Basic Skills in Higher Mathematics

1
Basic Skills in Higher Mathematics
Mathematics 1(H)Outcome 1
Robert GlenAdviser in Mathematics
2
Mathematics 1(Higher)

Outcome 1 Use the properties of the straight
line
Straight lines
3
Mathematics 1(Higher)

Outcome 1 Use the properties of the straight
line
PC Index
Click on the one you want
PC(a) Gradients and straight lines
PC(b) Gradients and angles
PC(c) Parallel and perpendicular
4
Mathematics 1(Higher)
Outcome 1 Use the properties of the straight
line
PC(a) Determine the equation of a
straight line given two points on
the line or one point and the gradient
Gradients
and straight lines
5
Mathematics 1(Higher)

Outcome 1 Use the properties of the straight
line
PC(a) Determine the equation of a
straight line given two points on
the line or one point and the gradient
Index Click on the section you
want
1 What is gradient?
2 The gradient of a line
3 The equation of a line given its gradient
and the intercept on the y - axis
4 The equation of a line given one point
on the line and the gradient
5 The equation of a line given two points
on the line
6
Mathematics 1(Higher)

Outcome 1 Use the properties of the straight
line
PC(a) Determine the equation of a
straight line given two points on
the line or one point and the gradient
Section 1
1 What is gradient?
7
Mathematics 1(Higher)
1.1
Outcome 1 Use the properties of the straight
line
PC(a) Determine the equation of a
straight line given two points on
the line or one point and the gradient
1 What is gradient?
The gradient (slope) of this roof is
2m
2
2m

3m
3
3m
8
Mathematics 1(Higher)
1.2
Outcome 1 Use the properties of the straight
line
PC(a) Determine the equation of a
straight line given two points on
the line or one point and the gradient
1 What is gradient?
3m
The gradient (slope) of this roof is
3m
3m

1
3m
9
Mathematics 1(Higher)
1.3
Outcome 1 Use the properties of the straight
line
PC(a) Determine the equation of a
straight line given two points on
the line or one point and the gradient
1 What is gradient?
The gradient (slope) of this roof is
3m
7m
3m
3

7m
7
10
Mathematics 1(Higher)
1.4
Outcome 1 Use the properties of the straight
line
PC(a) Determine the equation of a
straight line given two points on
the line or one point and the gradient
1 What is gradient?
3m
3

Gradient
7
7m
Check thisThe steeperthe slope, the greater
the gradient.
2m
2

Gradient
3m
3
3m

Gradient
1
3m
11
Mathematics 1(Higher)
1.5
Outcome 1 Use the properties of the straight
line
PC(a) Determine the equation of a
straight line given two points on
the line or one point and the gradient
1 What is gradient?
What is the gradient of this roof ?
5m
3m
5
4
A
B
4
5
4m
3
3
D
C
4
5
12
Mathematics 1(Higher)
1.6
Outcome 1 Use the properties of the straight
line
PC(a) Determine the equation of a
straight line given two points on
the line or one point and the gradient
1 What is gradient?
What is the gradient of this roof ?
5m
3m
5
4
A
B
4
5
4m
3
3
D
C
4
5
Click on the letter of the correct answer
13
Mathematics 1(Higher)
1.7
Outcome 1 Use the properties of the straight
line
PC(a) Determine the equation of a
straight line given two points on
the line or one point and the gradient
1 What is gradient?
Sorry, wrong answer
What is the gradient of this roof ?
5m
3m
5
4
A
B
4
5
4m
3
3
Gradient vertical horizontal
D
C
4
5
Have another go!
14
Mathematics 1(Higher)
1.8
Outcome 1 Use the properties of the straight
line
PC(a) Determine the equation of a
straight line given two points on
the line or one point and the gradient
1 What is gradient?
What is the gradient of this roof ?
5m
3m
5
4
A
B
4
5
4m
3
3
D
C
4
5
Click on the letter of the correct answer
15
Mathematics 1(Higher)
1.9
Outcome 1 Use the properties of the straight
line
PC(a) Determine the equation of a
straight line given two points on
the line or one point and the gradient
1 What is gradient?
What is the gradient of this roof ?
5m
3m
5
4
A
B
4
5
4m
3
3
D
C
Correct!
4
5
End of Section 1
16
Mathematics 1(Higher)

Outcome 1 Use the properties of the straight
line
PC(a) Determine the equation of a
straight line given two points on
the line or one point and the gradient
Section 2
2 The gradient of a line
17
Mathematics 1(Higher)
2.1
Outcome 1 Use the properties of the straight
line
PC(a) Determine the equation of a
straight line given two points on
the line or one point and the gradient
y
2 The gradient of a line
B
uphill
? Read all lines from left to right
A
? Line AB is uphill from left to right
x
? Line AB has a positive gradient
mAB ? 0
18
Mathematics 1(Higher)
2.2
Outcome 1 Use the properties of the straight
line
PC(a) Determine the equation of a
straight line given two points on
the line or one point and the gradient
y
2 The gradient of a line
P
B
uphill
? Read all lines from left to right
A
downhill
? Line PQ is downhill from left to right
Q
x
? Line PQ has a negative gradient
mPQ ? 0
19
Mathematics 1(Higher)
2.3
Outcome 1 Use the properties of the straight
line
PC(a) Determine the equation of a
straight line given two points on
the line or one point and the gradient
y
2 The gradient of a line
P
B
uphill
? Read all lines from left to right
A
downhill
Q
x
? Line PQ has a negative gradient
mPQ ? 0
? Line AB has a positive gradient
mAB ? 0
20
Mathematics 1(Higher)
2.4
Outcome 1 Use the properties of the straight
line
PC(a) Determine the equation of a
straight line given two points on
the line or one point and the gradient
y
2 The gradient of a line
B
(9, 6)
uphill
change in y
Gradient
change in x
3
A
39
(0, 3)
mAB
9
13

x
21
Mathematics 1(Higher)
2.6
Outcome 1 Use the properties of the straight
line
PC(a) Determine the equation of a
straight line given two points on
the line or one point and the gradient
y
2 The gradient of a line
B
(9, 6)
uphill
change in y
Gradient
1
change in x
3
1
A
3
1
39
(0, 3)
mAB
3
13

x
Note we could have measured the
gradient like this
22
Mathematics 1(Higher)
2.7
Outcome 1 Use the properties of the straight
line
PC(a) Determine the equation of a
straight line given two points on
the line or one point and the gradient
y
2 The gradient of a line
(0, 7)
P
change in y
Gradient
change in x
downhill
-6
-6 9
mPQ
Q
(9, 1)
23
x

-
9
23
Mathematics 1(Higher)
2.8
Outcome 1 Use the properties of the straight
line
PC(a) Determine the equation of a
straight line given two points on
the line or one point and the gradient
y
2 The gradient of a line
(0, 7)
P
change in y
-2
Gradient
change in x
3
downhill
-2
-6 9
mPQ
3
-2
Q
(9, 1)
23
3

-
x
Note we could have measured the
gradient like this
24
Mathematics 1(Higher)
2.9
Outcome 1 Use the properties of the straight
line
PC(a) Determine the equation of a
straight line given two points on
the line or one point and the gradient
y
2 The gradient of a line
B
uphill
(9, 6)
change in y
Gradient
change in x
6 - 3
A
(0, 3)
6 - 3 9 - 0
mAB
9 - 0
39

x
13

25
Mathematics 1(Higher)
2.10
Outcome 1 Use the properties of the straight
line
PC(a) Determine the equation of a
straight line given two points on
the line or one point and the gradient
y
2 The gradient of a line
(0, 7)
P
change in y
Gradient
change in x
downhill
1 - 7
1 - 7 9 - 0
mPQ
Q
(9, 1)
-6 9

9 - 0
x
23
-

26
Mathematics 1(Higher)
2.11
Outcome 1 Use the properties of the straight
line
PC(a) Determine the equation of a
straight line given two points on
the line or one point and the gradient
2 The gradient of a line
y
B (x2 , y2)
A formula to memorise
A (x1 , y1)
y2 - y1x2 - x1
x
mAB
27
Mathematics 1(Higher)
2.12
Outcome 1 Use the properties of the straight
line
PC(a) Determine the equation of a
straight line given two points on
the line or one point and the gradient
2 The gradient of a line
y
B (x2 , y2)
A formula to memorise
A (x1 , y1)
y2 - y1x2 - x1
x
mAB
28
Mathematics 1(Higher)
2.13
Outcome 1 Use the properties of the straight
line
PC(a) Determine the equation of a
straight line given two points on
the line or one point and the gradient
2 The gradient of a line
y
B (6 , 5)
1 Calculate the gradient of line AB
y2 - y1x2 - x1
A (2 , 3)
mAB
x
5 - 36 - 2

24

Did you getthis answer?
12

29
Mathematics 1(Higher)
2.14
Outcome 1 Use the properties of the straight
line
PC(a) Determine the equation of a
straight line given two points on
the line or one point and the gradient
2 The gradient of a line
y
2 Calculate the gradient of line CD.
D (6 , 2)
y2 - y1x2 - x1
x
mCD
C (2 , -1)
2 - (-1) 6 - 2

34

Did you getthis answer?
30
Mathematics 1(Higher)
2.15
Outcome 1 Use the properties of the straight
line
PC(a) Determine the equation of a
straight line given two points on
the line or one point and the gradient
2 The gradient of a line
y
E (-3 , 3)
3 Calculate the gradient of line EF.
y2 - y1 x2 - x1
x
mEF
F (5, -1)
-1 - 35 - (-3)

-4 8
Did you getthis answer?

12

-
End of Section 2
31
Mathematics 1(Higher)

Outcome 1 Use the properties of the straight
line
PC(a) Determine the equation of a
straight line given two points on
the line or one point and the gradient
Section 3
3 The equation of a linegiven its gradient and
theintercept on the y - axis
32
Mathematics 1(Higher)
3.1
Outcome 1 Use the properties of the straight
line
PC(a) Determine the equation of a
straight line given two points on
the line or one point and the gradient
3 The equation of a line given gradient and
intercept
y
(x, y)
Find the equation of line KL which has a gradient
of ½ and passes through the point (0, 3).
L
?
(0, 3)
?
m ½
K
y - 3x - 0
12
mKL

x
O
? y - 3 ½ x
The equation of KL is y ½ x 3
? y ½ x 3
33
Mathematics 1(Higher)
3.2
Outcome 1 Use the properties of the straight
line
PC(a) Determine the equation of a
straight line given two points on
the line or one point and the gradient
3 The equation of a line given gradient and
intercept
y
(x, y)
Find the equation of line KL which has a gradient
of ½ and passes through the point (0, 3).
L
?
(0, 3)
?
m ½
K
Formula y m x c
x
O
The equation of KL is y ½ x 3
34
Mathematics 1(Higher)
3.3
Outcome 1 Use the properties of the straight
line
PC(a) Determine the equation of a
straight line given two points on
the line or one point and the gradient
3 The equation of a line given gradient and
intercept
y
(x, y)
L
?
The equation of line with gradient m and
intercept c is
(0, c)
m
?
K
y m x c
x
O
Memorise this
35
Mathematics 1(Higher)
3.4
Outcome 1 Use the properties of the straight
line
PC(a) Determine the equation of a
straight line given two points on
the line or one point and the gradient
3 The equation of a line given gradient and
intercept
y
P
1 Find the equation of line PQ which has a
gradient of -2 and passes through the point (0,
5).
?
(0, 5)
m -2
?
(x, y)
Use the formula
x
O
Q
The equation of PQ is y -2 x 5
36
Mathematics 1(Higher)
3.5
Outcome 1 Use the properties of the straight
line
PC(a) Determine the equation of a
straight line given two points on
the line or one point and the gradient
3 The equation of a line given gradient and
intercept
F
y
?
(x, y)
2 Find the equation of line EF which has a
gradient of ¾ and passes through the point (0,
-3).
x
O
m ¾
?
(0, -3)
Use the formula
E
The equation of EF is y ¾ x - 3
37
Mathematics 1(Higher)

Outcome 1 Use the properties of the straight
line
PC(a) Determine the equation of a
straight line given two points on
the line or one point and the gradient
Section 4
4 The equation of a linegiven one point on the
line and the gradient
38
Mathematics 1(Higher)
4.1
Outcome 1 Use the properties of the straight
line
PC(a) Determine the equation of a
straight line given two points on
the line or one point and the gradient
4 The equation of a line given one point and the
gradient
y
L
Find the equation of the linethrough the point
(4, 3) with gradient 3.
(x, y)
?
m 3
?
(4, 3)
y - 3x - 4
mKL

3
x
O
? y - 3
3(x - 4)
K
? y - 3 3x - 12
The equation of KL is y 3x - 9
? y 3x
- 9
39
Mathematics 1(Higher)
4.2
Outcome 1 Use the properties of the straight
line
PC(a) Determine the equation of a
straight line given two points on
the line or one point and the gradient
4 The equation of a line given one point and the
gradient
y
L
Find the equation of the linethrough the point
(4, 3) with gradient 3.
(x, y)
?
m 3
?
(4, 3)
x
O
Formula y - b m (x - a)
K
The equation of KL is y 3x - 9
40
Mathematics 1(Higher)
4.3
Outcome 1 Use the properties of the straight
line
PC(a) Determine the equation of a
straight line given two points on
the line or one point and the gradient
4 The equation of a line given one point and the
gradient
y
L
(x, y)
?
The equation of the linethrough the point (a, b)
with gradient m is
m
(a, b)
?
y - b m (x - a)
x
O
K
Memorise this
41
Mathematics 1(Higher)
4.4
Outcome 1 Use the properties of the straight
line
PC(a) Determine the equation of a
straight line given two points on
the line or one point and the gradient
4 The equation of a line given one point and the
gradient
y
Q
1 Find the equation of the linethrough the point
(-1, 2) with gradient 2.
(x, y)
?
m 2
(-1, 2)
?
Use the formula
x
P
O
The equation of PQ is y 2 x 4
42
Mathematics 1(Higher)
4.5
Outcome 1 Use the properties of the straight
line
PC(a) Determine the equation of a
straight line given two points on
the line or one point and the gradient
4 The equation of a line given one point and the
gradient
y
Q
1 Find the equation of the linethrough the point
(-1, 2) with gradient 2.
(x, y)
?
(a, b)
m 2
y - b m (x - a)
(-1, 2)
?
y - 2
(x - (-1))
2
x
P
O
y - 2 2 (x 1)
y - 2 2 x 2
The equation of PQ is y 2 x 4
y 2 x
4
43
Mathematics 1(Higher)
4.6
Outcome 1 Use the properties of the straight
line
PC(a) Determine the equation of a
straight line given two points on
the line or one point and the gradient
4 The equation of a line given one point and the
gradient
y
2 Find the equation of the linethrough the point
(6, -2) with gradient ½.
N
m ½
?
x
O
(6, -2)
?
M
(x, y)
Use the formula
The equation of MN is 2y x - 10
44
Mathematics 1(Higher)
4.7
Outcome 1 Use the properties of the straight
line
PC(a) Determine the equation of a
straight line given two points on
the line or one point and the gradient
4 The equation of a line given one point and the
gradient
y
2 Find the equation of the linethrough the point
(6, -2) with gradient ½.
N
m ½
?
x
(x, y)
O
(6, -2)
?
y - b m (x - a)
M
y - (-2)
(a, b)
½
(x - 6)
y 2 ½ (x - 6)
Multiply both sides by 2to clear the fraction.
2y 4
x - 6
The equation of MN is 2y x - 10
2y x
- 10
or x - 2y - 10 0
45
Mathematics 1(Higher)
4.8
Outcome 1 Use the properties of the straight
line
PC(a) Determine the equation of a
straight line given two points on
the line or one point and the gradient
4 The equation of a line given one point and the
gradient
y
R
?
3 Find the equation of the linethrough the point
(-1, 4) with gradient 2/3 .
m -2/3
(-1, 4)
(x, y)
?
S
x
O
Use the formula
The equation of RS is 3y -2x 10
46
Mathematics 1(Higher)
4.9
Outcome 1 Use the properties of the straight
line
PC(a) Determine the equation of a
straight line given two points on
the line or one point and the gradient
4 The equation of a line given one point and the
gradient
y
R
?
3 Find the equation of the linethrough the point
(-1, 4) with gradient 2/3 .
m -2/3
(-1, 4)
(x, y)
(a, b)
?
y - b m (x - a)
x
O
S
y - 4
(x - (-1))
-2/3
y- 4 -2/3 (x 1)
Multiply both sides by 3to clear the fraction.
3y - 12
-2(x 1)
The equation of RS is 3y -2 x 10
3y
-2 x
10
or 2 x 2y - 10 0
47
Mathematics 1(Higher)

Outcome 1 Use the properties of the straight
line
PC(a) Determine the equation of a
straight line given two points on
the line or one point and the gradient
Section 5
5 The equation of a linegiven two points on the
line
48
Mathematics 1(Higher)
5.1
Outcome 1 Use the properties of the straight
line
PC(a) Determine the equation of a
straight line given two points on
the line or one point and the gradient
5 The equation of a line given two points on the
line
y
?
(6, 4)
B
Find the equation of the linejoining the points
A (3, 1) and B (6, 4) .
(3, 1)
(a, b)
m 1
?
A
x
O
Step 1
Calculate the gradient
Step 2
Calculate the equation
y2 - y1 x2 - x1
4 - 16 - 3
y - b m (x - a)
mAB

Choose A (3, 1) as thepoint on the line.i.e. a
3, b 1
y - 1
1
(x - 3)
33

y - 1 x - 3
(You get exactly thesame answer if youchoose B.)

1
y x - 2
49
Mathematics 1(Higher)
5.2
Outcome 1 Use the properties of the straight
line
PC(a) Determine the equation of a
straight line given two points on
the line or one point and the gradient
5 The equation of a line given two points on the
line
y
?
D
(5, 10)
1 Find the equation of the linejoining the
points C (1, 2) and D (5, 10) .
?
C
(1, 2)
x
O
Use the formula
The equation of AB is y 2x
50
Mathematics 1(Higher)
5.3
Outcome 1 Use the properties of the straight
line
PC(a) Determine the equation of a
straight line given two points on
the line or one point and the gradient
5 The equation of a line given two points on the
line
y
?
(5, 10)
D
1 Find the equation of the linejoining the
points C (1, 2) and D (5, 10) .
(a, b)
?
(1, 2)
C
x
O
Step 1
Calculate the gradient
Step 2
Calculate the equation
y2 - y1 x2 - x1
10 - 2 5 - 1
y - b m (x - a)
mAB

Choose C (1, 2) as thepoint on the line.i.e. a
1, b 2
y - 2
2
(x - 1)
84

y - 2 2 x - 2
(You get exactly thesame answer if youchoose B.)

2
y 2 x
51
Mathematics 1(Higher)
5.4
Outcome 1 Use the properties of the straight
line
PC(a) Determine the equation of a
straight line given two points on
the line or one point and the gradient
5 The equation of a line given two points on the
line
y
?
G
2 Find the equation of the linejoining the
points G (-3, 1) and H (5, -3) .
(-3, 1)
x
?
(5, -3)
H
Use the formula
The equation of AB is 2y - x - 1
52
Mathematics 1(Higher)
5.5
Outcome 1 Use the properties of the straight
line
PC(a) Determine the equation of a
straight line given two points on
the line or one point and the gradient
5 The equation of a line given two points on the
line
y
?
G
2 Find the equation of the linejoining the
points G (-3, 1) and H (5, -3) .
(-3, 1)
x
(a, b)
?
H
(5, -3)
Step 1
Calculate the gradient
Step 2
Calculate the equation
y2 - y1 x2 - x1
y - b m (x - a)
-3 - 15 - (-3)
mGH

Choose G (-3, 1) as the point on the line.i.e. a
-3, b 1
y - 1
-½
(x - (-3))
-4 8

2y - 2
- x
- 3
(You get exactly thesame answer if youchoose H.)
2y - x
- 1

-½
or x 2y 1 0
53
Mathematics 1(Higher)
5.6
Outcome 1 Use the properties of the straight
line
PC(a) Determine the equation of a
straight line given two points on
the line or one point and the gradient
5 The equation of a line given two points on the
line
y
?
G
2 Find the equation of the linejoining the
points G (-3, 1) and H (5, -3) .
x
(a, b)
?
(5, -3)
H
Step 2
Calculate the equation
A fuller explanation
y - b m (x - a)
y - 1 -½(x - (-3))
Multiply both sides by2 to clear the fraction.
y - 1 -½(x 3)
2y - 2 - x - 3
2y - x - 1
54
Mathematics 1(Higher)
5.7
Outcome 1 Use the properties of the straight
line
PC(a) Determine the equation of a
straight line given two points on
the line or one point and the gradient
y
y
(x2 , y2)
B
y2 - y1x2 - x1
?
(0, c)
m
?
mAB
?
y m x c
A (x1 , y1)
x
x
O
O
Summary
y
y
(x2 , y2)
1
Calculate m
(x , y)
?
?
y2 - y1x2 - x1
y - b m (x - a)
m
m
?
?
(a , b)
(x1 , y1)
2
y - b m (x - a)
(a, b)
x
x
O
O
55
Mathematics 1(Higher)

Outcome 1 Use the properties of the straight
line
PC(b) Find the gradient of a straight
line using m tan?
Gradientsand angles
? ?
56
Mathematics 1(Higher)
1.1


Outcome 1 Use the properties of the straight
line
PC(b) Find the gradient of a straight
line using m tan?
y
B
pq
mAB
p
tan? ?
? ?
A
q
x
O
57
Mathematics 1(Higher)
1.2


Outcome 1 Use the properties of the straight
line
PC(b) Find the gradient of a straight
line using m tan?
y
D
mCD
tan 35?
0.70 (to 2 dp)
35?
C
x
O
58
Mathematics 1(Higher)
1.3


Outcome 1 Use the properties of the straight
line
PC(b) Find the gradient of a straight
line using m tan?
y
E
Line EF is downhill,so its gradient is nottan
35?.
mEF
tan 145?
145?
35?
-0.70 (to 2 dp)
F
Always take the angle between the line and the
positive directionof the x-axis.
x
O
59
Mathematics 1(Higher)
1.4


Outcome 1 Use the properties of the straight
line
PC(b) Find the gradient of a straight
line using m tan?
y
1 What is the gradient of the line GH (to 2 dp)?
H
28?
mGH
tan 28?
G
0.53 (to 2 dp)
x
O
60
Mathematics 1(Higher)
1.5


Outcome 1 Use the properties of the straight
line
PC(b) Find the gradient of a straight
line using m tan?
y
2 What is the gradient of the line KL (to 2 dp)?
K
132?
48?
mKL
tan 132?
L
-1.11 (to 2 dp)
x
O
61
Mathematics 1(Higher)

Outcome 1 Use the properties of the straight
line
PC(c) Find the equation of a line
parallel to and perpendicular to a line
Parallel
and
perpendicular
62
Mathematics 1(Higher)

Outcome 1 Use the properties of the straight
line
PC(c) Find the equation of a line
parallel to and perpendicular to a line
Index Click on the section you want
1 Parallel lines
2 Perpendicular lines
3 Equations
63
Mathematics 1(Higher)
1.1


Outcome 1 Use the properties of the straight
line
PC(c) Find the equation of a line
parallel to and perpendicular to a line
Section 1
1 Parallel lines
64
Mathematics 1(Higher)
1.2


Outcome 1 Use the properties of the straight
line
PC(c) Find the equation of a line
parallel to and perpendicular to a line
Parallel lines have equal gradients
These lines are all parallel to each other
If one of the lines has agradient m, they all
havea gradient m.
65
Mathematics 1(Higher)
1.3


Outcome 1 Use the properties of the straight
line
PC(c) Find the equation of a line
parallel to and perpendicular to a line
The line y 2x 10 has a gradient of 2.
So any line parallel to this one has a gradient
of 2.
y
y 2x 10
10-
The line 2x - y 5 0 also belongs to this set
of parallel lines.Can you see why?
y 2x 5
5-
y 2x
0
x
2x - y 5 0 ?2x 5 y ? y 2x 5
y 2x - 5
-5-
y 2x - 10
-10-
66
Mathematics 1(Higher)
1.4


Outcome 1 Use the properties of the straight
line
PC(c) Find the equation of a line
parallel to and perpendicular to a line
1 Which of the following lines is/ are parallel
to the line y 3x - 5?
Click on the letterof a correct answer
y 3x - 1
y -3x 3
y 3x
A
C
B
NB There could be more than one rightanswer .
D
E
3x y 3
3x - y 3
67
Mathematics 1(Higher)
1.5


Outcome 1 Use the properties of the straight
line
PC(c) Find the equation of a line
parallel to and perpendicular to a line
1 Which of the following lines is/ are parallel
to the line y 3x - 5?
y 3x - 1
A
Correct!This line has a gradient of 3.
NB There could be more than one rightanswer .
Have another go!
68
Mathematics 1(Higher)
1.6


Outcome 1 Use the properties of the straight
line
PC(c) Find the equation of a line
parallel to and perpendicular to a line
1 Which of the following lines is/ are parallel
to the line y 3x - 5?
y -3x 3
B
Wrong!This line has a gradient of -3.
NB There could be more than one rightanswer .
Have another go!
69
Mathematics 1(Higher)
1.7


Outcome 1 Use the properties of the straight
line
PC(c) Find the equation of a line
parallel to and perpendicular to a line
1 Which of the following lines is/ are parallel
to the line y 3x - 5?
y 3x
C
Correct!This line has a gradient of 3.
NB There could be more than one rightanswer .
Have another go!
70
Mathematics 1(Higher)
1.8


Outcome 1 Use the properties of the straight
line
PC(c) Find the equation of a line
parallel to and perpendicular to a line
1 Which of the following lines is/ are parallel
to the line y 3x - 5?
Wrong!This line has a gradient of -3.
Click here to seeall the answers
NB There could be more than one rightanswer .
D
3x y 3
y -3x 3
Have another go!
71
Mathematics 1(Higher)
1.9


Outcome 1 Use the properties of the straight
line
PC(c) Find the equation of a line
parallel to and perpendicular to a line
1 Which of the following lines is/ are parallel
to the line y 3x - 5?
Correct!This line has a gradient of 3.
Click here to seeall the answers
E
3x - y 3
y 3x 3
Have another go!
72
Mathematics 1(Higher)
1.10


Outcome 1 Use the properties of the straight
line
PC(c) Find the equation of a line
parallel to and perpendicular to a line
1 Which of the following lines is/ are parallel
to the line y 3x - 5?
y 3x - 1
y -3x 3
y 3x
A
C
B
Key
Parallel toy 3 x - 5
D
E
3x y 3
3x - y 3
y 3x 3
Not parallel toy 3 x - 5
y -3x 3
73
Mathematics 1(Higher)
1.11


Outcome 1 Use the properties of the straight
line
PC(c) Find the equation of a line
parallel to and perpendicular to a line
2 Which of the following lines is/ are parallel
to the line x y 8?
Click on the letterof a correct answer
y x 5
y - x 1
y x
A
B
C
NB There could be more than one rightanswer .
D
E
x y 10
x - y 7
74
Mathematics 1(Higher)
1.12


Outcome 1 Use the properties of the straight
line
PC(c) Find the equation of a line
parallel to and perpendicular to a line
2 Which of the following lines is/ are parallel
to the line x y 8?
y x 5
A
Wrong!This line has a gradient of 1.
NB There could be more than one rightanswer .
Have another go
75
Mathematics 1(Higher)
1.13


Outcome 1 Use the properties of the straight
line
PC(c) Find the equation of a line
parallel to and perpendicular to a line
2 Which of the following lines is/ are parallel
to the line x y 8?
Click on the letterof a correct answer
y - x 1
B
Correct!This line has a gradient of -1.
NB There could be more than one rightanswer .
Have another go
76
Mathematics 1(Higher)
1.14


Outcome 1 Use the properties of the straight
line
PC(c) Find the equation of a line
parallel to and perpendicular to a line
2 Which of the following lines is/ are parallel
to the line x y 8?
Click on the letterof a correct answer
y x
C
Wrong!This line has a gradient of 1.
NB There could be more than one rightanswer .
Have another go
77
Mathematics 1(Higher)
1.15


Outcome 1 Use the properties of the straight
line
PC(c) Find the equation of a line
parallel to and perpendicular to a line
2 Which of the following lines is/ are parallel
to the line x y 8?
Correct!This line has a gradient of -1.
Click here to seeall the answers
NB There could be more than one rightanswer .
D
x y 10
y -x 10
Have another go
78
Mathematics 1(Higher)
1.16


Outcome 1 Use the properties of the straight
line
PC(c) Find the equation of a line
parallel to and perpendicular to a line
2 Which of the following lines is/ are parallel
to the line x y 8?
Wrong!This line has a gradient of 1.
Click here to seeall the answers
E
x - y 7
y x - 7
Have another go
79
Mathematics 1(Higher)
1.17


Outcome 1 Use the properties of the straight
line
PC(c) Find the equation of a line
parallel to and perpendicular to a line
2 Which of the following lines is/ are parallel
to the line x y 8?
y x 5
y - x 1
y x
A
B
C
Key
Parallel tox y 8
D
E
x y 10
x - y 7
y x - 7
Not parallel tox y 8
y -x 10
80
Mathematics 1(Higher)
1.18


Outcome 1 Use the properties of the straight
line
PC(c) Find the equation of a line
parallel to and perpendicular to a line
3 Which of the following lines is/ are parallel
to the line y ½ x - 3?
y 2x - 1
2y x
y ½ x 1
A
B
C
D
E
x - 2y 4
x - 2y 7 0
81
Mathematics 1(Higher)
1.19


Outcome 1 Use the properties of the straight
line
PC(c) Find the equation of a line
parallel to and perpendicular to a line
3 Which of the following lines is/ are parallel
to the line y ½ x - 3?
y 2x - 1
A
Wrong!This line has a gradient of 2.
Have another go
82
Mathematics 1(Higher)
1.20


Outcome 1 Use the properties of the straight
line
PC(c) Find the equation of a line
parallel to and perpendicular to a line
3 Which of the following lines is/ are parallel
to the line y ½ x - 3?
y ½ x 1
B
Correct!This line has a gradient of ½.
Have another go
83
Mathematics 1(Higher)
1.21


Outcome 1 Use the properties of the straight
line
PC(c) Find the equation of a line
parallel to and perpendicular to a line
3 Which of the following lines is/ are parallel
to the line y ½ x - 3?
2y x
C
Correct!This line has a gradient of ½.
y ½x
Have another go
84
Mathematics 1(Higher)
1.22


Outcome 1 Use the properties of the straight
line
PC(c) Find the equation of a line
parallel to and perpendicular to a line
3 Which of the following lines is/ are parallel
to the line y ½ x - 3?
Correct!This line has a gradient of ½.
Click here to seeall the answers
D
x - 2y 4
y ½ x - 2
Have another go
85
Mathematics 1(Higher)
1.23


Outcome 1 Use the properties of the straight
line
PC(c) Find the equation of a line
parallel to and perpendicular to a line
3 Which of the following lines is/ are parallel
to the line y ½ x - 3?
Correct!This line has a gradient of ½.
Click here to seeall the answers
E
x - 2y 7 0
y ½ x 3 ½
Have another go
86
Mathematics 1(Higher)
1.24


Outcome 1 Use the properties of the straight
line
PC(c) Find the equation of a line
parallel to and perpendicular to a line
3 Which of the following lines is/ are parallel
to the line y ½ x - 3?
y ½x
2y x
y ½ x 1
A
B
C
y 2x - 1
Key
Parallel toy ½ x - 3
D
E
x - 2y 4
x - 2y 7 0
y ½x - 2
Not parallel to y ½ x - 3
y ½ x 3 ½
87
Mathematics 1(Higher)
2.1


Outcome 1 Use the properties of the straight
line
PC(c) Find the equation of a line
parallel to and perpendicular to a line
Section 2
2 Perpendicular lines
88
Mathematics 1(Higher)
2.2


Outcome 1 Use the properties of the straight
line
PC(c) Find the equation of a line
parallel to and perpendicular to a line
y
B
CD is perpendicularto AB.
C
32
mAB
x
23
mCD
-
D
32
23
-
mAB ? mCD
?
-1
A
89
Mathematics 1(Higher)
2.3


Outcome 1 Use the properties of the straight
line
PC(c) Find the equation of a line
parallel to and perpendicular to a line
y
GH is perpendicularto EF.
F
G
34
mEF
x
43
mGH
-
34
43
-
?
mEF ? mGH
E
-1
H
90
Mathematics 1(Higher)
2.4


Outcome 1 Use the properties of the straight
line
PC(c) Find the equation of a line
parallel to and perpendicular to a line
y
Q
RS is perpendicularto PQ.
31
mPQ
R
x
13
-
mRS
S
31
13
?
-
mPQ ? mRS
-1
P
91
Mathematics 1(Higher)
2.5


Outcome 1 Use the properties of the straight
line
PC(c) Find the equation of a line
parallel to and perpendicular to a line
y
m1
If two lines with gradientsm1 and m2 are
perpendicularthen m1 m2 -1
x
m2
Memorise this
92
Mathematics 1(Higher)
2.8


Outcome 1 Use the properties of the straight
line
PC(c) Find the equation of a line
parallel to and perpendicular to a line
m
m
Summary
m
1
m
m
Parallel lines haveequal gradients.
y
m1
2
If two lines with gradientsm1 and m2 are
perpendicularthen m1 m2 -1.
x
m2
93
Mathematics 1(Higher)
2.6


Outcome 1 Use the properties of the straight
line
PC(c) Find the equation of a line
parallel to and perpendicular to a line
y
2
1 For each line write down the
gradient of any linea parallel to the
lineb perpendicular to the line
?
3
?
?
?
?
4
?
?
?
x
Here are the answers
?
1
Answers1 ½ , -2 2 -3, 1/3 3 3/4, -4/3 4
-1/3, 3
?
?
?
?
?
?
?
94
Mathematics 1(Higher)
2.7


Outcome 1 Use the properties of the straight
line
PC(c) Find the equation of a line
parallel to and perpendicular to a line
2 For each line write down the
gradient of any linea parallel to the
lineb perpendicular to the line
1 y 4x - 1
2 y ¾ x 5
3 y -5x
Here are the answers
4 x y 15
Answers1 4 , -¼ 2 ¾,
-4/3 3 -5, 1/5 4 -1, 1 5
½, -2 6 -3/5, 5/3
5 x - 2y 3 0
6 3x 5y 15
95
Mathematics 1(Higher)
3.1


Outcome 1 Use the properties of the straight
line
PC(c) Find the equation of a line
parallel to and perpendicular to a line
Section 3
3 Equations
96
Mathematics 1(Higher)
3.2


Outcome 1 Use the properties of the straight
line
PC(c) Find the equation of a line
parallel to and perpendicular to a line
AB has equation y 3x 5.Find the equation of
the line? parallel to AB through (1, -2)?
perpendicular to AB through (1, -2)
Click here for revisionof finding equationsof
straight lines
Parallel linemAB 3So mparallel 3Point on
line is (1, -2) y - b m (x - a) y - (-2)
3(x - 1) y 2 3x - 3 y 3x - 5
Perpendicular linemAB 3So mperp -1/3Point
on line is (1, -2) y - b m (x - a)
y - (-2) -1/3 (x - 1) 3y 6 - x
3 x 3y 3 0
97
Mathematics 1(Higher)
3.3


Outcome 1 Use the properties of the straight
line
PC(c) Find the equation of a line
parallel to and perpendicular to a line
Find the equation of the line
Answers 1 y 2x 3 2 y -4x 9 3 y
-x 4 y 2x -13 5 3x 2y -11 0
1 Through (0, 3), parallel to y 2x 1
2 Through (1, 5), perp to y ¼ x - 3
3 Through (-2, 2), parallel to x y 10
4 Through (5, -3), perp to y -½ x 7
5 Through (3, -1), parallel to
2x 3y 5 0