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Maths · Equations and graphs
Solving simultaneous equations graphically
Draw two straight lines on the same grid and read off the point where they cross to solve a pair of simultaneous equations.
Warm-up
Answer each one, then check.
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1
What is the gradient of \(y = 3x + 2\)?
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\(3\)
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2
Where does \(y = 3x + 2\) cross the y-axis?
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\((0, 2)\)
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3
Find \(y\) when \(x = 2\) in \(y = 2x - 1\).
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\(3\)
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4
What does "simultaneous" mean?
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Both equations true at the same time
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5
Rearrange \(x + y = 5\) to make \(y\) the subject.
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\(y = 5 - x\)
Learning Objectives
- 1Draw straight lines from equations.
- 2Find where two lines intersect.
- 3Read the solution to a pair of simultaneous equations.
- 4Check the solution by substituting.
GRAPHICAL SOLUTION
The coordinates of the point where two lines cross satisfy both equations at once.
The solution to \(y = 2x - 1\) and \(x + y = 5\) is the point \((2, 3)\), so \(x = 2\) and \(y = 3\).
Where the Lines Cross
The intersection gives both x and y.
Solving Graphically
Draw both lines carefully.
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1
Rearrange
Make each equation \(y = mx + c\) if you need to
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2
Make a table or use the gradient and intercept
Find at least three points per line
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3
Draw both lines
Use a ruler and extend across the grid
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4
Read the crossing point
Write down \(x\) and \(y\)
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5
Check
Substitute into both equations
Solve by Drawing
Solve \(y = 2x - 1\) and \(x + y = 5\) graphically.
Show the solutionHide the solution
- 1 Line 1 points \((0, -1)\), \((2, 3)\), \((4, 7)\)
- 2 Line 2: \(y = 5 - x\) \((0, 5)\), \((2, 3)\), \((5, 0)\)
- 3 Lines cross at \((2, 3)\)
- 4 Check line 1 \(2 \times 2 - 1 = 3\), correct
- 5 Check line 2 \(2 + 3 = 5\), correct
Answer\(x = 2\) and \(y = 3\)
No Solutions
Explain why \(y = 2x + 1\) and \(y = 2x + 5\) have no solutions.
Show the solutionHide the solution
- 1 Gradients Both are \(2\)
- 2 Parallel lines They never meet
AnswerThe lines are parallel, so there is no point that satisfies both equations.
Tips
Get an accurate answer.
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Ruler
Use a sharp pencil and a ruler.
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Check
Always substitute back to check.
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Rearranging
Get both equations into \(y = mx + c\) form before drawing.
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Estimates
A graph gives an approximate answer unless the point is on a grid crossing.
Find the Crossing Point
Draw \(y = x - 1\) and \(y = 5 - x\) on the same axes for \(x\) from 0 to 5. Read the solution.
1. Make a table of values for each line.
2. Read where they cross.
A good answer shows: The lines cross at \((3, 2)\), so \(x = 3\) and \(y = 2\).
Can I...?
- 1Draw a line from its equation.
- 2Rearrange to \(y = mx + c\).
- 3Draw two lines on one grid.
- 4Read the intersection.
- 5Write \(x\) and \(y\) values.
- 6Check by substitution.
- 7Explain why parallel lines have no solution.
- 8Draw accurately.
Summary & Exam Focus
- Solution = coordinates of the intersection.
- Check in both equations.
- Parallel lines have no solution.
- Rearrange first to help you draw.
Exam focus
On the grid, draw the graph of \(x + y = 5\). Hence solve \(y = 2x - 1\) and \(x + y = 5\). (4 marks) (4 marks)
Plot at least three points for each line. Circle the intersection.
Key terms
The vocabulary this lesson expects you to use. Each one is linked from the first place it appears above.
- Simultaneous equations
- Equations that are both true for the same values.
- Intersection
- The point where two lines cross.
- Gradient
- How steep a line is.
- Intercept
- Where a line crosses an axis.
- Parallel
- Lines with the same gradient.
- Solution
- The values of \(x\) and \(y\) that satisfy both equations.
Practice questions
Have a go at each one before you open its answer.
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Question 1 Draw 4 marks
The graph of \(y = 2x - 1\) is drawn on the grid. On the grid, draw the graph of \(x + y = 5\). Hence solve the simultaneous equations \(y = 2x - 1\) and \(x + y = 5\).
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Model answer
The line \(x + y = 5\) passes through \((0, 5)\), \((2, 3)\) and \((5, 0)\). The lines cross at \((2, 3)\), so \(x = 2\) and \(y = 3\).
Mark scheme
- Two correct points for the second line — M1
- Correct line drawn — A1
- Intersection identified — M1
- \(x = 2\), \(y = 3\) — A1
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Question 2 Solve 3 marks
Use a graph to solve \(y = x - 1\) and \(y = 5 - x\), for values of \(x\) from 0 to 5.
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Model answer
The lines cross at \((3, 2)\), so \(x = 3\) and \(y = 2\).
Mark scheme
- Table or points for both lines — M1
- Both lines drawn — A1
- \(x = 3\), \(y = 2\) — A1
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Question 3 Show that 3 marks
The lines \(y = 3x - 2\) and \(y = x + 4\) cross at the point \((3, 7)\). Show that \(x = 3\) and \(y = 7\) satisfy both equations.
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Model answer
\(3 \times 3 - 2 = 7\), and \(3 + 4 = 7\). Both are true.
Mark scheme
- Substitutes into first — M1
- Substitutes into second — M1
- Both correct — A1
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Question 4 Explain 2 marks
Explain why the simultaneous equations \(y = 2x + 1\) and \(y = 2x + 5\) have no solutions.
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Model answer
Both lines have gradient 2, so they are parallel and never meet.
Mark scheme
- Same gradient — M1
- Parallel, so no intersection — C1
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Question 5 Solve 3 marks
The lines \(y = 2x + 1\) and \(y = 7 - x\) are drawn on a grid. They meet at a single point. Find the coordinates of the point using algebra, and say how you could check this on the graph.
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Model answer
\(2x + 1 = 7 - x\), so \(3x = 6\), \(x = 2\), \(y = 5\). The point is \((2, 5)\). On the graph both lines pass through \((2, 5)\).
Mark scheme
- Equates the two \(y\) values — M1
- \(x = 2\) — A1
- \(y = 5\) — A1
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Question 6 Write down 2 marks
The graph shows two lines crossing at \((4, -1)\). Write down the solution to the pair of simultaneous equations.
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Model answer
\(x = 4\) and \(y = -1\)
Mark scheme
- \(x = 4\) — B1
- \(y = -1\) — B1
Quick check
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The solution of two simultaneous linear equations on a graph is...
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C: The point where the lines cross
The point where the two lines cross.
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Lines \(y = x + 1\) and \(y = x + 4\) are...
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B: Parallel
Parallel: same gradient, so they never meet.
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Two lines cross at \((2, 3)\). Then...
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A: \(x = 2,\ y = 3\)
\(x = 2\) and \(y = 3\).
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\(x + y = 5\) rearranges to...
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D: \(y = 5 - x\)
Subtract \(x\) from both sides.
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How many points do you need to draw a straight line accurately?
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C: 3
At least three, so you can check they line up.
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The lines \(y = 2x\) and \(y = 2x\) have...
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B: Infinitely many solutions
Infinitely many common points, because they are the same line.
Downloads
Free to keep, print and annotate.
- Solving simultaneous equations graphically.pptx Built from the lesson script on 30 September 2026. View
- Solving simultaneous equations graphically - Completed Notes.docx The full notes for the lesson, to revise from. Built from the lesson script on 30 September 2026. View
- Solving simultaneous equations graphically - Exam Questions.docx Built from the lesson script on 30 September 2026. View
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