Skip to content
VirtusAcademy

Linear Graphs

FoundationHigherAQAEdexcelOCR

Linear Graphs is a key algebra topic at GCSE Maths. This Foundation and Higher worksheet gives you exam-style questions on plotting straight-line graphs and finding gradients, 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. In y = mx + c, m is the gradient and c is the y-intercept.

Free downloads

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

A linear graph is a straight line, and its equation can always be written in the form \(y = mx + c\). Every point on the line satisfies the equation, and every solution of the equation lies on the line.

That correspondence is what makes graphs useful. Reading a value from the line is the same as solving the equation for that value, which is why graphical methods work for simultaneous equations and inequalities.

Some equations appear in other forms, such as \(ax + by = c\). These still give straight lines, and can either be rearranged into \(y = mx + c\) or plotted directly by finding where the line crosses each axis.

Revision notes

Plotting from y = mx + c

Start at the \(y\)-intercept, then use the gradient to find further points.

For \(y = 2x - 3\), start at \((0, -3)\), then move one right and two up to reach \((1, -1)\). Repeat and join the points.

Using intercepts for ax + by = c

Set \(x = 0\) to find where the line crosses the \(y\)-axis, and \(y = 0\) for the \(x\)-axis.

For \(2x + 3y = 12\): when \(x = 0\), \(y = 4\); when \(y = 0\), \(x = 6\). Plot \((0, 4)\) and \((6, 0)\) and join them.

Horizontal and vertical lines

The line \(y = 4\) is horizontal through \(4\) on the \(y\)-axis, and \(x = 3\) is vertical through \(3\) on the \(x\)-axis.

Students often swap these. Remember that \(y = 4\) means every point has a \(y\) value of \(4\), which forces the line to be flat.

Key points

  • A linear graph is always a straight line.
  • Every point on the line satisfies its equation.
  • Plot from the intercept using the gradient.
  • For \(ax + by = c\), find both axis intercepts.
  • \(y = k\) is a horizontal line.
  • \(x = k\) is a vertical line.

Worked examples

Example 1

Find where \(3x + 2y = 12\) crosses each axis.

Working

\[x = 0 \Rightarrow 2y = 12 \Rightarrow y = 6\]set x to zero for the y-intercept
\[y = 0 \Rightarrow 3x = 12 \Rightarrow x = 4\]set y to zero for the x-intercept
\[(0, 6) \text{ and } (4, 0)\]state both intercepts

Example 2

Describe the line \(y = -2\).

Working

\[\text{Every point has } y = -2\]the y value never changes
\[\text{Horizontal line through } (0, -2)\]state the line's position

Example 3

Does the point \((3, 7)\) lie on \(y = 2x + 1\)?

Working

\[2(3) + 1 = 7\]substitute x = 3 into the equation
\[\text{Yes, since } y = 7\]the calculated value matches, so the point lies on the line

Common mistakes

  • Swapping horizontal and vertical lines.

    y = 4 is horizontal, x = 4 is vertical. The named variable stays constant.

  • Reading the gradient from an unrearranged equation.

    3y = 6x + 9 has gradient 2, not 6.

  • Only finding one intercept.

    Two points are needed to draw the line, so find both.

  • Assuming a point lies on a line without checking.

    Substitute the coordinates and see whether the equation balances.

Exam tips

  • Use the intercept method for equations in the form ax + by = c.
  • Rearrange to y = mx + c if you prefer working from the gradient.
  • Test a point by substituting both coordinates.
  • Label the line with its equation on the graph.

Key terms

Linear graph
A graph that forms a straight line.
Intercept
Where a line crosses an axis.
Satisfy
To make an equation true when substituted.
Rearrange
To rewrite an equation in a different form.

Related topics

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