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Grade 5 · Foundation and Higher · free GCSE maths worksheet

Solving simultaneous equations graphically

Edexcel, AQA, OCR and Edexcel IGCSE

Reading the solution where two straight lines cross.

10 questions · 29 marks · type each answer for an instant mark, or print the sheet

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How to do it

  1. Draw both lines on the same grid (a table of values with 3 points for each)
  2. Find the point where the two lines cross
  3. Read its coordinates: the x-coordinate is x and the y-coordinate is y
  4. Check by putting x and y back into both equations
  5. Parallel lines never cross, so those equations have no solution

Watch out: Writing the answer the wrong way round: in (3, 2) x is 3 and y is 2.

Worked example

By drawing the graphs of y = x + 1 and y = 4 − 2x for values of x from 0 to 3, solve the simultaneous equations

y = x + 1

y = 4 − 2x

0123−2−112345xyy = x + 1y = 4 - 2x(1, 2)

Points on y = x + 1

(0, 1), (1, 2), (3, 4)

Points on y = 4 - 2x

(0, 4), (1, 2), (3, -2)

Draw both lines. They cross at (1, 2)

Check in both equations

1 + 1=2
4 - 2 × 1=2

x = 1, y = 2

The questions

1

The graphs of y = x + 1 and y = 7 − x are drawn on the grid.

Use the graphs to solve the simultaneous equations

y = x + 1

y = 7 − x

01234567012345678xyy = x + 1y = 7 - x
1 mark
Show answer
x = 3, y = 4
See the working

The lines cross at (3, 4)

Check

3 + 1=4
7 - 3=4

x = 3, y = 4

2

The graphs of y = 2x − 1 and x + y = 5 are drawn on the grid.

Use the graphs to solve the simultaneous equations

y = 2x − 1

x + y = 5

−112345−3−2−1123456Oxyy = 2x - 1x + y = 5
1 mark
Show answer
x = 2, y = 3
See the working

The lines cross at (2, 3)

Check

2 × 2 - 1=3
2 + 3=5

x = 2, y = 3

3

The graph of y = 2x is drawn on the grid.

  1. On the same grid, draw the graph of y = 6 − x for values of x from 0 to 6
  2. Use your graphs to solve the simultaneous equations y = 2x and y = 6 − x
0123456012345678xyy = 2x
3 marks
Show answer
a) Straight line through (0, 6), (3, 3) and (6, 0) b) x = 2, y = 4
See the working

a) Points on y = 6 - x

(0, 6), (3, 3), (6, 0)

Straight line through these points

b) The lines cross at (2, 4)

Check

2 × 2=4
6 - 2=4

x = 2, y = 4

4

The graph of x + y = 10 is drawn on the grid.

  1. Complete the table of values for y = 3x − 2
    x01234
    y17
    Write the missing values for x = 0, 2 and 4, in that order.
  2. On the grid, draw the graph of y = 3x − 2
  3. Use your graphs to solve the simultaneous equations x + y = 10 and y = 3x − 2
01234−2−112345678910xyx + y = 10
4 marks
Show answer
a) -2, 4, 10 b) Straight line through (0, -2), (1, 1), (2, 4), (3, 7) and (4, 10) c) x = 3, y = 7
See the working

a)

3 × 0 - 2=-2
3 × 2 - 2=4
3 × 4 - 2=10

-2, 4, 10

b) Plot every point in the table and join

Straight line from (0, -2) to (4, 10)

c) The lines cross at (3, 7)

Check

3 + 7=10
3 × 3 - 2=7

x = 3, y = 7

5
  1. On the grid, draw the graphs of 2x + 3y = 12 and y = x − 1 for values of x from 0 to 6
  2. Use your graphs to solve the simultaneous equations
    2x + 3y = 12
    y = x − 1
0123456−112345xy
4 marks
Show answer
a) 2x + 3y = 12 through (0, 4), (3, 2) and (6, 0); y = x - 1 through (0, -1), (3, 2) and (6, 5) b) x = 3, y = 2
See the working

a) 2x + 3y = 12

When x = 0

3y=12
y=4
When x = 6
12 + 3y=12
y=0

(0, 4), (6, 0)

y = x - 1

(0, -1), (3, 2), (6, 5)

Both lines drawn with a ruler

b) The lines cross at (3, 2)

Check

2 × 3 + 3 × 2=12
3 - 1=2

x = 3, y = 2

6

The graphs of y = 2x + 1 and y = 5 − x are drawn on the grid.

  1. Use the graphs to estimate the solution of the simultaneous equations y = 2x + 1 and y = 5 − x. Give each value to 1 decimal place.
  2. Explain why your answer to part (a) is only an estimate.
01230123456xyy = 2x + 1y = 5 - x
3 marks
Show answer
a) x = 1.3, y = 3.7 (accept x from 1.2 to 1.4 and y from 3.6 to 3.8) b) The lines do not cross on a grid point, so the values can only be read approximately
See the working

a) Read the crossing point carefully

one small square = 0.2

x ≈ 1.3, y ≈ 3.7

b)

The lines cross between grid lines

Values can only be read approximately

7

Maya wants to solve these simultaneous equations by drawing their graphs.

y = 2x + 3

4x − 2y = 2

Explain why she will not find a solution.

2 marks
Show answer
4x - 2y = 2 rearranges to y = 2x - 1. Both lines have gradient 2, so they are parallel and never cross, so there is no solution
See the working

Make y the subject

4x - 2y=2
+2y4x=2 + 2y+2y
-24x - 2=2y-2
÷22x - 1=y÷2

Both lines have gradient 2

Parallel lines never cross, so no solution

8

Two taxi firms work out their prices like this.

Firm A: £3 plus £2 per mile, so y = 2x + 3

Firm B: £6 plus £1.50 per mile, so y = 1.5x + 6

x is the number of miles and y is the cost in pounds. The graph shows both lines.

  1. For what distance do both firms charge the same?
  2. What is the cost for this distance?
  3. Which firm is cheaper for a 10 mile journey?
012345678910024681012141618202224Distance (miles)Cost (£)Firm AFirm B
4 marks
Show answer
a) 6 miles b) £15 c) Firm B (accept B)
See the working

a) The lines cross at (6, 15)

6 miles

b) Check

2 × 6 + 3=15
1.5 × 6 + 6=15

£15

c) 10 miles

A: 2 × 10 + 3=23
B: 1.5 × 10 + 6=21

Firm B

9

Aisha says the graphs of 3x + y = 11 and y = 2x − 4 cross at the point (2, 5).

  1. Is Aisha correct? You must show your working.
  2. Find the point where the two graphs cross.
3 marks
Show answer
a) No. (2, 5) is on 3x + y = 11, but 2 × 2 - 4 = 0, not 5, so it is not on y = 2x - 4 b) (3, 2) (accept x = 3, y = 2)
See the working

a) Test (2, 5) in both

3 × 2 + 5=11
2 × 2 - 4=0

0 is not 5, so (2, 5) is not on y = 2x - 4

No, Aisha is wrong

b) Put y = 2x - 4 into 3x + y = 11

3x + 2x - 4=11
5x - 4=11
+45x=15+4
÷5x=3÷5
y=2 × 3 - 4
=2

(3, 2)

10

Two numbers, x and y, add up to 9. Their difference, x − y, is 3.

  1. Write down two equations in x and y.
  2. On the grid, draw the graphs of your two equations for values of x from 0 to 9
  3. Use your graphs to find the two numbers.
0123456789−3−2−1123456789xy
4 marks
Show answer
a) x + y = 9 and x - y = 3 (in either order) b) x + y = 9 through (0, 9) and (9, 0); x - y = 3 through (3, 0) and (9, 6) c) x = 6, y = 3 (accept 6 and 3)
See the working

a)

x + y = 9

x - y = 3

b) x + y = 9

(0, 9), (4, 5), (9, 0)

x - y = 3 is y = x - 3

(3, 0), (6, 3), (9, 6)

c) The lines cross at (6, 3)

Check

6 + 3=9
6 - 3=3

x = 6, y = 3

Your score

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Every question here is original and written by Robinson Tuition. The grade follows the Edexcel grade descriptors and suits AQA and OCR too; this is not an exam-board paper. It is also on the Edexcel International GCSE (4MA1).

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