Download - Basic Skills in Higher Mathematics Robert Glen Adviser in Mathematics Mathematics 1(H) Outcome 1
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Basic Skills in
Higher Mathematics
Robert GlenAdviser in Mathematics
Mathematics 1(H)Outcome 1
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Mathematics 1(Higher)
Outcome 1 Use the properties of the straight line
Straight lines
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Mathematics 1(Higher)
Outcome 1 Use the properties of the straight line
PC Index
PC(a) Gradients and straight lines
PC(b) Gradients and angles
PC(c) Parallel and perpendicular
Click on the one you want
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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
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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
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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?
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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
2m
3m
The gradient (slope) of this roof is
2m
3m=
2
1 What is gradient?
3
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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
3m
3m3m
3m= 1
The gradient (slope) of this roof is
1 What is gradient?
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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
3m
7m3m
7m=
The gradient (slope) of this roof is
3
7
1 What is gradient?
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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
3m
7m
1 What is gradient?
2m
3m
3m
3m
=
= 3
7
= 2
3
Gradient
Gradient
Gradient 1
Check this:The steeperthe slope, the greater the gradient.
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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
3m
4m
5
4
What is the gradient of this roof ?
5m
A B
DC3 3
4
4 5
5
1 What is gradient?
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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
3m
4m
5
4
What is the gradient of this roof ?
5m
A B
DC3 3
4
4 5
5
Click on the letter of the correct answer
1 What is gradient?
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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
3m
4m
5
4
What is the gradient of this roof ?
5m
A B
DC3 3
4
4 5
5
Sorry, wrong answer
Have another go!
Gradient = vertical horizontal
1 What is gradient?
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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
3m
4m
5
4
What is the gradient of this roof ?
5m
A B
DC3 3
4
4 5
5
Click on the letter of the correct answer
1 What is gradient?
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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
3m
4m
5
4
What is the gradient of this roof ?
5m
A B
DC3 3
4
4 5
5
Correct!
1 What is gradient?
End of Section 1
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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
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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
Read all lines from left to right
Line AB is uphill from left to right
Line AB has a positive gradient mAB 0
A
B
y
x
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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
Read all lines from left to right
Line PQ is downhill from left to right
Line PQ has a negative gradient mPQ 0
A
BPy
Qx
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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
Read all lines from left to right
Line PQ has a negative gradient mPQ 0
Line AB has a positive gradient mAB 0
A
B
yP
Qx
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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
2 The gradient of a line
A
BGradient =
change in y
change in x
(9, 6)
(0, 3)mAB =39
13
=
3
9
y
x
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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
2 The gradient of a line
A
BGradient =
change in y
change in x
(9, 6)
(0, 3)mAB =39
13
=
Note: we could have measured the gradient like this
1
1
1
3
3
3
y
x
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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
x
Gradient = change in y
change in x
mPQ =-6 9
23
=Q
P
- 9
-6
(0, 7)
(9, 1)
y
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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
Gradient = change in y
change in x
mPQ =-6 9
23
=
Note: we could have measured the gradient like this
P
-
-2
-2
-2
3
3
3
y (0, 7)
Q (9, 1)x
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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
2 The gradient of a line
A
B (9, 6)
(0, 3)
Gradient = change in y
change in x
mAB =
6 - 3
9 - 0 6 - 3 9 - 0
= 3913
=
y
x
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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 = change in y
change in x
mPQ =
= 9 - 0
1 - 7 1 - 7 9 - 0-6 923
= -
yP
(0, 7)
Q (9, 1)x
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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
x
A formula to memorise
B (x2 , y2)
A (x1 , y1)
mAB =y2 - y1
x2 - x1
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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
x
A formula to memorise
B (x2 , y2)
A (x1 , y1)
mAB =y2 - y1
x2 - x1
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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
x
1 Calculate the gradient of line AB
B (6 , 5)
A (2 , 3)mAB =y2 - y1
x2 - x1
=5 - 36 - 2
= 24
= 12
Did you getthis answer?
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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
x
2 Calculate the gradient of line CD. D (6 , 2)
C (2 , -1)mCD =
y2 - y1
x2 - x1
=2 - (-1) 6 - 2
= 34
Did you getthis answer?
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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
x
3 Calculate the gradient of line EF.
F (5, -1)
E (-3 , 3)
mEF = y2 - y1
x2 - x1
=-1 - 35 - (-3)
= -4 8
= - 12 End of Section 2
Did you getthis answer?
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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
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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
xO
(0, 3) m = ½
(x, y)
K
LFind the equation of line KL which has a gradient of ½ and passes through the point (0, 3).
mKL =y - 3x - 0
= 12
y - 3 = ½ x
y = ½ x + 3The equation of KL is y = ½ x + 3
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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
xO
(0, 3) m = ½
(x, y)
K
LFind the equation of line KL which has a gradient of ½ and passes through the point (0, 3).
The equation of KL is y = ½ x + 3
Formula: y = m x + c
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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
xO
(0, c) m
(x, y)
K
LThe equation of line with gradient m and intercept c is:
y = m x + c
Memorise this
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1 Find the equation of line PQ which has a gradient of -2 and passes through the point (0, 5).
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
xO
(0, 5)
m = -2
P
Q
The equation of PQ is y = -2 x + 5
(x, y)Use the formula
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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
y
xO
(0, -3)
m = ¾
E
F2 Find the equation of line EF which has a gradient of ¾ and passes through the point (0, -3).
The equation of EF is y = ¾ x - 3
(x, y)
Use the formula
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Mathematics 1(Higher) 3.6
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
3 The equation of a line given gradient and intercept
You should now do Section A1 questions 1 - 10 on page 3 of
the Basic Skills booklet.
End of Section 3
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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
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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
x
(4, 3)
K
L (x, y)
O
Find the equation of the linethrough the point (4, 3) with gradient 3.
mKL =y - 3x - 4
= 3
y - 3 =
y - 3 = 3x - 12
y = 3x The equation of KL is y = 3x - 9
m = 3
3(x - 4)
- 9
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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
x
(4, 3)
K
L (x, y)
O
Find the equation of the linethrough the point (4, 3) with gradient 3.
The equation of KL is y = 3x - 9
m = 3
Formula: y - b = m (x - a)
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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
x
(a, b)
K
L (x, y)
O
The equation of the linethrough the point (a, b) with gradient m is :
m
y - b = m (x - a)
Memorise this
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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 gradienty
x
(-1, 2)
P
Q
(x, y)
O
1 Find the equation of the linethrough the point (-1, 2) with gradient 2.
The equation of PQ is y = 2 x + 4
m = 2
Use the formula
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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 gradienty
x
(-1, 2)
P
Q
(x, y)
O
1 Find the equation of the linethrough the point (-1, 2) with gradient 2.
The equation of PQ is y = 2 x + 4
m = 2 y - b = m (x - a)
y - 2 =
y - 2 = 2 (x + 1)
y - 2 = 2 x + 2
y = 2 x
(a, b)
(x - (-1))2
+ 4
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2 Find the equation of the linethrough the point (6, -2) with gradient ½.
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 gradienty
x (6, -2)
M
N
(x, y)
Om = ½
Use the formula
The equation of MN is 2y = x - 10
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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 gradienty
x (6, -2)
M
N
(x, y)O
2 Find the equation of the linethrough the point (6, -2) with gradient ½.
The equation of MN is 2y = x - 10
m = ½
y - b = m (x - a)
y - (-2) =
y + 2 = ½ (x - 6)
2y + 4 =
2y = x
(a, b)
or x - 2y - 10 = 0
Multiply both sides by 2to clear the fraction.
½ (x - 6)
x - 6
- 10
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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 gradienty
x
(-1, 4)
R
S
(x, y)
O
3 Find the equation of the linethrough the point (-1, 4) with gradient 2/3 .
The equation of RS is 3y = -2x + 10
m = -2/3
Use the formula
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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 gradienty
x
(-1, 4)
R
S
(x, y)
O
3 Find the equation of the linethrough the point (-1, 4) with gradient 2/3 .
The equation of RS is 3y = -2 x + 10
m = -2/3
y - b = m (x - a)
y - 4 =
3y - 12 =
3y =
(a, b)
or 2 x + 2y - 10 = 0
Multiply both sides by 3to clear the fraction.
-2/3(x - (-1))
y- 4 = -2/3 (x + 1)
-2(x + 1)
-2 x + 10
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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
You should now do Section A1 questions 11 - 20 on page 3 of
the Basic Skills booklet.
End of Section 4
Mathematics 1(Higher) 4.10
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
4 The equation of a line given one point and the gradient
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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
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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
Find the equation of the linejoining the points A (3, 1) and B (6, 4) .
Step 1 Calculate the gradient
mAB = y2 - y1
x2 - x1
=4 - 16 - 3
= 33
= 1
Step 2 Calculate the equation
y - b = m (x - a)
y - 1 =
y - 1 = x - 3
y = x - 2
Choose A (3, 1) as thepoint on the line.i.e. a = 3, b = 1
(You get exactly thesame answer if youchoose B.)
y
x
(6, 4)
A
B
(3, 1)
O
(a, b)
m = 1
1 (x - 3)
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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
The equation of CD is y = 2x
Use the formula
1 Find the equation of the linejoining the points C (1, 2) and D (5, 10) .
y
x
D
OC
(5, 10)
(1, 2)
Answer coming up!
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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
Step 1 Calculate the gradient
mAB = y2 - y1
x2 - x1
=10 - 2 5 - 1
= 84
= 2
Step 2 Calculate the equation
y - b = m (x - a)
y - 2 =
y - 2 = 2 x - 2
y = 2 x
Choose C (1, 2) as thepoint on the line.i.e. a = 1, b = 2
(You get exactly thesame answer if youchoose B.)
(a, b)1 Find the equation of the linejoining the points C (1, 2) and D (5, 10) .
y
xO
(5, 10)
(1, 2)
D
C
2 (x - 1)
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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
2 Find the equation of the linejoining the points G (-3, 1) and H (5, -3) .
x
(5, -3)
G
H
(-3, 1)
The equation of GH is 2y = - x - 1
Use the formula
y
Answer coming up!
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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
Step 1 Calculate the gradient
mGH = y2 - y1
x2 - x1
= -3 - 15 - (-3)
= -4 8
= -½
Step 2 Calculate the equation
y - b = m (x - a)
y - 1 =
2y - 2 =
2y = - x
Choose G (-3, 1) as the point on the line.i.e. a = -3, b = 1
(You get exactly thesame answer if youchoose H.)
(a, b)
or x + 2y +1 = 0
2 Find the equation of the linejoining the points G (-3, 1) and H (5, -3) .
x
G
H (5, -3)
(-3, 1)
y
-½(x - (-3))
- x - 3
- 1
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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
Step 2 Calculate the equation
y - b = m (x - a)
y - 1 = -½(x - (-3))
2y - 2 = - x - 3 2y = - x - 1
Multiply both sides by2 to clear the fraction.
A fuller explanation
y - 1 = -½(x + 3)
(a, b)
2 Find the equation of the linejoining the points G (-3, 1) and H (5, -3) .
x
(5, -3)
G
H
y
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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
mAB =y2 - y1
x2 - x1
y
x
(x2 , y2)
A (x1 , y1)
By
x
m
y = m x + c
(0, c)
O O
y
xO
y
xO
y - b = m (x - a)
(a , b)
(x , y)
(x1 , y1)
(x2 , y2)
m
1 Calculate m
m =y2 - y1
x2 - x1
2 y - b = m (x - a) (a, b)
Summary
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Mathematics 1(Higher) 5.8
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
5 The equation of a line given two points on the line
You should now do Sections A2 and A3 on page 3 of
the Basic Skills booklet.
End of Section 5
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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
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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
x
A
B
O
p
q
mAB =pq
= tan
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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
x
C
D
O
mCD=
35
= 0.70 (to 2 dp)
tan 35
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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
x
E
F
O
mEF =
35= -0.70 (to 2 dp)
tan 145
Line EF is downhill,so its gradient is nottan 35.
145
Always take the angle
between the line and the positive directionof the x-axis.
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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
mGH =
= 0.53 (to 2 dp)
tan 28
1 What is the gradient of the line GH (to 2 dp)?
x
28G
H
y
O
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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
x
K
L
O
mKL =48
= -1.11 (to 2 dp)
tan 132132
2 What is the gradient of the line KL (to 2 dp)?
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Mathematics 1(Higher) 1.6 Outcome 1 Use the properties of the straight line
PC(b) Find the gradient of a straight line using m = tan
You should now do the questions on page 7 of
the Basic Skills booklet.
End of PC(b)
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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
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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
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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
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These lines are all parallel to each other
If one of the lines has agradient m, they all havea gradient m.
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
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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 = 2x + 10
y = 2x + 5
y = 2x
y = 2x - 5
y = 2x - 10
x
y
The line 2x - y + 5 = 0 also belongs to this set of parallel lines.Can you see why?
2x - y + 5 = 0 2x + 5 = y y = 2x + 5
10-
5-
0
-5-
-10-
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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?
y = 3x - 1 y = -3x + 3 y = 3x
3x + y = 3 3x - y = 3
A C
Click on the letterof a correct answer
NB There could be more than one rightanswer .
B
D E
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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
NB There could be more than one rightanswer .
Correct!This line has a gradient of 3.
Have another go!
A
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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?
NB There could be more than one rightanswer .
Have another go!
Wrong!This line has a gradient of -3.
y = -3x + 3B
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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
NB There could be more than one rightanswer .
Correct!This line has a gradient of 3.
Have another go!
C
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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?
NB There could be more than one rightanswer .
3x + y = 3
Wrong!This line has a gradient of -3.
Have another go!
y = -3x +3
D
Click here to seeall the answers
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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.
Have another go!
Click here to seeall the answers
y = 3x +3
3x - y = 3E
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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?
Parallel toy = 3 x - 5
Not parallel toy = 3 x - 5
Key
y = -3x +3
y = 3x - 1 y = -3x + 3 y = 3x
3x + y = 3 3x - y = 3
A CB
D E
y = 3x +3
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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?
y = x + 5y = - x +
1 y = x
x + y = 10 x - y = 7
Click on the letterof a correct answer
NB There could be more than one rightanswer .
A
D
B
E
C
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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?
NB There could be more than one rightanswer .
Wrong!This line has a gradient of +1.
Have another go
y = x + 5A
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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?
y = - x + 1
Click on the letterof a correct answer
NB There could be more than one rightanswer .
Correct!This line has a gradient of -1.
Have another go
B
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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
NB There could be more than one rightanswer .
Wrong!This line has a gradient of +1.
Have another go
y = x C
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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?
NB There could be more than one rightanswer .
Correct!This line has a gradient of -1.
Have another go
y = -x +10
x + y = 10D
Click here to seeall the answers
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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.
Have another go
Click here to seeall the answers
y = x - 7
x - y = 7E
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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
Parallel tox + y = 8
Not parallel tox + y = 8
Key
2 Which of the following lines is/ are parallel to the line x + y = 8?
y = -x +10
y = x + 5y = - x +
1 y = x
x + y = 10 x - y = 7
A
D
B
E
C
y = x - 7
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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
y = ½ x + 1 2y = x
x - 2y = 4 x - 2y + 7= 0
A
D
B
E
C
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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?
Wrong!This line has a gradient of 2.
Have another go
y = 2x - 1A
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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?
Correct!This line has a gradient of ½.
Have another go
y = ½ x + 1B
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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
y = ½x
2y = x C
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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.
Have another go
y = ½ x - 2x - 2y = 4D
Click here to seeall the answers
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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
Have another go
y = ½ x + 3 ½
x - 2y + 7= 0E
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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?
Parallel toy = ½ x - 3
Not parallel to y = ½ x - 3
Key
y =½x
y = ½x - 2
y = 2x - 1 y = ½ x + 1 2y = x
x - 2y = 4 x - 2y + 7= 0
A
D
B
E
C
y = ½ x + 3 ½
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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
Continue with Section 2Perpendicular lines
End of Section 1
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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
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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
x
y
A
BC
D
mAB =32
CD is perpendicularto AB.
mCD =23
-
mAB mCD = 32
23
-
= -1
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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
x
y
E
FG
H
mEF =34
GH is perpendicularto EF.
mGH =43
-
mEF mGH = 34
43
-
= -1
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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
x
y
P
Q
R
S
mPQ =31
RS is perpendicularto PQ.
mRS =13
-
mPQ mRS = 31
13
-
= -1
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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
x
ym1
m2
If two lines with gradientsm1 and m2 are perpendicularthen m1 × m2 = -1
Memorise this
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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
If two lines with gradientsm1 and m2 are perpendicularthen m1 × m2 = -1.
Parallel lines haveequal gradients.
Summarym
m
m mm
x
ym1
m2
1
2
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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
x
y 1 For each line write down the gradient of any linea parallel to the lineb perpendicular to the line
1 Answers1 ½ , -2
2 -3, 1/3
3 3/4, -4/3
4 -1/3, 3
23
4
Here are the answers
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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
Answers1 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 = ¾ x + 5
6 3x + 5y = 15
3 y = -5x
4 x + y = 15
5 x - 2y + 3 = 0
Here are the answers
2 For each line write down the gradient of any linea parallel to the lineb perpendicular to the line
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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
You should now do Section C1 on page 11 of the Basic Skills booklet.
End of Section 2
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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
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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 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
Click here for revisionof finding equationsof straight lines
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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:
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 +75 Through (3, -1), parallel to 2x + 3y + 5 =0
Answers
1 y = 2x +3
2 y = -4x + 9
3 y = -x
4 y = 2x -13
5 3x + 2y -11 = 0
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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
You should now do Sections C2 and C3 on page 11 of the Basic Skills
booklet.
End of PC(c)