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Graphically solving two linear graphs that intersect

I can recognise that the point of intersection of two linear graphs satisfies both relationships and hence represents the solution to both those equations.

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Year 9

Graphically solving two linear graphs that intersect

I can recognise that the point of intersection of two linear graphs satisfies both relationships and hence represents the solution to both those equations.

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Lesson details

Key learning points

More than one linear graph can be drawn on the same set of axes.
A pair of coordinates which lay on the line satisfy the equation of the line.
The pair of coordinates for the point of intersection satisfy both equations.
Graphical software can be used to show this visually.

Keywords

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

Common misconception

There is only one way to solve linear equations; by manipulation and isolating the variable(s).

When we plot more than one linear equation on a graph we can use the intersection to find the solution.

Encourage pupils to verbally explain why it is that the intersection of two linear equations is the solution and why no other point is.

Teacher tip

Licence

This content is © Oak National Academy Limited (2025), licensed on Open Government Licence version 3.0 except where otherwise stated. See Oak's terms & conditions (Collection 2).

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Prior knowledge starter quiz

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6 Questions

Q1.
The graph of a __________ equation is a straight line.

arithmetic

Correct answer: linear

quadratic

variable

Q2.
Which of these lines has the least steep gradient?

Line a

Line b

Correct answer: Line c

Q3.
Use the graph of $$y =3x-4$$ to solve the equation $$3x-4=2$$. The solution is $$x=$$ .

Correct Answer: 2, two

Q4.
Use the graph of $$y =7.4 x+12.95$$ to estimate the value of $$7.4\times 4+12.95$$.

Correct answer: 43

Q5.
Aisha is plotting the graph of the line $$3y + 4x=24$$. She only wants to plot two points. Which of these points would be the best points for Aisha to use?

Correct answer: When $$x=0$$

When $$x=2$$

When $$x=4$$

Correct answer: When $$x=6$$

When $$x=8$$

Q6.
Select the points which lie both on the line $$y=2x+1$$ and on the line $$y=3x-2$$.

(1, 3)

(4, 10)

(0, -2)

Correct answer: (3, 7)

(2, 5)

Assessment exit quiz

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6 Questions

Q1.
The point where the lines $$y=2x+4$$ and $$y=2-x$$ meet is called the .

answer

Correct answer: intersection

solution

variable

Q2.
Izzy plots the graphs of $$y=2x+1$$ and $$y=3x-2$$ on the same coordinate grid. How many points will lie on both lines?

Correct Answer: 1, 1 point , one point

Q3.
The linear equations $$y = 2x + 5$$ and $$y = 2 − x$$ intersect at the point with coordinates (−1, 3). What is the solution to the equation $$2x+5=2-x$$?

Correct answer: $$x=-1$$

$$x=3$$

$$y=-1$$

$$y=3$$

Q4.
Use the graph to solve the simultaneous equations $$y=2x-1$$ and $$y=3x-2$$.

$$x=2$$, $$y=3$$

$$x=7$$, $$y=-2$$

$$x=1$$, $$y=1$$

Correct answer: $$x=-2$$, $$y=7$$

$$x=3$$, $$y=2$$

Q5.
The diagram shows 6 points and the the lines $$y=2x-1$$ and $$y=5-x$$. Which points satisfy the inequality $$y>2x-1$$ ?

A and C

C and D

A and D

Correct answer: A, B and C

D, E and F

Q6.
The diagram shows 6 points and the the lines $$y=2x-1$$ and $$y=5-x$$. The point that is the region $$y>5-x$$ and $$y<2x-1$$ is .

Correct Answer: D, d

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UnitMaths / Graphical representations

Maths

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Graphically solving two linear graphs that intersect

Graphically solving two linear graphs that intersect

Lesson slides

Lesson details

Key learning points

Keywords

Common misconception

Licence

Lesson video

Worksheet

Prior knowledge starter quiz

6 Questions

Q1.The graph of a __________ equation is a straight line.

Q2.Which of these lines has the least steep gradient?

Q3.Use the graph of $$y =3x-4$$ to solve the equation $$3x-4=2$$. The solution is $$x=$$ .

Q4.Use the graph of $$y =7.4 x+12.95$$ to estimate the value of $$7.4\times 4+12.95$$.

Q5.Aisha is plotting the graph of the line $$3y + 4x=24$$. She only wants to plot two points. Which of these points would be the best points for Aisha to use?

Q6.Select the points which lie both on the line $$y=2x+1$$ and on the line $$y=3x-2$$.

Assessment exit quiz

6 Questions

Q1.The point where the lines $$y=2x+4$$ and $$y=2-x$$ meet is called the .

Q2.Izzy plots the graphs of $$y=2x+1$$ and $$y=3x-2$$ on the same coordinate grid. How many points will lie on both lines?

Q3.The linear equations $$y = 2x + 5$$ and $$y = 2 − x$$ intersect at the point with coordinates (−1, 3). What is the solution to the equation $$2x+5=2-x$$?

Q4.Use the graph to solve the simultaneous equations $$y=2x-1$$ and $$y=3x-2$$.

Q5.The diagram shows 6 points and the the lines $$y=2x-1$$ and $$y=5-x$$. Which points satisfy the inequality $$y>2x-1$$ ?

Q6.The diagram shows 6 points and the the lines $$y=2x-1$$ and $$y=5-x$$. The point that is the region $$y>5-x$$ and $$y<2x-1$$ is .

Lesson appears in

UnitMaths / Graphical representations

Maths

Q1.
The graph of a __________ equation is a straight line.

Q2.
Which of these lines has the least steep gradient?

Q3.
Use the graph of $$y =3x-4$$ to solve the equation $$3x-4=2$$. The solution is $$x=$$ .

Q4.
Use the graph of $$y =7.4 x+12.95$$ to estimate the value of $$7.4\times 4+12.95$$.

Q5.
Aisha is plotting the graph of the line $$3y + 4x=24$$. She only wants to plot two points. Which of these points would be the best points for Aisha to use?

Q6.
Select the points which lie both on the line $$y=2x+1$$ and on the line $$y=3x-2$$.

Q1.
The point where the lines $$y=2x+4$$ and $$y=2-x$$ meet is called the .

Q2.
Izzy plots the graphs of $$y=2x+1$$ and $$y=3x-2$$ on the same coordinate grid. How many points will lie on both lines?

Q3.
The linear equations $$y = 2x + 5$$ and $$y = 2 − x$$ intersect at the point with coordinates (−1, 3). What is the solution to the equation $$2x+5=2-x$$?

Q4.
Use the graph to solve the simultaneous equations $$y=2x-1$$ and $$y=3x-2$$.

Q5.
The diagram shows 6 points and the the lines $$y=2x-1$$ and $$y=5-x$$. Which points satisfy the inequality $$y>2x-1$$ ?

Q6.
The diagram shows 6 points and the the lines $$y=2x-1$$ and $$y=5-x$$. The point that is the region $$y>5-x$$ and $$y<2x-1$$ is .