KS3 Maths · Year 8 · Chapter 11: Solving equations graphically · Lesson 1

Graphs from equations in the form ay ± bx = c

Year 8About 70 minutesNo calculator48 marks

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Date

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Quick questions to start.

A1
Needed today
Work out the value of $5 y - 2 x$ when $x = 3$ and $y = 4$ .
[1 mark]
Answer
A2
Needed today
Solve $- 4 x = 18$ .
[1 mark]
Answer
A3
From chapter 10
A cylinder has radius $3$ cm and height $10$ cm. Work out its volume. Leave your answer in terms of $π$ .
[1 mark]
Answer
A4
From chapter 9
Write $0.00072$ in standard form.
[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

Find where the line $4 y + 3 x = 36$ crosses each axis.

Hint On the

y

-axis

x = 0

, so cover up the

3 x

. On the

x

-axis

y = 0

, so cover up the

4 y

B1
Find where the line $5 y + 3 x = 30$ crosses each axis.
[2 marks]
Answer $(\dots\dots, \dots\dots)$ and $(\dots\dots, \dots\dots)$

Example 2

Find where the line $3 y - 2 x = 18$ crosses each axis.

Hint Keep the minus sign with the

2 x

: when

y = 0

you solve

- 2 x = 18

B2
Find where the line $2 y - 5 x = 20$ crosses each axis.
[2 marks]
Answer $(\dots\dots, \dots\dots)$ and $(\dots\dots, \dots\dots)$

Example 3

Draw the graph of $4 y + 6 x = 18$ on the grid.

Hint

4 y = 18

gives

y = 4.5

: plot it halfway between

4

and

5

B3
Draw the graph of $4 y + 5 x = 10$ on the grid.
[3 marks]

Example 4

The graph of $y = 2 x - 1$ is drawn. Use the graph to solve (a) $2 x - 1 = 4$ (b) $2 x - 1 = - 2$

Hint Find

4

on the

y

-axis, go across to the line, then down to the

x

-axis.

B4
The graph of $2 x + y = 6$ is drawn. Use the graph to solve (a) $6 - 2 x = 3$ (b) $6 - 2 x = - 1$
[2 marks]
Answer(a) $\dots\dots$ (b) $\dots\dots$

Example 5

Rearrange $2 y - 8 x = 6$ into the form $y = m x + c$ . Write down the gradient of the line.

Hint Add

8 x

to both sides, then divide every term by

2

B5
Rearrange $4 y - 2 x = 12$ into the form $y = m x + c$ . Write down the gradient of the line.
[2 marks]
Answer

Practice

Year 8

C1
Which two of these lines are parallel? $A$ : $2 y - 4 x = 6$ $B$ : $3 y + 6 x = 12$ $C$ : $4 y - 8 x = 20$
[2 marks]
Answer $\dots\dots$ and $\dots\dots$
C2
Which of the points $(2, 6)$ , $(- 2, 0)$ and $(4, 8)$ lie on the line $2 y - 3 x = 6$ ?
[2 marks]
Answer

C3
Which of these is the equation of the line on the grid? Give a reason. $A$ : $4 x - 3 y = 12$ $B$ : $3 x - 4 y = 12$ $C$ : $4 x + 3 y = 12$
[2 marks]
Answer
C4
A line crosses the axes at $(0, 5)$ and $(- 2, 0)$ . Write its equation in the form $a y - b x = c$ , where $a$ , $b$ and $c$ are whole numbers.
[2 marks]
Answer

Exam-style questions

GCSE Foundation

D1
Leo says that the line $3 y - 4 x = 24$ crosses the $x$ -axis at $(6, 0)$ .
Leo is wrong. Explain his mistake, and write down where the line does cross the $x$ -axis.
[3 marks]
Answer $(\dots\dots, \dots\dots)$

D2
This question is about the line $3 y + 2 x = 15$ .
(a)
Find where the line crosses each axis.
[2 marks]
Answer $(\dots\dots, \dots\dots)$ and $(\dots\dots, \dots\dots)$
(b)
Draw the line on the grid.
[2 marks]
(c)
Use your line to find the value of $y$ when $x = 3$ .
[1 mark]
Answer
Total for D2: 5 marks
D3
The graph of $2 y - x = 4$ is drawn on the grid.
(a)
Use the graph to find $x$ when $y = 3.5$ .
[1 mark]
Answer
(b)
Write down where the line crosses the $x$ -axis.
[1 mark]
Answer
(c)
Rearrange $2 y - x = 4$ into the form $y = m x + c$ .
[2 marks]
Answer
Total for D3: 4 marks
D4
The line $5 y + 4 x = 60$ and the two axes make a triangle. Work out the area of the triangle.
[4 marks]
Not drawn accurately
Answer
D5
Children's tickets for a show cost £ $4$ and adult tickets cost £ $6$ . A group spends exactly £ $48$ . The number of children, $x$ , and the number of adults, $y$ , fit the equation $4 x + 6 y = 48$ .
(a)
Find where the graph of $4 x + 6 y = 48$ crosses each axis.
[2 marks]
Answer $(\dots\dots, \dots\dots)$ and $(\dots\dots, \dots\dots)$
(b)
What does the point $(12, 0)$ mean for the group?
[1 mark]
(c)
The group has $4$ adults. How many children are in the group?
[2 marks]
Answer
Total for D5: 5 marks

Extension

Stretch

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

E1
The line $p x + q y = 24$ crosses the $x$ -axis at $(8, 0)$ . The line and the two axes make the shaded triangle, of area $24$ . Find $p$ and $q$ .
[4 marks]
Not drawn accurately
Hint 1 · What to try
At $(8, 0)$ , $y = 0$ , so $8 p = 24$ .
Hint 2 · The first line
The triangle's base is $8$ . Its height is where the line crosses the $y$ -axis.
Hint 3 · The full method
$\frac{1}{2} \times 8 \times h = 24$ gives $h = 6$ , so $6 q = 24$ .
Answer $p = \dots\dots$ $q = \dots\dots$

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can find where a line such as $3 y - 2 x = 18$ crosses each axis, and draw it.
I can solve an equation by reading across from a graph.
I can rearrange $a y \pm b x = c$ into $y = m x + c$ to find the gradient.

Answers and mark scheme

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