14 Linear Law

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WAJA 2009

ADDITIONAL MATHEMATICS

FORM FIVE

( Students Copy )

Name: ___________________________

Class : ___________________________

www.cikgurohaiza.com

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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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( 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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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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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


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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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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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


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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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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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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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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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


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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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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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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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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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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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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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


b) xxy 73 72 xx

y

Y m X c


c) 8yx =

Y m X c

y 8

y 2x 5

y x

y

x

1 x2

-


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d) xy3

7 2 = 7 +

Y m X c

3x 3


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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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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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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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 ]

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14 Linear Law