2. Linear Law

Chapter 2 - LINEAR LAW

1.1 Draw the line of best fit 1

2

3

4

1.2 Write the equation for the line of best fit of the following graphs. 1

.

[ ]

2

[ ]

3

[y=2x2+6]

4

[y=-3x2+11]

5

[ ]

6

[ ]

Linear law 1

y

x

P(0,3)

Q(6,13)

.

.

XX

X

XX

X

XP(0,5)

Q(2,0)

y

x

.

.

X

X

XX

X

P(-1,4)

Q(5,16)

x2

y

.

.

xx

x

xx

A(3,0)

B(5,9)

x

x

x

xs

t

y2

x

y

x2

log 10 y

log 10 x

y

x

y

x2

P(1,8)

Q(3,2)

.

.

x

x xx

x

7

[ p = ]

8

[ ]

1.3 Determine the values of variables from lines of best fit 1

The diagram below shows a line of best fit. From the graph, find

i. the value of y when x = 0.5ii. the value of x when y = 7

iii.

[2.8, 3.3]

2

The diagram below shows a line of best fit. From the graph, find

i. the value of t when w = 38ii. the value of w when t = 1.6

[3.6,22]

3

The diagram below shows a line of best fit obtained by plotting the graph of d against t. The line intersects the vertical and the horizontal axes at points (0,2) and (6,0) respectively. Find

i. the equation of best fitii. the value of t when d=3

iii. the value of d when t=4

[ ]

4

Two variables, p and q are known to be linearly related as shown by the line of best fit in the diagram below. The line passes through points (1.6, 6) and (13.6 , 30). Determine

i. the equation of best fitii. the value of q when p= 15

iii. the value of p when q = 5

Linear law 2

B(5,3)

x

xp

q

A(2,)

d(0,2)

(6,0)

t

p(13, 30)

(1, 6)

q

x

x

x

x

x

x

x

1 2 3 4

2

4

6

8

x

y

x

x

x

x

100

20

30

1 2 3 4 t

W

40 x

[p=2q+4, 5.5, 14]2.1 Reduce non linear relations to linear form Reduce each of the equations to the form Y=m X+C where a and b are constants.

Non-linear equation Linear equation Y X m C1 y = a x2 + b x x

2 y = ax3 + bx2 x

3y =

y

4y =

xy x2

5xy =

y

6 x +by = axy

7 xy x2

8 xy

9 y = abx log10 y x

10 y = axb log10 y log10 x

11 y =a2xb log10 y log10 x

12 PV=a P

Linear law 3

2.2 Determine values of constants of non-linear relations given lines of best fit1 The diagram below shows the line of best fit for the

graph of y2 against x. Determine the non-linear equation connecting y and x.

[y2=-2x+4]

2 The diagram below shows the line of best fit for the

graph of y2 against . Determine the non-linear equation

connecting y and x.

[ ]

3 The diagram below shows the line of best fit for the

graph of against x. Determine the non-linear

equation connecting y and x.

[ ]

4 The diagram below shows the line of best fit for the

graph of against x. Determine the non-linear equation connecting y and x.

[ ]

Linear law 4

P(0,4)

Q(2,0)

y2

x

X

XP(0,2)

Q(2,12)y2

x

1

X

P(3,0)

Q(6,12)2x

y

xX

X

(2,4)

2x

y

(4, 5)

x0

5 The diagram below shows the line of best fit for the graph of log10 y against x. Determine the non-linear equation connecting y and x.

[log10 y = 3x ]

6 The diagram below shows the line of best fit for the graph of log10 y against log 10 x. Determine the non-linear equation connecting y and x.

[ log 10 y = 2 log10 x+2]7 The diagram below shows the line of best fit for the

graph of xy against x. Determine the relation between y and x.

[ ]

8 The diagram below shows the straight line graph of against x. Express y in terms of x.

[ ]

9 The diagram below shows the straight line graph of

against x. Express y in terms of x.

10 The diagram below shows the line of best fit for the graph of x2y against x. Determine the relation between y and x.

Linear law 5

(0,0)

(2,6)y10log

xX

X

(0,2)

(2,6)

y10log

x10log

X

X

(4,10)

x

(1, 1)

xy

P(2,4)

Q(4,12)

xX

Xxy

(4,10)

x

(1, 1)

x 2 y

P(2,4)

Q(4,12)

xX

Xx2 y

[ ] [ ]

11 The diagram below shows the line when against x

is drawn. Express y as a function of x.

[ ]

12The diagram below shows the line when against x is

drawn. Express y as a function of x.

[ ]

13

The diagram below shows the line of best fit for the

graph of against . Determine the relation

between y and x.

[ ]

14 The diagram below shows the line when against x is

drawn. Express y as a function of x.

[ ]

15The diagram below shows the line when

against x is drawn. Express y in terms of x.

16The diagram below shows the line when against x is

drawn. Determine the non-linear equation connecting y and x

Linear law 6

(12, -1)

5

x

y

x(12, -1)

5

x

y

x0

(0,8)

(-4,0)

x

y 1

x

(8,10)

(4,2)

y

1

x0

P(3,0)

Q(6,12)2x

y

xX

X

(2,4)

2x

y

(4, 5)

x0

3

(8, -3)

x

y

x

x

y

x0

[y=2x2+8x-1]

[ ]

17The diagram below shows the line when against x

is drawn. Determine the non-linear equation connecting y and x

[

]

18

The diagram below shows the line of best fit for the graph of log10 y against x. Determine the relation between y and x.

[ y = 10 3x ]

19 The diagram below shows part the graph of log10 y against x. Form the equation that connecting y and x.

[]

20

The diagram below shows the line of best fit for the graph of log10 y against log 10 x. Determine the relation between y and x.

[y= 100x2]

21 The diagram below shows part the graph of log10 y against log10 x. Form the equation that connecting y and x.

[ ]

22

The diagram below shows part the graph of log 2 y against log2 x. Determine the relation between y and x.

[ ]

Linear law 7

(3,6)

(0,3)

y

x

x0(0,0)

(2,6)y10log

xX

X

6

-3

log10 x

log10 y

2

(5,6)

log2 x

log2 y

0

log10 y

(3,4)

(4,0)

x (0,2)

(2,6)

y10log

x10log

X

X

2.3 Determine values of constants of non-linear relations when given lines of best fitExample :The variables x and y are related

by the equation , 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 .

Find the values of a and b.

1 From the graph identify the representation of y-axis and x-axisY = , X =

2 Reduce the equation given to linear form, Y = mX + [ then ]

3 Compare with Y = mX + c Y = , m = , X = , c =

4 Find m from the graph :

5 Find c, substitute X = 1, Y = 35, m = 5 into the equation Y = mX +

c

6 Find the variables a and b :

Enrichment Exercise1.The variables x and y are related by the equation

. Find

(i) the value of k and c.(ii) the value ofd x when y =2

2. The figure below shows part of a straight line graph drawn to represent the equation of . Find the value of a and b.

3. The figure below shows part of a straight line graph drawn to represent the equation of . Find the value of a and b.

4. The figure below shows part of a straight line graph drawn to represent the equation of . Find the value of a and b.

Linear law 8

5. The figure below shows part of a straight line graph drawn to represent the equation of . Find the value of a and b.

6. The diagram below represents a linear graph of log y against x. It is known that the variables x and y are related by the equation y a where a and b are constants. Find the values of a and b.

7. The diagram below represents a line of against

It is known that the variables x and y are related by the

equation where p and q are constants. Find the

values of p and q.

8. Diagram shows a straight line graph of against

Given that x and y are related by the equation

where p and q are constants, find the values of p and q.

9. Diagram shows a straight line graph of against

Given that x and y are related by the equation

10. Diagram shows a straight line graph of against

Given that x and y are related by the equation

Linear law 9

O

(4,6)

(1,3) x²

(0,2)

10log x

10log y(4,10)

2x

y

x

(0,2)

(6,5)

where a and b are constants, find the values of a and b. where a and b are constants, find the values of a and b.

Further practices

The graph shows against . It is known that and are

related by . Find the value of and .

The diagram shows part of the graph against .

Given that , find the values of and .

•

1. Diagram 2 shows a straight line graph of .

Given that , calculate the value of and of . [3

The variables x and y are related by the equation, where k is a constant.

(a) Convert the equation to linear form.(b) Diagram 3 shows the straight line obtained by plotting against . Find the value of (i) (ii) . [4

Linear law 10

• (2, )

• (, 1)

O

x

(2, h)

• (, 3)

O

2x

y

(-1,p)(q,2)

x

x

y

(6 ,p)

(q ,2)

2x

SPM Past Year SeriesSPM 2003 P 1 Q10x and y are related by the equation , where p and

q are constants. A straight line is obtained by plotting

against x, as shown in the diagram below.

Calculate the values of p and q. [4 marks ]

[p= -2, q =13]

SPM 2004 P1Q13

Diagram below shows a straight line graph of against x

Given that y= 6x-x2, calculate the value of k and h [3 marks]

[h=3, k=4]

SPM 2005 P 1 Q13The variables x and y are related by the equation y=kx4, where k is a constant.

(a) Convert the equation y=kx4 to linear form.(b) Diagram below shows the straight line obtained by

plotting log10y against log10x

Find the value of(i) log10 k(ii) h [4 marks]

[3, 11]

SPM 2006P1 Q11

The first diagram shows the curve y = 3x2+5. The second graph shows the straight line graph obtained when y = -3 x2 +5 is expresses in the linear form Y= 5X+c. Express X and Y in terms of a and/or y [3 marks]

X=9/25 x2, Y=-3/5 y

SPM 2007 P1 Q12 SPM2008 P1 Q12

The variables x and y are related by the equation y = ,

where k is a constant. Diagram below shows the straight line graph obtained by plotting log10y against x.

Linear law 11

(2,9)

(6,1)

x0

(2, k)

(h, 3)

x0

(2, h )

(0, 3)

10log x0

The variables x and y are related by the equation y2=2x(10-x).

A straight line graph is obtained by plotting against x

Find the value of p and of q [3 marks]

[10,14]

(a) Express the equation y = ,in its linear form used to

obtain the straight line graph shown in the diagram above.

(b) Find the value of k [4 marks]

[log10 y= -xlog105 +log10k, 0.01]

2.3 Obtain information from (i) lines of best fit (ii) equations of lines of best fit.

A. Using a graph paper 1. Identify the graph to be drawn 2. Change the non-linear function with variables x and y to a linear form Y = mX + c 3. Construct a table for X and Y 4. Plot the linear graph using the scale given by the question & label both axes 5. Draw line of best fit 6. Determine : gradient , m and Y-intercept , c from the graph to find the values oj constants in the non-linear equation.

Example: 1. Use graph paper to answer this question.

The table below records the values of an experiment for two variables x and y which are related by where p

and q are constants.x 0.8 1 1.3 1.4 1.5 1.7y 108.75 79 45.38 36.5 26.67 8.19

(a) Plot xy against x3 using scale 2 cm represents 1 unit in x-axis and 2 cm represents 10 units for y-axis. Hence, draw the line of best fit

[5marks ] (b) From the graph, estimate the value of

(i) p and q

(ii) x when y= [5marks] Answer:p=-16.67, q=95, x=1.458

2. SPM 2004 2 Q 7

Linear law 12

Use graph paper to answer this question. Table below shows the values of two variables, x and y, obatained from an experiment. Variables x and y are related by the equation y = p k x , where p and k are constants.

x 2 4 6 8 10 12y 3.16 5.50 5.50 16.22 28.84 46.77

(a) Plot log10 y against x by using a scale of 2 cm to 2 units on the x-axis and 2 cm to 0.2 unit on the y-axis. Hence, draw the line of best fit [4 marks ](b) Use your graph from (a) to find the value of

(i) p(ii) k [ 6 marks]

Answer :p =1.820, k =1.309

3. SPM 2005 P 2 Q 7

Use graph paper to answer this question. Table below shows the values of two variables, x and y, obtained from experiment. The variables x and y are

related by the equation , where p and r are constants.

x 1.0 2.0 3.0 4.0 5.0 5.5y 5.5 4.7 5.0 6.5 7.7 8.4

(a) Plot xy against x2, by using a scale of 2 cm to 5 units on both axes. Hence,draw the line of best fit. [5 marks](b) Use the graph from (a) to find the value of

(i) p(ii) r [5 marks] Answer :[ p=1.37, r=5.48]

3. SPM 2006 P2 Q7 Use graph paper to answer this question. Table below shows the values of two variables, x and y, obtained from an experiment. Variables x and y are related by the equation , where p and k are constants.

x 1 2 3 4 5 6y 4.0 5.7 8.7 13.2 20.0 28.8

(a) Plot log y against (x+1), using a scale of 2 cm to 1 unit on the (x+1) –axis and 2 cm to 0.2 unit on the log y-axis.Hence, draw the line of best fit. [5 marks]

(b) Use you graph from (a) to find the values of(i) p(ii) k [5 marks]Answers: 1.778, 1.483

4.SPM 2007 P2Q7 Use Graph paper to answer this question.Table below shows the values of two variables, x and y, obtained from an experiment.

Variables x and y are related by the equation , where p and k are constants.

x 2 3 4 5 6 7y 8 13.2 20 27.5 36.6 45.5

(a) Plot against x, using a scale of 2cm to 1 unit on both axes. Hence, draw the line of best fit.

[4marks](b) Use your graph in part (a) to find the value of

(i) p,(ii) k,(iii) y when x=1.2 [6marks]Answers : 0.754, 0.26, 4.2

5. SPM 2008 P2 Q8

Linear law 13

Table below shows the values of two variables, x and y , obtained from an experiment. Variables x and yare related by the equation y = hk2x, where h and k are constants.

x 1.5 3.0 4.5 6.0 7.5 9.0y 2.51 3.24 4.37 5.75 7.76 10.00

(a) based on the Table above, construct a table for the values of log10 y. [ 1 mark](b) Plot log10 y against x, using a scale of 2cm to 1 unit on the x-axis and 2 cm to 0.1 unit on the log10 y-axis Hence, draw the line of best fit. [4 marks](c) Use the graph in part (b) to find the value of (i) x when y = 4.8 (ii) h (iii) k [ 5 marks]Answers : 4.95, 1.905, 1.096

12 SPM 2009 P2 Q 8Use Graph paper to answer this question.Table below shows the values of two variables, x and y, obtained from an experiment.

The Variables x and y are related by the equation , where p and k are constants.

x 1.5 2.0 3.0 4.0 5.0 6.0y 2.502 0.770 0.465 0.385 0.351 0.328

a) Based on the Table above, construct a table for the values of and . [2 marks]

b) Plot against , using a scale of 2cm to 1 unit on both axes. Hence, draw the line of best fit. [5marks]

c) Use your graph in part (a) to find the value of(i) k,

(ii) p [5marks] Answers : { (c) (i) 0.256 (ii) – 1.331} Answer for 2.1

Non-linear equation Linear equation Y X m C1 y = a x2 + b x x a b

2 y = ax3 + bx2 x a b

3y =

y b a

4y =

xy x2 a b

5xy =

y b a

6 x +by = axy -b a

7 xy x2 -3 5

8 xy ab -a

9 y = abx log10 y x log 10 b log10 a

10 y = axb log10 y log10 x b log10 a

11 y =a2xb log10 y log10 x b 2log10 a

12 PV=a P a 0

Linear law 14