GCSE Maths · Higher · Chapter 10: Linear graphs

Gradients, equations of lines and real-life graphs

Grades 4–9About 80 minutesNo calculator91 marks

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Before you start

Grades 4–5

You will need these in every question below.

A1
Work out the value of $2 x - 7$ when $x = - 3$ .
[1 mark]
Answer

A2
Write down the coordinates of the point where the line $y = 5 x - 8$ crosses the $y$ -axis.
[1 mark]
Answer
A3
Rearrange $4 x + 2 y = 18$ to make $y$ the subject.
[2 marks]
Answer
A4
The point $(3, k)$ lies on the line $y = 2 x - 9$ . Work out the value of $k$ .
[1 mark]
Answer
A5
A line goes up 6 squares for every 4 squares it goes across. Work out its gradient. Give your answer as a fraction in its simplest form.
[1 mark]
Answer

Examples, then your turn

Grades 5–6

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

Example 1

Work out the gradient of the line through $(2, 3)$ and $(5, 12)$ .

Hint Gradient is the change in

y

divided by the change in

x

B1
Work out the gradient of the line through $(- 1, 7)$ and $(3, - 5)$ .
[2 marks]
Answer

Example 2

Write down the gradient and the $y$ -intercept of the line $3 y = 6 x - 12$ .

Hint Divide every term by 3 to get the equation in the form

y = m x + c

B2
Write down the gradient and the $y$ -intercept of the line $2 y + 5 x = 8$ .
[2 marks]
Answergradient $\dots\dots$ , $y$ -intercept $\dots\dots$

Example 3

A line has gradient 5 and passes through $(2, 3)$ . Work out its equation.

Hint Put

x = 2

and

y = 3

into

y = 5 x + c

to find

c

B3
Work out the equation of the line through $(1, 10)$ and $(4, 1)$ .
[3 marks]
Answer

Example 4

Work out the equation of the line that is perpendicular to $y = 2 x + 1$ and passes through $(4, 3)$ .

Hint The gradients of perpendicular lines multiply to

- 1

, so the new gradient is

- \frac{1}{2}

B4
Work out the equation of the line that is perpendicular to $y = 3 x - 2$ and passes through $(6, 1)$ .
[3 marks]
Answer

Practice

Grades 5–7

Each question asks for something different. Show your working.

C1
For the line $y = 3 - 2 x$ , work out the value of $y$ when $x = - 2$ , when $x = 0$ and when $x = 3$ .
[2 marks]
Answer

C2
Work out the equation of the line shown on the grid.
[3 marks]
Answer
C3
The line $4 x + 5 y = 20$ crosses the $x$ -axis at $A$ and the $y$ -axis at $B$ . Write down the coordinates of $A$ and of $B$ .
[2 marks]
Answer $A$ $\dots\dots$ , $B$ $\dots\dots$
C4
A line is parallel to $3 y = 15 x + 2$ and passes through $(2, 1)$ . Work out its equation.
[3 marks]
Answer
C5
$A$ is the point $(- 2, 1)$ and $B$ is the point $(4, 4)$ .
(a)
Work out the gradient of $A B$ .
[2 marks]
Answer
(b)
Work out the equation of the line through $A$ and $B$ .
[2 marks]
Answer
(c)
Show that the point $(10, 7)$ lies on the same line.
[2 marks]
Total for C5: 6 marks
C6
The graph is used to change between pounds (£) and euros (€).
(a)
Use the graph to change £50 into euros.
[1 mark]
Answer€ $\dots\dots$
(b)
Lena changes €150 into pounds. Work out how many pounds she gets.
[2 marks]
Answer£ $\dots\dots$
Total for C6: 3 marks
C7
Water drains out of a tank. The graph shows the depth of water in the tank.
(a)
Work out the gradient of the line.
[2 marks]
Answer
(b)
Explain what the gradient tells you about the water.
[1 mark]
Total for C7: 3 marks
C8
Use the graph to solve the simultaneous equations $y = x + 1$ and $y = 7 - 2 x$ .
[2 marks]
Answer $x = \dots\dots$ , $y = \dots\dots$
C9
Work out the coordinates of the point where the line $y = 3 x - 4$ meets the line $y = 8$ .
[2 marks]
Answer

Exam-style questions

Grades 6–9

Answer every question in the space given. Show your working: most of the marks are for method.

D1
$A$ is the point $(1, 2)$ and $B$ is the point $(7, 10)$ . Work out the equation of the perpendicular bisector of $A B$ .
[5 marks]
Answer

D2
Amir says, "The lines $2 y = 6 x + 5$ and $y = 3 x - 2$ cross at exactly one point."
Show that Amir is wrong.
[2 marks]
D3
An energy company has two tariffs. Tariff $A$ costs a fixed £18 each month plus 20p for each unit used. Tariff $B$ costs 28p for each unit used and has no fixed charge. The cost for a month in which $u$ units are used is £ $C$ .
(a)
Write down an equation for $C$ in terms of $u$ for each tariff.
[2 marks]
Answer $A$ : $\dots\dots$ , $B$ : $\dots\dots$
(b)
For how many units do the two tariffs cost the same?
[2 marks]
Answer
(c)
In one month Kim pays £84 on tariff $B$ . How much less would she have paid that month on tariff $A$ ?
[2 marks]
Answer£ $\dots\dots$
Total for D3: 6 marks
D4
The line $3 x + 4 y = 24$ crosses the $x$ -axis at $P$ and the $y$ -axis at $Q$ . $O$ is the origin. Work out the area of triangle $O P Q$ .
[3 marks]
Answer
D5
The points $(2, 5)$ , $(6, 13)$ and $(k, 21)$ lie on a straight line. Work out the value of $k$ .
[3 marks]
Answer
D6
Line $L_{1}$ passes through $(0, 3)$ and $(6, 0)$ . Line $L_{2}$ is perpendicular to $L_{1}$ and passes through the origin. Work out the coordinates of the point where $L_{1}$ and $L_{2}$ cross.
[4 marks]
Answer
D7
The graph shows the charge made by a plumber for a job. The charge is a fixed call-out fee plus an amount for each hour. Work out the call-out fee and the amount charged for each hour.
[3 marks]
Answercall-out fee £ $\dots\dots$ , each hour £ $\dots\dots$
D8
Work out the equation of the line that passes through $(- 2, 7)$ and is parallel to the line through $(1, 1)$ and $(4, - 5)$ .
[3 marks]
Answer
D9
$A$ is $(0, 0)$ , $B$ is $(4, 2)$ and $C$ is $(1, 8)$ . Show that angle $A B C$ is a right angle.
[3 marks]
D10
The line $y = 4 x + c$ passes through the point where the lines $y = 3 x - 1$ and $y = 9 - 2 x$ cross. Work out the value of $c$ .
[3 marks]
Answer

Extension

Grade 9 and beyond

No route is given. Spend five minutes on a problem before you look at a hint. The hints are on the last page of the worksheet, and you can use one at a time.

E1
The lines $y = 2 x + 1$ and $y = 7 - x$ and the $y$ -axis make a triangle. Work out the area of the triangle.
[3 marks]
Hint 1 · What to try
Find the three corners of the triangle. Two of them are on the $y$ -axis.
Hint 2 · The first line
The lines cross where $2 x + 1 = 7 - x$ , so at $(2, 5)$ . They meet the $y$ -axis at $(0, 1)$ and $(0, 7)$ .
Hint 3 · The full method
Use the side on the $y$ -axis as the base: it has length 6, and the height is the distance from the $y$ -axis to $(2, 5)$ .
Answer

E2
The lines $k x + 2 y = 7$ and $3 x - 6 y = 4$ are perpendicular. Work out the value of $k$ .
[3 marks]
Hint 1 · What to try
Write each line in the form $y = m x + c$ .
Hint 2 · The first line
The gradients are $- \frac{k}{2}$ and $\frac{1}{2}$ .
Hint 3 · The full method
Perpendicular gradients multiply to $- 1$ , so $- \frac{k}{2} \times \frac{1}{2} = - 1$ .
Answer
E3
$A (1, 1)$ and $B (5, 3)$ are two corners of a square $A B C D$ , next to each other. The letters go anticlockwise round the square. Work out the coordinates of $C$ and of $D$ .
[4 marks]
Hint 1 · What to try
To get from $A$ to $B$ you go 4 across and 2 up.
Hint 2 · The first line
$B C$ is perpendicular to $A B$ and the same length. Going anticlockwise, you go 2 back and 4 up.
Hint 3 · The full method
$C$ is 2 left and 4 up from $B$ . $D$ is 2 left and 4 up from $A$ .
Answer $C$ $\dots\dots$ , $D$ $\dots\dots$
E4
A straight line passes through $(2, 9)$ . It makes a triangle with the positive $x$ -axis and the positive $y$ -axis, and the area of the triangle is 36. Work out the equation of the line.
[4 marks]
Hint 1 · What to try
Call the gradient $- t$ , with $t$ positive. Write down where the line meets each axis in terms of $t$ .
Hint 2 · The first line
It meets the $y$ -axis at $9 + 2 t$ and the $x$ -axis at $2 + \frac{9}{t}$ . So $\frac{1}{2} (9 + 2 t) (2 + \frac{9}{t}) = 36$ .
Hint 3 · The full method
This gives $4 t + \frac{81}{t} = 36$ , so $4 t^{2} - 36 t + 81 = 0$ , which is $(2 t - 9)^{2} = 0$ .
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can draw a line from a table or from its equation, and find its gradient and $y$ -intercept.
I can find the equation of a line from a graph or two points, and use parallel and perpendicular gradients.
I can read conversion and rate graphs, and solve simultaneous equations from where two lines cross.

Answers and mark scheme

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