EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
Solving simultaneous equations graphically
Equations and graphs · Lesson 1 of 6
Warm-up
Answer each one, then check.
1. What is the gradient of y = 3x + 2?
3
2. Where does y = 3x + 2 cross the y-axis?
(0, 2)
3. Find y when x = 2 in y = 2x − 1.
3
4. What does "simultaneous" mean?
Both equations true at the same time
5. Rearrange x + y = 5 to make y the subject.
y = 5 − x
Learning Objectives
1. Draw straight lines from equations.
2. Find where two lines intersect.
3. Read the solution to a pair of simultaneous equations.
4. Check 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
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The intersection gives both x and y. |
Two straight lines y equals 2x minus 1 and x plus y equals 5 crossing at the point (2, 3).
Solving Graphically
Draw both lines carefully.
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1 Rearrange Make each equation y = mx + c if you need to |
2 Make a table or use the gradient and intercept Find at least three points per line |
3 Draw both lines Use a ruler and extend across the grid |
4 Read the crossing point Write down x and y |
5 Check Substitute into both equations |
Solve by Drawing
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Solve y = 2x − 1 and x + y = 5 graphically. |
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 × 2 − 1 = 3, correct
5. Check line 2
2 + 3 = 5, correct
Answer: x = 2 and y = 3
No Solutions
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Explain why y = 2x + 1 and y = 2x + 5 have no solutions. |
1. Gradients
Both are 2
2. Parallel lines
They never meet
Answer: The lines are parallel, so there is no point that satisfies both equations.
Tips
Get an accurate answer.
▸ Ruler. Use a sharp pencil and a ruler.
▸ Check. Always substitute back to check.
▸ Rearranging. Get both equations into y = mx + c form before drawing.
▸ Estimates. A graph gives an approximate answer unless the point is on a grid crossing.
Key Terms
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Simultaneous equations Equations that are both true for the same values. |
Intersection The point where two lines cross. |
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Gradient How steep a line is. |
Intercept Where a line crosses an axis. |
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Parallel Lines with the same gradient. |
Solution The values of x and y that satisfy both equations. |
Your Task: Find the Crossing Point
12 minutes
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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...?
☐ Draw a line from its equation.
☐ Rearrange to y = mx + c.
☐ Draw two lines on one grid.
☐ Read the intersection.
☐ Write x and y values.
☐ Check by substitution.
☐ Explain why parallel lines have no solution.
☐ Draw accurately.
Summary
✓ Solution = coordinates of the intersection.
✓ Check in both equations.
✓ Parallel lines have no solution.
✓ Rearrange first to help you draw.
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EXAM FOCUS On the grid, draw the graph of x + y = 5. Hence solve y = 2x − 1 and x + y = 5. (4 marks) Plot at least three points for each line. Circle the intersection. |