Example – Using two points

A line goes through (2, 9) and (3, 7).

a Find the gradient of the line.

b Find the equation of the line.

Solution

a

2 3

1

−2

0

(3, 7) (2, 9)

x y

7 9

y x

Gradient=change in change in

=–2 1

=–2

b The equation of a straight line is

y

=

mx

+

c

. The gradient is −2 so

m

^{= −2.}

So

y

= −2

x

+

c

.

You can fi nd the value of

c

by substituting the co-ordinates of one point on the line into

y

= −2

x

+

c

.

For the point (2, 9):

substituting

x

^{= 2,}

y

^{= 9 into}

y

^{= −2}

x

⁺

c

gives 9 = −2 × 2 +

c

9 = −4 +

c

13 =

c

So

c

^{= 13}

The equation of the line is

y

= −2

x

+ 13.

Always start with a sketch.

You could use (3, 7) instead.

You can write this as

y

= 13 − 2

x

.

‘Downhill lines’ have a negative gradient.

Practising skills

1 The lines

y

^{= 5}

x

^{+ 2 and}

y

^{= 5}

x

– 2 are parallel. Match these lines into pairs that are parallel and fi nd the odd one out.

a

y

= 3

x

+ 7 e

y

^{= –5}

x

^{+ 9}

b

y

= 7

x

+ 3 f

y

^{= 3}

x

^{+ 9}

c

y

= –3

x

+ 2 g

y

^{= –3}

x

^{– 10}

d

y

= 7

x

– 5 h Write the equation of a line that is parallel to the odd one out.

2 All of the lines in this question have a gradient of 2.

a Draw the graphs of the lines that go through these points.

i (1, 5) ii (1, 12) iii (1, 1) iv (5, 7) v (–2, 1)

b Find their

y

-intercepts.

c Write down their equations.

3 A line has equation

y

^{= 3}

x

⁺

k.

a What is the gradient of the line?

The line passes through the point (2, 11).

b How does this give you the equation 11 = 6 +

k

?

c Find the value of

k

and write down the equation of the line.

d Where does the line cross the

y

axis?

4 Find the equation of the line that has a gradient of a 3 and goes through (1, 1)

b 7 and goes through (2, 5) c 2 and goes through (10, 1) d 1

2 and goes through (2, 2) e –12 and goes through (2, 2).

5 A line goes through the points (2, 1) and (4, 9).

a Show that the gradient of the line is 4.

b Find the equation of the line with gradient 4 that passes through (2, 1).

c Check that (4, 9) lies on your line.

6 For the following pairs of points

a fi nd the gradient of the line joining them.

b fi nd the equation of the line through them.

c check that both points lie on the lines by substituting

x

^{- and}

y

^-co-ordinates into the equation.

i (1, 1) to (5, 7) iv (–2, 3) to (5, 7)

ii (1, 3) to (5, 7) v (2, 9) to (5, 7)

iii (3, 4) to (5, 7)

Unit 5 Finding equations of straight lines Band h

Developing fl uency

1 These two lines meet at (–1, 2). One has a gradient of 2 and the other has a gradient of 3.

a Find the equation of each line.

b Find the co-ordinates of the points where they cross the

y

^axis.

c State whether these points are on the red line, the blue line or neither.

i (1, 6)

2 Write down the equation of a straight-line graph that fi ts into each section of this two-way table.

Gradient is 3 Gradient is –7

y

-intercept is 6

c Find the angle between the two lines.

4 Find the equations of these lines.

a Through (1, 2) and (3, 4).

Which lines are the same as each other?

Reasoning

5 The co-ordinates of point A are (

a

,

b

) and the co-ordinates of point B are (

b

,

a

).

Is it always, sometimes or never true that the line through A and B has a negative gradient?

If you think always or never, you must explain how you can be so certain.

If you think sometimes, then you must explain when the statement is and isn’t true.

6 A triangle is drawn on a co-ordinate grid. It has vertices at (1, 1), (6, 6) and (1, 11).

a Find the equations of the three lines used to make the triangle.

b Draw the triangle on graph paper, using the same scales for the

x

axis and the

y

^axis.

c Describe the triangle.

d Find the area of the triangle.

7 ABCD is a quadrilateral. The equations of its sides are:

AB

y

^{= –3}

x

^{+ 7}

BC

y

^{= 2}

x

^{– 3}

DC

y

= –2

x

+ 13 DA

y

^{= 3}

x

^{– 17}

a Draw the four lines on a graph.

b Find the co-ordinates of A, B, C and D.

c What sort of quadrilateral is ABCD?

d The diagonals AC and BD meet at E. What are the co-ordinates of E?

e What are the equations of AC and BD?

Problem solving

1

l

^and

m

are two straight lines.

l

has a gradient of 2 and crosses the

y

axis at (0, –1).

m

has a gradient of –3 and crosses the

y

axis at (0, 4).

a Find the equations of

l

^and

m

^.

b Draw lines

l

^and

m

on a graph.

c Find the co-ordinates of their point of intersection.

2

p

^and

q

are two straight lines.

p

has a gradient of 3 and passes through the point (3, 2).

q

has a gradient of –2 and passes through the point (1, –4).

a Draw lines

p

^and

q

on a graph.

b Find the equations of

p

^and

q

.

c Write down the co-ordinates of R, the point of intersection of

p

^and

q

^.

d Show that R lies on the line

y

^{= –4}

x

3 A straight line

r

is parallel to the line

y

⁼

x

+ 3 and passes through the point (0, 5).

Another straight line

s

is parallel to the line

x

⁺

y

= 5 and passes through the point (0, 1).

a Find the equations of the lines

r

^and

s

^.

b Find, by drawing, the co-ordinates of the point of intersection of the two straight lines.

c Substitute the

x

^{- and}

y

-values you found in part b into the equations of the lines

r

^and

s

^.

How does this check your answers?

ReasoningExam-styleExam-styleExam-styleExam-style

Unit 5 Finding equations of straight lines Band h 4 A straight line

l

is parallel to the line

y

= 4

x

– 3 and passes through the point (–1, 2).

Another straight line

m

is parallel to the line 4

x

^{+ 3}

y

= 2 and passes through the point (3, –2).

a Draw the lines

l

and

m

on a graph.

b Find the equations of

l

^and

m

^.

c Find the point of intersection of

l

and

m

.

d Use your answer to part c to check your equation of

l

^and

m

^.

5 a On the same axes, draw the lines

p

^:

y

^{= 1}

q

^:

x

^{= 3}

r

:

y

= 3

x

+ 1

b Write down co-ordinates of the point, C, where line

p

^{and line}

q

^meet.

c Line

m

is parallel to

r

and passes through C.

Write down the equation of line

m

.

d Find the area of the region bounded by the

y

-axis and the lines

q

^,

r

^and

m

^.

6 Line

l

passes through the points (−1, 6) and (1, 2).

Line

m

has equation

y

^{+ 2}

x

^{= 9.}

a Show that the lines

l

^and

m

are parallel.

b Draw lines

l

^and

m

on the same co-ordinate axes.

Line

n

has gradient 12 and the same

y

-intercept as line

l

^.

c Find the equation of line

n

.

d By substituting for

x

^and

y

, show that the point (2, 5) lies on both lines

m

^and

n

^.

e Find the area of the triangle bounded by line

m

, line

n

and the

y

axis.

Exam-styleExam-styleExam-style

Reviewing skills

1 Find the equations of these lines.

a Parallel to

y

^{= 2}

x

^{+ 3 with}

y-

intercept 5.

b Gradient –1 through the point (5, 2).

c Through the points (–1, –4) and (2, 5).

2 a Draw a graph showing the lines

y

^{= 2}

x

^{+ 3 and}

y

^{= –}

x

^{+ 6.}

b Find the co-ordinates of the point of intersection, P.

c Show algebraically that the line joining (0, 7) to (3.5, 0) passes through P.

3 A is (0, 0), B is (3, 4), C is (9, 4) and D is (6, 0).

a Show that the quadrilateral ABCD is a parallelogram, but not a rhombus.

b Find the equations of the lines AC and BD.

c Use algebra to show that the point E (4.5, 2) lies on both lines AC and BD.

d Show points A, B, C, D and E on a graph. Show also the parallelogram ABCD and its diagonals.

3 Unit 6 Quadratic functions

In document Mathematics for Edexcel GCSE - Foundation 2-Higher 1 (2015) (Page 128-133)