AQA Level 2 Further Maths · Level 2 Certificate · Chapter 3: Algebra III · Lesson 4

Finding the equation of a line

About 75 minutesNo calculator61 marks

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Date

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The first two questions are what today's lesson needs. The others bring back earlier topics.

A1
Needed today
Write $y - 5 = - 3 (x + 2)$ in the form $y = m x + c$ .
[2 marks]
Answer $y = \dots\dots$
A2
Needed today
Work out the gradient of the line through $(- 1, 8)$ and $(5, - 4)$ .
[2 marks]
Answer $\dots\dots$
A3
Last lesson
Work out where the line $4 x - 3 y = 24$ meets each axis.
[2 marks]
Answer $(\dots, \dots)$ and $(\dots, \dots)$
A4
From chapter 2
Write $x^{2} + 10 x + 18$ in the form $(x + a)^{2} + b$ .
[2 marks]
Answer $\dots\dots$

Example, then your turn

Level 2

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

Example 1

Work out the equation of the line with gradient $- 4$ that passes through $(4, - 3)$ . Give your answer in the form $y = m x + c$ .

Hint A line with gradient

m

through

(x_{1}, y_{1})

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

B1
Work out the equation of the line with gradient $5$ that passes through $(- 2, - 1)$ . Give your answer in the form $y = m x + c$ .
[2 marks]
Answer $y = \dots\dots$

Example 2

Work out the equation of the line with gradient $- \frac{2}{3}$ that passes through $(6, - 1)$ . Give your answer in the form $p x + q y + r = 0$ , where $p$ , $q$ and $r$ are integers.

Hint Write

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

, then multiply every term by

3

to clear the fraction.

B2
Work out the equation of the line with gradient $\frac{3}{4}$ that passes through $(- 2, 5)$ . Give your answer in the form $p x + q y + r = 0$ , where $p$ , $q$ and $r$ are integers.
[3 marks]
Answer $\dots\dots = 0$

Example 3

Work out the equation of the line through $(- 3, 8)$ and $(5, - 4)$ . Give your answer in the form $p x + q y + r = 0$ , where $p$ , $q$ and $r$ are integers.

Hint Work out the gradient first. Then use either point in

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

, and check with the other.

B3
Work out the equation of the line through $(- 4, - 3)$ and $(2, 1)$ . Give your answer in the form $p x + q y + r = 0$ , where $p$ , $q$ and $r$ are integers.
[3 marks]
Answer $\dots\dots = 0$

Example 4

The diagram shows the lines $L_{1}$ and $L_{2}$ . Work out the equation of each line.

Hint

L_{1}

meets the axes at two points you can read: use them for the gradient.

L_{2}

is horizontal.

B4
The diagram shows the lines $L_{3}$ and $L_{4}$ . Work out the equation of each line. Give the equation of $L_{3}$ in the form $p x + q y + r = 0$ , where $p$ , $q$ and $r$ are integers.
[3 marks]
Answer $L_{3}$ : $\dots\dots = 0 L_{4}$ : $\dots\dots$

Example 5

The length, $L$ cm, of a spring is a linear function of the mass, $m$ grams, hanging from it. With $20$ g hanging, $L = 13$ ; with $50$ g hanging, $L = 17.5$ . Work out a formula for $L$ in terms of $m$ . Write down the length with no mass, and work out the mass that makes the spring $22$ cm long.

Hint The two readings are two points

(m, L)

. Find the gradient, then use one point.

B5
A print shop charges $£ C$ for $n$ posters, where $C$ is a linear function of $n$ . $40$ posters cost $£26$ and $100$ posters cost $£50$ . Work out a formula for $C$ in terms of $n$ . Write down the fixed set-up charge, and work out how many posters cost $£70$ .
[4 marks]
Answer $C = \dots\dots £ \dots\dots \dots\dots$ posters

Practice

Level 2

C1
Work out the equation of the line that is parallel to $4 x + 2 y = 7$ and passes through $(- 2, 7)$ . Give your answer in the form $y = m x + c$ .
[3 marks]
Answer $y = \dots\dots$
C2
Work out the equation of the line through the origin and the point $(- 6, 4)$ . Give your answer in the form $p x + q y = 0$ , where $p$ and $q$ are integers.
[2 marks]
Answer $\dots\dots = 0$

C3
A line meets the $y$ -axis at $(0, - 4)$ and passes through $(6, 5)$ . Work out its equation in the form $p x + q y + r = 0$ , where $p$ , $q$ and $r$ are integers.
[2 marks]
Answer $\dots\dots = 0$
C4
A line has gradient $4$ and passes through $(- 1, - 5)$ . Does the point $(3, 12)$ lie on the line? Show how you decide.
[2 marks]
C5
Work out the equation of the line through $(\frac{1}{2}, 3)$ and $(2, 0)$ . Give your answer in the form $y = m x + c$ .
[2 marks]
Answer $y = \dots\dots$

Exam-style questions

AQA exam style

D1
A line has gradient $- 2$ and passes through $(- 3, 4)$ . Which is its equation? Circle your answer. $y = - 2 x - 2 y = - 2 x + 10 y = 4 x - 2 y = - 2 x + 5$
[1 mark]

D2
Tom wants the equation of the line through $A (4, - 1)$ and $B (- 2, 8)$ . He writes: "Gradient $= \frac{- 2 - 4}{8 - ( - 1 )} = - \frac{2}{3}$ , so the line is $y + 1 = - \frac{2}{3} (x - 4)$ ."
Tom is wrong.
(a)
Explain Tom's mistake, and work out the correct gradient.
[2 marks]
Answergradient $\dots\dots$
(b)
Work out the equation of the line $A B$ . Give your answer in the form $p x + q y + r = 0$ , where $p$ , $q$ and $r$ are integers.
[2 marks]
Answer $\dots\dots = 0$
Total for D2: 4 marks
D3
The line through the points $(2, 9)$ and $(6, k)$ is parallel to the line $3 x + 2 y = 5$ .
(a)
Work out the value of $k$ .
[3 marks]
Answer $k = \dots\dots$
(b)
Work out the equation of the line through $(2, 9)$ and $(6, k)$ . Give your answer in the form $p x + q y + r = 0$ , where $p$ , $q$ and $r$ are integers.
[2 marks]
Answer $\dots\dots = 0$
Total for D3: 5 marks
D4
The line $L$ passes through the points $A (- 3, 7)$ and $B (5, 1)$ . $L$ meets the $x$ -axis at $P$ .
Not drawn accurately
(a)
Work out the equation of $L$ . Give your answer in the form $p x + q y + r = 0$ , where $p$ , $q$ and $r$ are integers.
[3 marks]
Answer $\dots\dots = 0$
(b)
Work out the coordinates of $P$ .
[1 mark]
Answer $(\dots\dots, \dots\dots)$
(c)
The point $C (k, - 2)$ lies on $L$ . Work out the value of $k$ .
[2 marks]
Answer $k = \dots\dots$
Total for D4: 6 marks
D5
An oven is switched on. For the first $15$ minutes its temperature, $T^{\circ}$ C, is a linear function of the time, $t$ minutes. When $t = 4$ , $T = 74$ . When $t = 10$ , $T = 146$ .
(a)
Show that $T = 12 t + 26$ .
[3 marks]
(b)
Write down the temperature of the oven when it was switched on.
[1 mark]
Answer $\dots\dots^{\circ}$ C
(c)
Work out the time at which the temperature reaches $200^{\circ}$ C.
[2 marks]
Answer $\dots\dots$ minutes
Total for D5: 6 marks

Extension

Stretch

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

E1
A straight line passes through the point $(3, 2)$ . It meets the positive $x$ -axis and the positive $y$ -axis. The sum of the two intercepts is $12$ . Work out the two possible equations of the line, each in the form $p x + q y = r$ , where $p$ , $q$ and $r$ are integers.
[5 marks]
Hint 1 · What to try
Call the intercepts $a$ and $12 - a$ , and write the line as $\frac{x}{a} + \frac{y}{12 - a} = 1$ .
Hint 2 · The first line
The point $(3, 2)$ is on the line, so $\frac{3}{a} + \frac{2}{12 - a} = 1$ .
Hint 3 · The full method
Multiply by $a (12 - a)$ : $3 (12 - a) + 2 a = a (12 - a)$ , so $a^{2} - 13 a + 36 = 0$ and $a = 4$ or $a = 9$ . The intercepts are $4$ and $8$ , giving $2 x + y = 8$ , or $9$ and $3$ , giving $x + 3 y = 9$ .
Answer $\dots\dots$ and $\dots\dots$

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can write the equation of a line from its gradient and one point, using $y - y_{1} = m (x - x_{1})$ .
I can find the equation of the line through two points, or read it from a diagram.
I can give an equation in the form $p x + q y + r = 0$ with integer coefficients.
I can build a linear model from two readings and use it.

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