top of page

20.3 Solving Simultaneous Equations Graphically

GCSE Maths Foundation · Edexcel · Chapter 20: More Algebra

A free GCSE Maths Foundation lesson for Edexcel (9–1) from OBA School: using the point where two straight lines cross. Watch the video, read the explanation and worked examples, then try the revision questions. You can work through the full lesson, with interactive activities, practice quizzes and printable worksheets, in our free online course.

What you will learn

  • Understand that the solution of two simultaneous equations is the point where their graphs cross
  • Draw straight-line graphs from equations such as y = 2x − 1 and x + y = 5
  • Read and check solutions from a graph
  • Recognise when there is no solution, and solve real-life problems graphically

Key words

  • Simultaneous equations: Two equations with the same two unknowns, which are both true at the same time.
  • Point of intersection: The point where two graphs cross. Its coordinates fit both equations.
  • Gradient: The steepness of a line. In y = mx + c the gradient is m.
  • Intercept: Where a line crosses an axis. In y = mx + c the y-intercept is c.

The solution is where the lines cross

Every point on a line fits its equation. The point where two lines cross is on both lines, so it fits both equations at once. Its x- and y-coordinates are the solution.

Worked example: read and check

The lines cross at (3, 4), so x = 3 and y = 4.
Check: y = x + 1 gives 3 + 1 = 4 ✓. y = 7 − x gives 7 − 3 = 4 ✓

Answer: x = 3, y = 4

Drawing the lines

For y = mx + c, make a table with three x values. For an equation such as x + y = 5, the quickest way is the cover-up method: put x = 0 to find where it crosses the y-axis, and y = 0 to find where it crosses the x-axis.

Worked example: solve x + y = 5 and y = 2x − 1

x + y = 5: when x = 0, y = 5; when y = 0, x = 5. Join (0, 5) and (5, 0).
y = 2x − 1: x = 0, 1, 3 gives y = −1, 1, 5.
The lines cross at (2, 3). Check: 2 + 3 = 5 ✓ and 2 × 2 − 1 = 3 ✓

Answer: x = 2, y = 3

Key fact: Always check your answer by putting the values back into both equations. A graph only gives exact answers if the lines cross on a grid point.

When there is no solution

If two lines have the same gradient but different intercepts, they are parallel. They never cross, so the equations have no solution.

Worked example: can these be solved?

y = 2x + 1 and y = 2x − 3 both have gradient 2.
They are parallel lines, 4 units apart on the y-axis.

Answer: No solution

Real-life problems

Graphs are useful for comparing two plans or prices. The crossing point shows where they cost the same.

Worked example: phone plans

Plan A costs £10 a month plus £2 per GB: y = 10 + 2x.
Plan B costs £4 per GB: y = 4x.
The lines cross at (5, 20): both plans cost £20 for 5 GB.
For more than 5 GB, Plan A is cheaper; for less than 5 GB, Plan B is cheaper.

Answer: Same cost at 5 GB (£20)

Common mistakes

  • Writing the solution the wrong way round. A point (3, 4) means x = 3 and y = 4.
  • Not drawing the lines long enough to cross.
  • Plotting x + y = 5 as if it were y = x + 5.
  • Not checking the answer in both equations.

Revision questions

Try these, then check your answers below each one.

Question 1. The lines y = x + 2 and y = 6 - x are drawn. Where do they cross? (Choose from: (2, 4), (4, 2), (2, 6), (−2, 0).)

Answer: (2, 4)

Working: At (2, 4): 2 + 2 = 4 ✓ and 6 - 2 = 4 ✓.

Question 2. Do the simultaneous equations y = 4x + 1 and y = 4x - 5 have a solution?

Answer: No

Working: Both lines have gradient 4, so they are parallel and never cross. There is no solution.

Question 3. The point (3, 1) is where two lines cross. Which pair of lines could they be? (Choose from: y = x - 2 and x + y = 4, y = 2x and x + y = 4, y = x + 2 and x + y = 4, y = 3x and x - y = 2.)

Answer: y = x - 2 and x + y = 4

Working: 3 - 2 = 1 ✓ and 3 + 1 = 4 ✓. Each other pair fails at least one check.

Related lessons

Continue in the full course

This lesson is section 20.3 (EGCMF2003) of our free GCSE Maths Foundation course. In the full course you get every video for this section, an animated interactive lesson, a self-marking practice quiz, a printable worksheet with a mark scheme, and your progress saved. Go to the full GCSE Maths Foundation course.

 

 

 

 

 

 

 

bottom of page