the second rearranges to \(y = 2x - 3\).

Writing the equation of a parallel line

Keep the gradient and find the new intercept by substituting the point the line passes through.

For a line parallel to \(y = 4x + 1\) through \((2, 3)\): the gradient is \(4\), so \(3 = 4(2) + c\), giving \(c = -5\) and the equation \(y = 4x - 5\).

Parallel versus identical

Parallel lines share a gradient but must have different intercepts.

If both match, the equations describe the same line. \(y = 2x + 3\) and \(2y = 4x + 6\) look different but are identical once rearranged.

Key points

  • Parallel lines have equal gradients.
  • Rearrange to \(y = mx + c\) before comparing.
  • The y-intercepts must be different.
  • Equal gradient and intercept means the same line.
  • Keep the gradient and find a new \(c\) from a given point.
  • Parallel lines never meet.

Worked examples

Example 1

Are \(y = 5x - 2\) and \(2y = 10x + 6\) parallel?

Working

\[2y = 10x + 6 \Rightarrow y = 5x + 3\]rearrange the second equation
\[\text{Both have gradient } 5\]compare the gradients
\[\text{Yes, and the intercepts differ}\]confirm they are parallel, not identical

Example 2

Find the equation of the line parallel to \(y = 3x + 7\) passing through \((1, 2)\).

Working

\[m = 3\]parallel lines share the gradient
\[2 = 3(1) + c \Rightarrow c = -1\]substitute the point to find c
\[y = 3x - 1\]write the equation

Example 3

Explain why \(y = 2x + 4\) and \(3y = 6x + 12\) are not parallel.

Working

\[3y = 6x + 12 \Rightarrow y = 2x + 4\]rearrange the second equation
\[\text{Both equations are identical}\]same gradient and same intercept, so it is one line

Common mistakes

  • Comparing gradients without rearranging.

    2y = 10x + 6 has gradient 5, not 10. Divide through first.

  • Calling identical lines parallel.

    Parallel lines must be distinct, so the intercepts have to differ.

  • Changing the gradient when finding a parallel line.

    The gradient stays the same; only the intercept changes.

  • Substituting the point into the wrong equation.

    Use the new equation with the shared gradient to solve for c.

Exam tips

  • Rearrange both equations before comparing gradients.
  • Keep m and solve for c when given a point.
  • Check the intercepts differ before calling lines parallel.
  • Substitute the point back to verify your final equation.

Key terms

Parallel
Lines with the same gradient that never meet.
Gradient
The steepness of a line.
Identical
Describing the same line despite different-looking equations.
Substitute
To replace variables with known values.

Related topics

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