Transcript No Slide Title

Basic Skills
in
Higher Mathematics
Mathematics 1(H)
Outcome 1
Robert Glen
Adviser in Mathematics
Mathematics 1(Higher)
Outcome 1
Use the properties of the straight line
Straight
lines
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
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
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
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?
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
=
3m
3
2m
3m
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
= 1
3m
3m
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
3
=
7m
7
3m
7m
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
Check this:
The steeper
the slope,
the greater
the gradient.
7m
2m
2
Gradient =
3
3m
3m
Gradient = 1
3m
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 ?
A
5
4
B
5m
4
5
3m
4m
C
3
4
D
3
5
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 ?
A
5
4
B
5m
4
5
3m
4m
C
3
4
D
3
5
Click on the letter of
the correct answer
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?
What is the gradient
of this roof ?
A
5
4
B
Sorry, wrong answer
5m
4
5
3m
4m
C
3
4
D
Have another go!
3
5
Gradient = vertical
horizontal
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 ?
A
5
4
B
5m
4
5
3m
4m
C
3
4
D
3
5
Click on the letter of
the correct answer
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 ?
A
5
4
B
5m
4
5
3m
4m
C
3
4
D
3
5
Correct!
End of Section 1
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
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
2 The gradient of a line
y
B
 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
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
2 The gradient of a line
y
P
B
 Read all lines from
left to right
A
 Line PQ is downhill from
left to right
Q
x
 Line PQ has a negative gradient
mPQ  0
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
2 The gradient of a line
y
P
B
 Read all lines from
left to right
A
Q
x
 Line PQ has a negative gradient
mPQ  0
 Line AB has a positive gradient
mAB  0
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)
change in y
Gradient =
change in x
mAB =
3
9
= 1
3
A
(0, 3)
3
9
x
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)
change in y
Gradient =
change in x
mAB =
3
9
A
(0, 3)
1
1
1
3
3
3
= 1
3
Note: we could have measured
the gradient like this
x
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
2 The gradient of a line
change in y
Gradient =
change in x
y
(0, 7)
P
-6
mPQ = -6
9
= - 2
3
9
Q (9, 1)
x
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
2 The gradient of a line
change in y
Gradient =
change in x
y
(0, 7)
P
-2
3
-2
mPQ = -6
9
= - 2
3
Note: we could have measured
the gradient like this
3
-2
3
Q (9, 1)
x
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 (9, 6)
change in y
Gradient =
change in x
mAB =
=
=
6-3
9-0
3
9
1
3
A
(0, 3)
6-3
9-0
x
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
2 The gradient of a line
Gradient =
mPQ =
=
=
y
(0, 7)
P
change in y
change in x
1-7
1-7
9-0
-6
9
- 2
3
9-0
Q (9, 1)
x
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)
mAB =
y2 - y1
x2 - x1
x
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)
mAB =
y2 - y1
x2 - x1
x
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 - y1
A (2 , 3)
mAB = x - x
2
1
5-3
= 6-2
2
=
Did you get
4
this answer?
1
=
2
x
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.
y2 - y1
mCD = x - x
2
1
2 - (-1)
= 6-2
Did you get = 3
4
this answer?
D (6 , 2)
x
C (2 , -1)
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.
mEF =
=
Did you get =
this answer?
=
y2 - y1
x2 - x1
-1 - 3
5 - (-3)
-4
8
- 1
2
x
F (5, -1)
End of Section 2
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 line
given its gradient and the
intercept on the y - axis
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
Find the equation of line
(x, y) L

KL which has a gradient
(0, 3)
of ½ and passes through
 m=½
the point (0, 3).
K
y-3
1
=
x-0
2
 y-3=½x
mKL =

y =½x+3
O
The equation of KL is
y =½x+3
x
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
Find the equation of line
(x, y) L

KL which has a gradient
(0, 3)
of ½ and passes through
 m=½
the point (0, 3).
K
Formula:
y=mx+c
O
The equation of KL is
y =½x+3
x
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
(0, c)
m
gradient m and intercept c

K
is:
y=mx+c
O
Memorise this
x
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

(0, 5) m = -2
of -2 and passes through the
point (0, 5).
 (x, y)
Use the formula
O
Q
The equation of PQ is
y = -2 x + 5
x
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
O m=¾
the point (0, -3).
(0, -3)
Use the formula
E
The equation of EF is
y =¾x-3
x
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
3 The equation of a line given gradient and intercept
End of Section 3
You should now do Section A1
questions 1 - 10 on page 3 of
the Basic Skills booklet.
3.6
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 4
4 The equation of a line
given one point on the line
and the gradient
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 line
(x, y)
through the point (4, 3) with
m=3
gradient 3.
(4, 3)
mKL = y - 3 = 3
x-4
O
x
 y - 3 = 3(x - 4)
 y - 3 = 3x - 12

y = 3x - 9
K
The equation of KL is
y = 3x - 9
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 line
(x, y)
through the point (4, 3) with
m=3
gradient 3.
(4, 3)
Formula:
y - b = m (x - a)
O
K
The equation of KL is
y = 3x - 9
x
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 line
m
through the point (a, b) with
gradient m is :
 (a, b)
y - b = m (x - a)
O
x
K
Memorise this
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 line
(x, y)

through the point (-1, 2) with
gradient 2.
m=2
(-1, 2)
Use the formula

x
P
O
The equation of PQ is
y=2x+4
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 line
(x, y)

through the point (-1, 2) with
gradient 2.
(a, b) m = 2
y - b = m (x - a)
(-1, 2)

y - 2 = 2 (x - (-1))
x
P
O
y - 2 = 2 (x + 1)
y-2=2x+2
y=2x+4
The equation of PQ is
y=2x+4
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 line
N
through the point (6, -2) with
m=½
gradient ½.

O
x
(6, -2)

M
(x, y)
Use the formula
The equation of MN is
2y = x - 10
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
4.7
4 The equation of a line given one point and the gradient
y
2 Find the equation of the line
N
through the point (6, -2) with
m=½
gradient ½.

O (x, y)
x
(6, -2)

y - b = m (x - a)
M
y - (-2) = ½ (x - 6)
(a, b)
y + 2 = ½ (x - 6) Multiply both sides by 2
to clear the fraction.
2y + 4 = x - 6
The equation of MN is
2y = x - 10
2y = x - 10
or x - 2y - 10 = 0
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
R y
 m = -2/3
3 Find the equation of the line
through the point (-1, 4) with (-1, 4)
(x, y)
gradient 2/3 .

S x
O
Use the formula
The equation of RS is
3y = -2x + 10
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
4.9
4 The equation of a line given one point and the gradient
y
R
 m = -2/3
3 Find the equation of the line
(-1, 4)
through the point (-1, 4) with
(x, y)
gradient 2/3 .
(a, b)

y - b = m (x - a)
O
x
S
y - 4 = -2/3(x - (-1))
y- 4 = -2/3 (x + 1) Multiply both sides by 3
to 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
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
4 The equation of a line given one point and the gradient
End of Section 4
You should now do Section A1
questions 11 - 20 on page 3 of
the Basic Skills booklet.
4.10
4.9
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 line
given two points on the line
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
 B (6, 4)
Find the equation of the line
joining the points A (3, 1)
(a, b) (3, 1) m = 1
and B (6, 4) .
A
x
O
Step 1 Calculate the gradient
y2 - y1
4-1
mAB = x - x = 6 - 3
2
1
3
=
3
= 1
Step 2
Calculate the equation
y - b = m (x - a)
y - 1 = 1 (x - 3)
y-1=x-3
y=x-2
Choose A (3, 1) as the
point on the line.
i.e. a = 3, b = 1
(You get exactly the
same answer if you
choose B.)
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 line
joining the points C (1, 2)

C
(1, 2)
and D (5, 10) .
O
x
Use the formula
Answer coming up!
The equation of CD is
y = 2x
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
 D (5, 10)
1 Find the equation of the line
(a, b)
joining the points C (1, 2)

C
(1, 2)
and D (5, 10) .
O
x
Step 1
Calculate the gradient
y2 - y1 10 - 2
mAB = x - x = 5 - 1
2
1
8
=
4
= 2
Step 2
Calculate the equation
y - b = m (x - a)
y - 2 = 2 (x - 1)
y-2=2x-2
y=2x
Choose C (1, 2) as the
point on the line.
i.e. a = 1, b = 2
(You get exactly the
same answer if you
choose B.)
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
2 Find the equation of the line G 
joining the points G (-3, 1) (-3, 1)
and H (5, -3) .
x

H (5, -3)
Use the formula
Answer coming up!
The equation of GH is
2y = - x - 1
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
2 Find the equation of the line G 
joining the points G (-3, 1) (-3, 1)
(a, b)
and H (5, -3) .
x

H(5, -3)
Step 1
Calculate the gradient
Step 2
Calculate the equation
y2 - y1
y - b = m (x - a)
mGH = x - x = -3 - 1
2
1
5 - (-3) y - 1 = -½(x - (-3))
-4
=
2y - 2 = - x - 3
8
2y = - x - 1
= -½
or x + 2y +1 = 0
Choose G (-3, 1) as
the point on the line.
i.e. a = -3, b = 1
(You get exactly the
same answer if you
choose H.)
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
2 Find the equation of the line G 
joining the points G (-3, 1)
(a, b)
and H (5, -3) .
x

H (5, -3)
Step 2
Calculate the equation
A fuller explanation
y - b = m (x - a)
y - 1 = -½(x - (-3))
y - 1 = -½(x + 3)
2y - 2 = - x - 3
2y = - x - 1
Multiply both sides by
2 to clear the fraction.
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
mAB =
y2 - y1
x2 - x1

A (x1 , y1)
O
(0, c)

m
y=mx+c
x
Summary
y
y
2 y - b = m (x - a)
x
x
O
1 Calculate m
y2 - y1
m = x -x
2
1
(x , y)

y - b = m (x - a)  m
(a , b)
O
y
(x2 , y2)B

O
(x2 , y2)


(x1 , y1)
(a, b)
x
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
5 The equation of a line given two points on the line
End of Section 5
You should now do Sections A2
and A3 on page 3 of
the Basic Skills booklet.
5.8
Mathematics 1(Higher)
Outcome 1
Use the properties of the straight line
PC(b)
Find the gradient of a straight line using m = tan
Gradients
and
angles

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
p
mAB =
q
p
= tan 
A

q
O
x
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)
C
O
35
x
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
Line EF is downhill, E
so its gradient is not
tan 35.
mEF = tan 145
35
= -0.70 (to 2 dp)
145
F
Always take the angle
between the line and
the positive direction
of the x-axis.
O
x
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)
O
x
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 K
the line KL (to 2 dp)?
48
mKL = tan 132
132
L
= -1.11 (to 2 dp)
O
x
Mathematics 1(Higher)
Outcome 1
Use the properties of the straight line
PC(b)
Find the gradient of a straight line using m = tan
End of PC(b)
You should now do the questions
on page 7 of
the Basic Skills booklet.
1.6
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
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
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
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 a
gradient m, they all have
a gradient m.
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.
The line 2x - y + 5 = 0
also belongs to this
set of parallel lines.
Can you see why?
So any line parallel to this
one has a gradient of 2.
y
y = 2x + 10
10y = 2x + 5
5-
y = 2x
0
x
2x - y + 5 = 0
2x + 5 = y
 y = 2x + 5
y = 2x - 5
y = 2x - 10
-5-10-
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?
A
D
y = 3x - 1
3x + y = 3
B
E
y = -3x + 3
3x - y = 3
Click on the letter
of a correct answer
C
y = 3x
NB There could be
more than one right
answer .
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?
A
y = 3x - 1
Correct!
This line has
a gradient of 3.
Have another go!
NB There could be
more than one right
answer .
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?
B
y = -3x + 3
Wrong!
This line has a
gradient of -3.
Have another go!
NB There could be
more than one right
answer .
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?
Correct!
This line has
a gradient of 3.
C
y = 3x
NB There could be
more than one right
answer .
Have another go!
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.
D
3x + y = 3
y = -3x +3
Have another go!
Click here to see
all the answers
NB There could be
more than one right
answer .
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.
E
3x - y = 3
y = 3x +3
Have another go!
Click here to see
all the answers
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?
A
y = 3x - 1
B
y = -3x + 3
C
y = 3x
Key
D
3x + y = 3
y = -3x +3
E
3x - y = 3
y = 3x +3
Parallel to
y=3x-5
Not parallel to
y=3x-5
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?
A
D
y=x+5
x + y = 10
B y=-x+1
E
x-y=7
Click on the letter
of a correct answer
C
y=x
NB There could be
more than one right
answer .
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?
A
y=x+5
Wrong!
This line has a
gradient of +1.
Have another go
NB There could be
more than one right
answer .
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 letter
of a correct answer
B y=-x+1
Correct!
This line has
a gradient of -1.
Have another go
NB There could be
more than one right
answer .
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 letter
of a correct answer
C
Wrong!
This line has a
gradient of +1.
Have another go
y=x
NB There could be
more than one right
answer .
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.
D
x + y = 10
y = -x +10
Have another go
Click here to see
all the answers
NB There could be
more than one right
answer .
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.
E
x-y=7
y=x-7
Have another go
Click here to see
all the answers
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?
A
y=x+5
B y=-x+1
C
y=x
Key
D
x + y = 10
y = -x +10
E
x-y=7
y=x-7
Parallel to
x+y=8
Not parallel to
x+y=8
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?
A
y = 2x 1
D
x - 2y = 4
B y=½x+1
E x - 2y + 7= 0
C
2y = x
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?
A
y = 2x 1
Wrong!
This line has a
gradient of 2.
Have another go
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?
B y=½x+1
Correct!
This line has
a gradient of ½.
Have another go
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?
Correct!
This line has
a gradient of ½.
Have another go
C
2y = x
y = ½x
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.
D
x - 2y = 4
Have another go
y=½x-2
Click here to see
all the answers
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.
E x - 2y + 7= 0
y=½x+3½
Have another go
Click here to see
all the answers
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
A
y = 2x - 1
B y=½x+1
C
2y = x
Key
D
x - 2y = 4
y = ½x - 2
E x - 2y + 7= 0
y=½x+3½
Parallel to
y=½x-3
Not parallel to
y=½x-3
Mathematics 1(Higher)
1.25
Outcome 1
Use the properties of the straight line
PC(c)
Find the equation of a line parallel to and perpendicular to a line
End of Section 1
Continue with Section 2
Perpendicular lines
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
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
CD is perpendicular
to AB.
mAB =
B
C
3
2
x
mCD = - 2
3
D
mAB  mCD = 3  - 2
2
3
= -1
A
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 perpendicular
to EF.
mEF =
3
4
F
G
x
mGH = - 4
3
3  -4
mEF  mGH =
4
3
= -1
E
H
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
RS is perpendicular
to PQ.
3
mPQ =
1
Q
R
x
mRS = - 1
3
S
3  -1
mPQ  mRS =
1
3
= -1
P
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 gradients
m1 and m2 are perpendicular
then m1 × m2 = -1
x
m2
Memorise this
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
Summary
1
Parallel lines have
equal gradients.
y
2
m m
m
m
m
m1
x
m2
If two lines with gradients
m1 and m2 are perpendicular
then m1 × m2 = -1.
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
1 For each line write 2 
down the gradient of
any line

a parallel to the line
 
b perpendicular to the line




Answers
1 ½ , -2



2 -3, 1/3


3 3/4, -4/3

3
4

Here are the answers
4 -1/3, 3
y
1
x
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 line
a parallel to the line
b perpendicular to the line
Here are the answers
Answers
1 4 , -¼
2 ¾, -4/3
3 -5, 1/5
4 -1, 1
5 ½, -2
6 -3/5, 5/3
1 y = 4x - 1
2 y=¾x+5
3 y = -5x
4 x + y = 15
5 x - 2y + 3 = 0
6 3x + 5y = 15
Mathematics 1(Higher)
2.9
Outcome 1
Use the properties of the straight line
PC(c)
Find the equation of a line parallel to and perpendicular to a line
End of Section 2
You should now do Section C1 on
page 11 of the Basic Skills booklet.
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
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)
Parallel line
mAB = 3
So mparallel = 3
Point on line is (1, -2)
y - b = m (x - a)
y - (-2) = 3(x - 1)
y + 2 = 3x - 3
y = 3x - 5
Click here for revision
of finding equations
of straight lines
Perpendicular line
mAB = 3
So mperp = -1/3
Point on line is (1, -2)
y - b = m (x - a)
y - (-2) = -1/3 (x - 1)
3y + 6 = - x + 3
x + 3y + 3 = 0
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 Through (0, 3), parallel to y = 2x +1
1 y = 2x +3
2 Through (1, 5), perp to y = ¼ x - 3
2 y = -4x + 9
3 Through (-2, 2), parallel to x + y = 10
3 y = -x
4 Through (5, -3), perp to y = -½ x +7
4 y = 2x -13
5 Through (3, -1), parallel to
2x + 3y + 5 =0
5 3x + 2y -11 = 0
Mathematics 1(Higher)
3.4
Outcome 1
Use the properties of the straight line
PC(c)
Find the equation of a line parallel to and perpendicular to a line
End of PC(c)
You should now do Sections C2 and
C3 on page 11 of the Basic Skills
booklet.