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WAJA 2009
ADDITIONAL MATHEMATICS
FORM FIVE
( Students Copy )
Name: ___________________________
Class : ___________________________
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WAJA 2009 ADDITIONAL MATHEMATICS (FORM 5) Chapter 2: Linear Law
2
Learning Objective:
2.Understand and use the concept of lines of best fit
Learning Outcome:
2.1.1 Draw lines of best fit by inspection of given data
1) Which one between the pair is the best fit line? Tick the appropriate cup. Then writedown the reason for your choice in the rectangular box.
Example:
Draw a line of best fit by inspecting the given data on the graph
( i )
xxx
x
xx
x
y
x
Criteria the line of best fit:1.points lie as close as possible to the line
2.line pass through as many points as possible
3.number of points above and below should be the same
distance from the linex
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WAJA 2009 ADDITIONAL MATHEMATICS (FORM 5) Chapter 2: Linear Law
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( ii )
( iii )
2) Draw a line of best fit by inspecting the given data on the graph for each of the following
graph
(a) (b)
xx
x xx
x x
x
0 0
x
x
y
x
y
x
x
x
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WAJA 2009 ADDITIONAL MATHEMATICS (FORM 5) Chapter 2: Linear Law
4
c)
Learning Outcome:
2.1.2 Write equation for lines of best fit
1. Match the correct answer
0x
y
x
x
xx
0x
y
x
xx
x x
x
x
(d)
y=mx+c is the linear equation of a straight line
21
21
12
12
xx
yyor
xx
yy
c
m=gradient
c= -i ntercept
is a value of y where the graph cuts the y-axis
a) find y-intercept given y=2x+8
b) find the gradient given y=-5x-8
c) find y-intercept from the graph
below
2
d) find the gradient from the graph below
P(4,2)
Q(8,14)
e) find the value of y-intercept
from the graph below
(5,0)
(0,-3)
-3
3
8
2
-5
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WAJA 2009 ADDITIONAL MATHEMATICS (FORM 5) Chapter 2: Linear Law
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2. Write the equation of the line of best fit for each of the following graphsExample:
i) ii)
Find m:04
26
m
1
Find m:18
61
m
7
5
Find c:
y-intercept, c = 2
Find c: cmtP
c )1(756
7
47c
Substitute into cmxy
The equation of the line is 2 xy
Substitute into cmxy
The equation of the line :7
47
7
5 tP
a) b)
2
0
x
y
4 6
x
x
x
F
X0
(2,2)
(8,10)x
x
x
x
[Ans :3
2
3
4
xF ]
0 V
(1,5)
x
(6,1)x
x
x
P
[Ans :
5
29
5
4
VP ]
0
(1,6)
(( 8,1)x
x
x
x
t
P
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WAJA 2009 ADDITIONAL MATHEMATICS (FORM 5) Chapter 2: Linear Law
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3. Determine the horizontal and vertical axes of the following graph
Example:
Plot graph y2
against x
a) Plot graph y against x2
b) Plot graph P against V
x
y2
c) Plot graph xy againstx
1d) Plot graph a against
b
1
horizontal axes
vertical axes
0
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WAJA 2009 ADDITIONAL MATHEMATICS (FORM 5) Chapter 2: Linear Law
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4. Based on the table given, plot the points and draw a line of best fit. Hence write the
equation for the line of best fit.
Example:
i) The values of variables G and H in an experiment are given in the table below
G 0.3 0.6 0.8 0.9 1.0
H 0.35 0.50 0.59 0.65 0.7
a) Plot G against H and draw a line of best fit
b) Write an equation for the line of best fit
Solution:Plot the graph:
Find m from the graph:01
22.07.0
m
48.0
Find c from the graph: c = 0.21
Substitute into cmxy
21.048.0 HG
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.00.90.80.70.60.50.40.30.20.1
G
H
(0,0.22)
(1.0,0.7)
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WAJA 2009 ADDITIONAL MATHEMATICS (FORM 5) Chapter 2: Linear Law
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a) The table below represents the experimental values of two variables L and W.
W 0.1 0.2 0.3 0.4 0.5
L 10.8 11.3 12.1 12.4 12.7
i) Plot L against W and draw the line of best fit
ii) Write an equation for the line of best fit
b) The table below represents the experimental values of two variables p and q.
p 1.0 2.8 5.6 8.4 11.5 14.2
q 2.2 3.4 5.7 7.4 9.7 11.2
i) Plot q against p and draw the line of best fit
ii) Write an equation for the line of best fit
Learning Outcomes:
2.1.3 Determine values of variables
2.1.3 ( a )Determine values of variables from lines of best fit
1. Determine the values of the variables from the given lines of best fit.
Example:
[Ans : m = 5.3, c = 10.2, L = 5.3W+10.2]
[Ans : m = 0.7, c = 1.4, q = 0.7p + 1.4]
y
x00
5
4
3
1
2
31 2
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WAJA 2009 ADDITIONAL MATHEMATICS (FORM 5) Chapter 2: Linear Law
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Two variables, x and y, are related by the line of best fit as shown in the graph above. Fromthe graph, determine the value of
( i ) y when x = 1.4
( ii ) x when y = 4.4
Solution
From the graph,
( i ) when x = 1.4 , y = 2.9
( ii ) when y = 4.4 , x = 2.2
y=2.9
x=1.4
y=4.4
00
5
4
3
1
2
31 2
y
x
x =2.2
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WAJA 2009 ADDITIONAL MATHEMATICS (FORM 5) Chapter 2: Linear Law
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a) Two variables , x and y, are related by the line of best fit as shown in the graph above. From
the graph , determine the value of
( i ) y when x = 0.8
( ii ) y when x = 2.2
( iii ) x when y = 1.2
b) The table below shows the corresponding values of the variables T and V obtained froman experiment.
T 20 40 60 80 100 120
V 122 130 136 148 154 163
i) Plot a graph of V against T and draw a line of best fit.
ii) Use the graph obtained in ( a ) to determine the value ofa) V when T = 72
b) T when V = 150
00
3
1
2
31 2
y
x
[ Answer: a) i) 2.2 ii) 1.4 iii) 2.6 b) ii) a) 143 b) 88 ]
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WAJA 2009 ADDITIONAL MATHEMATICS (FORM 5) Chapter 2: Linear Law
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2.1.3 ( b ) Determine values of variables from the equations of lines of best fit
Example :
yP(0.2,16.5)
Q(0.8,9.0)x
0
The diagram above shows a line of best fit obtained by plotting a graph of y against x. Theline passes through points P(0.2,16.5) and Q(0.8,9.0)
i) Find the equation of the line of best fit
ii) Determine the value of( a) y when x = 0.7
( b ) x when y = 22
Solution
i)
Find m:
5.128.02.0
0.95.16
m
Find c: cxy 5.12
At point P(0.2,16.5), c )2.0(5.125.16
19
5.25.16
c
c
Substitute into cmxy
195.12 xy
ii)
195.12 xy
i) When 7.0x , 19)7.0(5.12 y25.10
ii) When 22y , 195.1222 x
35.12 x
24.0x
X
x
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WAJA 2009 ADDITIONAL MATHEMATICS (FORM 5) Chapter 2: Linear Law
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a) The diagram above shows part of a line of best fit obtained by plotting a graph ofp against q.
The line passes through points (2,0.5) and (5,3).i) Find the equation of the line of best fit
ii) Determine the value of
( a ) q when p=0
( b ) p when q = -1
b) Two variables, v and t, are known to be linearly related as shown by the line of best fit in the
graph above. The line passes through points (2,2) and (3,0.5).
i) Determine the linear equation relating v and t
ii) Find the value of v when( a) t= 0( b) t= 5
(5,3)
2,0.5
q
p
(3,0.5)
(2,2)
t
v
[ Answer : a) i) 1.18.0 qp ii) (a)1.38 (b)-1.9
b) i) 55.1 tv ii) (a)5 (b) -2.5 ]
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WAJA 2009 ADDITIONAL MATHEMATICS (FORM 5) Chapter 2: Linear Law
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Learning Objective:
2.2Apply linear law to non-linear relations
Learning Outcome:
2.2.1 Reduce non-linear relations to linear form
1. Plot the graphs based on the given tables of values.
52 xy 52 xy
x 0 1 2 3 4
y -5 -4 -1 4 11
What do you observe?
Quadratic graphs can be drawn as linear graphs if we change the representation ofx-
axis, that is x x2
.
x 0 1 4 9 16
y -5 -4 -1 4 11
y
x
0 252015105
15
10
5
-5
20
y
x2
0 252015105
15
10
5
-5
20
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WAJA 2009 ADDITIONAL MATHEMATICS (FORM 5) Chapter 2: Linear Law
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b) 12
xy
1)1
(2 x
y
x 0.5 1 2 3 4
y 5 3 2 1.7 1.5
Again, what do you observe?
Reciprocal graphs can be drawn as linear graphs if we change the representation ofx-axis,
that is x 1x .
Conclusion: A non-linear equation can be reduced to a linear form Y= mX+ c.
x
y
0
5
4
3
2
1
1 2 3 4
y
1
x0
5
4
3
2
1
1 2 3 4
1 2 1 0.5 0.33 0.25
y 5 3 2 1.7 1.5
0.5 1 1.5 2
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WAJA 2009 ADDITIONAL MATHEMATICS (FORM 5) Chapter 2: Linear Law
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Linear Form of Equation
2. Identify .,,, cmXY based on the following linear form.
Example:
i)
ii)
Fill in the blank with the correct answer :
L inear equation Y m X c
85 2 xy
22 xy y x
y 7 7 0
xx
y 52
12
38 xxy
2
1
2
31
xy
Variables for the x-axis
cmXY
Variables for the Y-axiscoefficient=1
m=gradient
Constant (no variable)
y-intercept
or mXY , c=0
32 2 xy
Y= m=2 X=x2
c=-3
53 2 xy
yY
m=3 X=x2
c=-5
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WAJA 2009 ADDITIONAL MATHEMATICS (FORM 5) Chapter 2: Linear Law
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3. Reduce the following non linear equation to linear equation in the form of cmXY .
Hence identify .,,, cmXY
Example:
i) 532 2 xy
Create the y-intercept
x
x
x
x
x
yx
532:
2
532
xx
y
Create coefficient ofy=1 & Arrange in the form Y=mX+c
2
5
2
3:2 x
x
y
2
5
2
3 x
y
Compare to Y=mX+c
2
5
2
3 x
y
2
5,,
2
3, cxXm
yY
ii) xx
y 372
Create the y-intercept
xxxxyx 37
: 2
22 37 xxy
Arrange in the form Y=mX+c
73 22 xxy
Compare to Y=mX+c
73 22 xxy
7,,3, 22 cxXmxyY
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WAJA 2009 ADDITIONAL MATHEMATICS (FORM 5) Chapter 2: Linear Law
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4. Fill in the blank with correct answer
Example:
Non Linear Equation Linear Form Equation
a) 52 2 xy = - 5
Y m X c
2 -5
Non Linear Equation Linear Form Equation
b) xxy 73 72 xx
y
Y m X c
Non Linear Equation Linear Form Equation
c) 8yx =
Y m X c
y 8
y 2x 5
y x
y
x
1 x2
-
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WAJA 2009 ADDITIONAL MATHEMATICS (FORM 5) Chapter 2: Linear Law
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Non Linear Equation Linear Form Equation
d) xy3
7 2 = 7 +
Y m X c
3x 3
Non Linear Equation Linear Form Equation
e) 235 xxy x
x
y35
Y m X C
Non Linear Equation Linear form equation
f) 749
xy1
9
41
xy
Y m X C
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WAJA 2009 ADDITIONAL MATHEMATICS (FORM 5) Chapter 2: Linear Law
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Non Linear Equation Linear Form Equation
g) xaby ylog +
Y m X C
alog
Learning Outcomes:
2.2.2 Determine values of constants on non-linear relations given:
2.2.2a) Determine values of constants from lines of best fit
1. Find the value of the following situation.
Example:
The variables x and y are related by the equationa
xby where a and b are constants.The
diagram below shows part of a line of best fit obtained by plotting a graph of xy against x2
.Find
the values of a and b.
xy
(4,50)
(1,35)
x2
0
From the graph identify the representation of y-axis and x-axis2, xXxyY
Reduce the equation given to linear form , Y = mX +c:
bxa
bx
bbyb :
ybx
a
b
x
xbx
ax
b
xxyx :
21 x
bxy
b
a
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WAJA 2009 ADDITIONAL MATHEMATICS (FORM 5) Chapter 2: Linear Law
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Compare Y=mX+c
b
ac
bmxXxyY ,
1,,
2
Find m from the graph:
514
3550
m
Find c , substitute X=1 , Y=35 , m=5 into the equation cmXY
Y mX+c35=5(1)+c
c=30
Find the variables a and b
bm
1 =5
5
1b
a) Diagram shows a straight line graph ofx
yagainst 2x
x
y
2x
Given that qxpxy 3 where p and q are constants, find the values ofp and q.
[p=1, q=2]
0
(4,6)
(1,3)
6
30
5
1
30
a
ab
a
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WAJA 2009 ADDITIONAL MATHEMATICS (FORM 5) Chapter 2: Linear Law
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b) Diagram shows a straight line graph ofx
yagainst
2
1
x.The variables x and y are related
by the equationb
axy where a and b are constants. Find the values a and b.
y
(3,7)
(1,3)
[ a = 1 , b = 2]
2.2.2b)Determine values of constants from data
1. Find the values of the constants based on the table of values.
Example:
x 1 2 3 4 5
y 1.00 2.83 3.81 5.00 5.90
The table above shows experimental values of two variables, x and y. The variables x and y are
known to be related by the equationx
kxhy where h and k are constants.
( i )Plot a graph of xy against x and draw a line of best fit.
( ii )From the graph obtained in ( b ) , find the values of h and k.
Solution:
From the graph identify the representation of y-axis and x-axis
Y= xy , X=x
Construct a new table
X=x 1 2 3 4 5
xyY 1.00 4.00 6.60 10.00 13.19
2
1
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Plot the points and draw the graph
Reduce the equation given to linear form , cmXY
x : xx
kxxhxy
khxxy
Compare cmXY
kchmxXxyY ,,,
Find m and y-interceptfrom the graph :
077.3
077.3
077.34.14
210
h
hm
m y-intercept=c=-2c=k=-2
2 k
-2
Graph of y x against x
x
y x
14
12
10
8
6
4
2
1 2 3 4 50
10 2 = 8
4 1.4 = 2.6
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a) Table shows the values of two variables , x and y,obtained from an experiment.The
variables x and y are related by the equation ,px
rpxy where p and r are constants.
x 1.0 2.0 3.0 4.0 5.0 5.5
y 5.5 4.7 5.0 6.5 7.7 8.4
( i ) Plot xy against x2
, by using a scale of 2 cm to 5 units on both axes.Hence , draw the line of
best fit.
( ii ) Use the graph from ( a ) to find the value of
( a ) p,
( b ) r. [ p = 1.3778 , r = 5.5111 ]
b) Table shows the values of two variables , x and y,obtained from an experiment.The variables
x and y are related by the equation ,2 2 xk
pkxy where p and k are constants.
x 2 3 4 5 6 7
y 8 13.2 20 27.5 36.6 45.5
( i )Plotx
yagainst x , using a scale of 2 cm to 1 unit on both axes. Hence, draw the line of best
fit.
( ii )Use the graph from ( a ) to find the value of
( a ) p,
( b ) k.
( c ) y when x = 1.2 [ p = 0.7763 , k = 0.2875 ,y = 4.08 ]