A-Level Maths (Edexcel 9MA0) · Pure Year 1 · Chapter 5: Straight line graphs · Lesson 1

Equations of straight lines

70 marks

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Quick questions to start.

A1
Needed today
Make $y$ the subject of $5 x - 3 y + 12 = 0$ .
[2 marks]
Answer $y = \dots\dots$
A2
GCSE Higher
Solve the simultaneous equations $y = 2 x - 1$ and $3 x + 2 y = 19$ .
[2 marks]
Answer $x = \dots\dots$ $y = \dots\dots$
A3
From chapter 1
Simplify $\frac{18}{6} + 54$ .
[2 marks]
Answer
A4
Last lesson
$y = f (x)$ has a maximum point at $(2, 5)$ . Write down the maximum point of $y = f (x - 3) + 1$ .
[2 marks]
Answer $(\dots\dots, \dots\dots)$

A-Level

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

Example 1

Show that the points $A (- 1, - 4)$ , $B (2, 2)$ and $C (7, 12)$ are collinear.

Hint Equal gradients for

A B

and

B C

, with

B

shared, means one line.

B1
Show that $P (3 a, - a)$ , $Q (a, 3 a)$ and $R (- 2 a, 9 a)$ are collinear, where $a \neq = 0$ .
[4 marks]

Example 2

The line $L$ has equation $4 x - 6 y + 9 = 0$ . Find its gradient and where it crosses each axis.

Hint Rearrange into the form

y = m x + c

. On the

x

-axis

y = 0

B2
The line $L$ has equation $6 x + 4 y - 15 = 0$ . Find its gradient and where it crosses each axis.
[4 marks]
Answer

Example 3

Find an equation of the line through $(- 3, 4)$ and $(5, - 2)$ , in the form $a x + b y + c = 0$ with $a$ , $b$ and $c$ integers.

Hint Find the gradient first. Then use

y - y_{1} = m (x - x_{1})

and clear the fraction.

B3
Find an equation of the line through $(- 4, - 1)$ and $(2, 3)$ , in the form $a x + b y + c = 0$ with $a$ , $b$ and $c$ integers.
[4 marks]
Answer

A-Level

C1
The lines $y = 3 x - 11$ and $2 x + 5 y - 13 = 0$ meet at the point $A$ . The point $B$ has coordinates $(- 3, 6)$ . Find an equation of the line through $A$ and $B$ . Give your answer in the form $a x + b y + c = 0$ , where $a$ , $b$ and $c$ are integers.
[4 marks]
Answer

C2
The line $5 x + 2 y - 30 = 0$ crosses the $x$ -axis at the point $A$ . The line $l$ passes through $A$ and has gradient $- \frac{3}{4}$ . Find an equation of $l$ in the form $a x + b y + c = 0$ , where $a$ , $b$ and $c$ are integers.
[3 marks]
Answer
C3
The line with gradient $- \frac{2}{5}$ that crosses the $y$ -axis at $(0, - 3)$ has equation $a x + 10 y + c = 0$ . Find the value of $a$ and the value of $c$ .
[2 marks]
Answer $a = \dots\dots$ $c = \dots\dots$
C4
The line through $(- 6, 5)$ and $(3, - 1)$ crosses the $x$ -axis at $P$ and the $y$ -axis at $Q$ . Find the coordinates of $P$ and of $Q$ .
[4 marks]
Answer $P (\dots\dots, 0)$ $Q (0, \dots\dots)$

Edexcel exam style

D2
The line $l_{1}$ passes through the points $A (- 4, 11)$ and $B (6, - 4)$ .
Not drawn accurately
(a)
Find an equation of $l_{1}$ in the form $a x + b y + c = 0$ , where $a$ , $b$ and $c$ are integers.
[3 marks]
Answer
(b)
The line $l_{2}$ has equation $3 x - 4 y + 2 = 0$ . The lines $l_{1}$ and $l_{2}$ meet at $C$ . Find the coordinates of $C$ .
[3 marks]
Answer $(\dots\dots, \dots\dots)$
(c)
Hence find the area of the triangle formed by $l_{1}$ , $l_{2}$ and the $y$ -axis.
[4 marks]
Answer
Total for D2: 10 marks
D3
The straight line $l$ passes through the points $A (2, 3)$ and $B (3 + 3, 4 + 3)$ . Find the exact $x$ -coordinate of the point where $l$ crosses the $x$ -axis. Give your answer in the form $p + q 3$ , where $p$ and $q$ are rational numbers.
[6 marks]
Answer $x = \dots\dots$
D4
The straight line $l$ passes through $A (1, p)$ and $B (p, 9)$ , where $p$ is a constant and $p \neq = 1$ . The gradient of $l$ is $p$ .
(a)
Show that $p^{2} = 9$ .
[3 marks]
(b)
Hence find the two possible equations of $l$ .
[3 marks]
Answer $y = \dots\dots$ or $y = \dots\dots$
Total for D4: 6 marks

Stretch

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

Colour one circle on each line.

	Not yet	Nearly	Yes
I can find the gradient through two points.
I can find a line's equation from a point and a gradient, or two points.
I can write a line as ax + by + c = 0 and read off its gradient.
I can find where lines meet each other and the axes.

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