Graphical Simultaneous Equations
Graphical Simultaneous Equations is a key algebra topic at GCSE Maths. This Foundation and Higher worksheet gives you exam-style questions on solving simultaneous 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. The solution is the point where the two graphs cross.
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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.
Topic overview
Simultaneous equations can be solved graphically by drawing both lines and reading where they cross. The point of intersection satisfies both equations at once, which is exactly what a solution means.
The coordinates of that point give the answer: the \(x\) value and the \(y\) value together. Because you are reading from a drawing, answers are approximate unless the crossing point falls neatly on grid lines.
The method also reveals when there is no solution. Parallel lines never meet, so a pair of equations with the same gradient but different intercepts has no solution at all — something the algebraic method shows less obviously.
Revision notes
Drawing both lines
Rearrange each equation into \(y = mx + c\) if needed, then plot each line using a table of values or the gradient and intercept.
Draw both on the same axes with a ruler, extending them far enough that the crossing point is clearly visible.
Reading the solution
The coordinates of the intersection are the solution. Write them as \(x = \ldots\) and \(y = \ldots\), not just as a coordinate pair, unless the question asks otherwise.
If the lines cross at \((2, 5)\), the solution is \(x = 2\), \(y = 5\).
No solution and infinite solutions
Parallel lines never meet, so there is no solution. Identical lines overlap completely, so every point on the line is a solution.
Check the gradients: equal gradients with different intercepts means no solution.
Key points
- The solution is where the two lines cross.
- The intersection satisfies both equations.
- Rearrange into \(y = mx + c\) before plotting.
- Give both the \(x\) and \(y\) values.
- Parallel lines mean no solution.
- Graphical answers are approximate.
Worked examples
Example 1
Two lines cross at \((3, 7)\). Write down the solution of the simultaneous equations.
Working
Example 2
Solve \(y = x + 1\) and \(y = 2x - 1\) graphically.
Working
Example 3
Explain why \(y = 2x + 1\) and \(y = 2x - 4\) have no solution.
Working
Common mistakes
Giving only the x value.
A solution to simultaneous equations is a pair, so both x and y are needed.
Reading the intersection inaccurately.
Extend both lines fully and read carefully against the grid.
Plotting only two points per line without checking.
A third point catches arithmetic errors before you draw.
Not recognising parallel lines.
Equal gradients mean the lines never meet, so there is no solution.
Exam tips
- Rearrange both equations into y = mx + c before plotting.
- Extend the lines beyond the crossing point.
- State both x and y in your answer.
- Check your solution satisfies both original equations.
Key terms
- Intersection
- The point where two lines cross.
- Simultaneous equations
- Equations true at the same time for the same unknowns.
- Parallel
- Having the same gradient and never meeting.
- Solution
- The pair of values satisfying both equations.
Related topics
Written and reviewed against the current AQA, Edexcel and OCR specifications. Spotted an error? Let us know.