You can use your knowledge of linear graphs to draw the graphs of both equations on the same axes.
If there exists a solution, the point of intersection is that solution.
If the lines are parallel then there are no solutions.
If the lines lie on top of each other, there are infinitely many solutions.

Common misconception

Equations always have one solution.

Some equations have one solution but, some have no solutions, some have two solutions, some have more than two solutions.

Keywords

Linear - The relationship between two variables is linear if, when plotted on a pair of axes, a straight line is formed.

If you have access to technology, encourage pupils to explore graphing and see if they can come up with their own equations which have no solutions and ones which have infinitely many solutions.

Teacher tip

Licence

This content is © Oak National Academy Limited (2024), licensed on Open Government Licence version 3.0 except where otherwise stated. See Oak's terms & conditions (Collection 2).

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Starter quiz

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6 Questions

Q1.

Using substitution, what is the value of $$x$$ for simultaneous equations $$6x + 4y = 32$$ and $$x + 3y = 17$$?

Correct Answer: 2, x = 2

Q2.

Using substitution, what is the value of $$x$$ for simultaneous equations $$6x + 4y = 64$$ and $$x + 3y = 34$$?

Correct Answer: 4, x = 4

Q3.

Using substitution, what is the value of $$y$$ for simultaneous equations $$10x + 8y = 56$$ and $$x + 3y = 10$$?

Correct Answer: 2, y = 2

Q4.

Using substitution, what is the value of $$y$$ for simultaneous equations $$14x + 4y = 6$$ and $$2x = 3 - y$$?

Correct Answer: 5, y = 5

Q5.

Using substitution, what is the value of $$y$$ for simultaneous equations $$14x + 4y = 72$$ and $$2x + y = 15$$?

Correct Answer: 11, y = 11

Q6.

Using substitution, what is the value of $$y$$ for simultaneous equations $$2y - 2x = -2$$ and $$3x + y = 19$$?

Correct Answer: 4, y = 4

Exit quiz

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6 Questions

Q1.

If A and B are simultaneous equations such that A: $$x + 2y = 21$$ and B: $$3x + y = 43$$ where will coordinate pair $$(1, 10)$$ go in the Venn diagram?

Correct answer: In A but not B

In B but not A

The intersection of A and B

Outside of both A and B

Q2.

If A and B are simultaneous equations such that A: $$2x + 2y = 10$$ and B: $$3x + y = 7$$ where will coordinate pair $$(1, 4)$$ go in the Venn diagram?

In A but not B

In B but not A

Correct answer: The intersection of A and B

Outside of both A and B

Q3.

If A and B are simultaneous equations such that A: $$2y - 2x = 8$$ and B: $$3x + y = 4$$ where will coordinate pair $$(0, 0)$$ go in the Venn diagram?

In A but not B

In B but not A

The intersection of A and B

Correct answer: Outside of both A and B

Q4.

If A and B are simultaneous equations such that A: $$3x + y = 40$$ and B: $$7x + y = 88$$ where will coordinate pair $$(12, 4)$$ go in the Venn diagram?

In A but not B

In B but not A

Correct answer: The intersection of A and B

Outside of both A and B

Q5.

If A and B are simultaneous equations such that A: $$3x + y = 13$$ and B: $$7x + 6y = 34$$ where will coordinate pair $$(5, -2)$$ go in the Venn diagram?

Correct answer: In A but not B

In B but not A

The intersection of A and B

Outside of both A and B

Q6.

If A and B are simultaneous equations such that A: $$3x + y = 9$$ and B: $$4x +3y = 7$$ where will coordinate pair $$(1, 1)$$ go in the Venn diagram?

In A but not B

Correct answer: In B but not A

The intersection of A and B

Outside of both A and B