KS3 Maths · Year 8 · Chapter 11: Solving equations graphically · Lesson 2

Solving simultaneous equations by drawing graphs

Year 8About 70 minutesNo calculator45 marks

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Date

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Quick questions to start.

A1
Last lesson
Where does the line $2 y + 3 x = 12$ cross the $x$ -axis?
[1 mark]
Answer
A2
Needed today
Work out the value of $3 x - 2 y$ when $x = 4$ and $y = - 2$ .
[1 mark]
Answer
A3
From chapter 10
A cylinder has radius $5$ cm and height $4$ cm. Work out its curved surface area. Leave your answer in terms of $π$ .
[1 mark]
Answer
A4
Year 7
Solve $5 x - 7 = 2 x + 8$ .
[1 mark]
Answer

Example, then your turn

Year 8

Your teacher works through each example with you. Then try the one beside it.

Example 1

The graphs of $y = x + 1$ and $x + y = 7$ are drawn. Use them to solve the simultaneous equations.

Hint The solution is the one point that is on both lines.

B1
The graphs of $y = 3 x - 2$ and $x + 2 y = 10$ are drawn. Use them to solve the simultaneous equations.
[2 marks]
Answer $x = \dots\dots$ $y = \dots\dots$

Example 2

Draw the graphs of $2 x + y = 8$ and $y = x + 2$ on the grid. Use them to solve the simultaneous equations.

Hint

2 x + y = 8

crosses the axes at

(0, 8)

and

(4, 0)

. For

y = x + 2

, find two points such as

(0, 2)

and

(3, 5)

B2
Draw the graphs of $x + 2 y = 8$ and $y = 2 x - 1$ on the grid. Use them to solve the simultaneous equations.
[3 marks]
Answer $x = \dots\dots$ $y = \dots\dots$

Example 3

Is $x = 4$ , $y = - 1$ the solution of $2 x + 3 y = 5$ and $x - y = 5$ ?

Hint Put the values into both equations. Both must be true.

B3
Is $x = 3$ , $y = 2$ the solution of $3 x - y = 7$ and $x + 2 y = 8$ ? Show how you decide.
[2 marks]

Example 4

Explain why $y = 2 x + 1$ and $2 y - 4 x = 6$ have no solution.

Hint Rearrange the second equation into the form

y = m x + c

B4
How many solutions do $x + 2 y = 6$ and $2 x + 4 y = 12$ have? Give a reason.
[2 marks]

Example 5

The graphs of $y = 3 x$ and $x + y = 2$ are drawn. Use them to solve the simultaneous equations.

Hint The lines cross between two gridlines: read the halfway value.

B5
The graphs of $y = x - 1$ and $x + y = 6$ are drawn. Use them to solve the simultaneous equations.
[2 marks]
Answer $x = \dots\dots$ $y = \dots\dots$

Practice

Year 8

C1
The graphs of $x = 3$ and $y = 2 x - 1$ are drawn. Write down the solution of the simultaneous equations.
[2 marks]
Answer $x = \dots\dots$ $y = \dots\dots$
C2
Two numbers $x$ and $y$ add up to $11$ . Their difference, $x - y$ , is $5$ . Write down two equations, and solve them.
[3 marks]
Answer

C3
The lines $x + y = 3$ and $y = m x + 6$ cross at the point $(- 1, 4)$ . Work out the value of $m$ .
[2 marks]
Answer
C4
Which pair of equations has the solution $x = 2$ , $y = - 1$ ? $A$ : $x + y = 1$ and $y = x - 3$ $B$ : $x + y = 1$ and $y = 2 x + 1$ $C$ : $x - y = 3$ and $x + y = 3$
[2 marks]
Answer

Exam-style questions

GCSE Foundation

D1
Ana says, "The solution of $x + y = 6$ and $y = x + 2$ is $x = 4$ , $y = 2$ , because $4 + 2 = 6$ ."
Explain why Ana is wrong. Find the correct solution.
[4 marks]
Answer $x = \dots\dots$ $y = \dots\dots$

D2
At a museum, $2$ adult tickets and $1$ child ticket cost £ $20$ altogether. An adult ticket costs £ $4$ more than a child ticket. Let a child ticket cost £ $x$ and an adult ticket cost £ $y$ .
(a)
Explain why $x + 2 y = 20$ . Write down a second equation.
[1 mark]
Answer
(b)
Draw the graphs of $x + 2 y = 20$ and $y = x + 4$ on the grid.
[2 marks]
(c)
Use your graphs to find the cost of each ticket.
[2 marks]
Answerchild £ $\dots\dots$ adult £ $\dots\dots$
Total for D2: 5 marks
D3
The lines $A$ : $y = x + 1$ , $B$ : $x + y = 5$ and $C$ : $y = - 1$ are drawn on the grid.
(a)
Solve the simultaneous equations $y = x + 1$ and $x + y = 5$ .
[1 mark]
Answer $x = \dots\dots$ $y = \dots\dots$
(b)
Solve the simultaneous equations $y = x + 1$ and $y = - 1$ .
[1 mark]
Answer $x = \dots\dots$ $y = \dots\dots$
(c)
The three lines make a triangle. Work out its area.
[2 marks]
Answer
Total for D3: 4 marks
D4
Draw the graphs of $3 x + 2 y = 18$ and $y = x - 1$ on the grid. Use them to solve the simultaneous equations.
[4 marks]
Answer $x = \dots\dots$ $y = \dots\dots$

Extension

Stretch

No route is given. The hints are at the end of the sheet. Use one at a time.

E1
The lines $k x + 2 y = 12$ and $y = 2 x - 6$ meet at a point on the $x$ -axis. Work out the value of $k$ . Then find where the line $k x + 2 y = 12$ crosses the $y$ -axis.
[4 marks]
Not drawn accurately
Hint 1 · What to try
On the $x$ -axis $y = 0$ . Find where $y = 2 x - 6$ crosses it.
Hint 2 · The first line
$y = 2 x - 6$ crosses the $x$ -axis at $(3, 0)$ , so $(3, 0)$ is on $k x + 2 y = 12$ too.
Hint 3 · The full method
$3 k = 12$ . On the $y$ -axis $x = 0$ , so $2 y = 12$ .
Answer $k = \dots\dots$ $(\dots\dots, \dots\dots)$

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can draw two lines on one grid and read where they cross.
I can check a solution by putting it into both equations.
I can explain why parallel lines give no solution.

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