22 Jan 2010
LINEAR LAW
The line best fit is a straight line that has the following properties,
- The straight line is drawn in such a way that it passes through as many points
as possible
- The number of points that do not lie on the straight line drawn should be more
or less the same on both sides of the straight line.
The steps to find the values of constant is a non-linear function are as follows:
i. Reduce the non-linear function with variables x and y to the linear form
, where X and Y represent the functions of x and y or both.
ii. Prepare a table for the values of X and Y
iii. Choose a suitable scale such that the graph drawn is as large as possible and
label both axes.
iv. Plot the graph of Y against X and draw the line of best fit.
v. Construct a right-angled triangle, such that two vertices are on the line of best
fit, to calculate the gradient, m, of the straight line.
vi. Determine the Y- intercept, c, from the straight-line graph
A few tips to do the process of reducing non – linear to linear form.
- make sure to create one of the constant on the right hand side is free from x to
take a value of c (y-intercept).
- The coefficient of y must be 1.
1
Reduce each of the following equations to the linear form. Hence, state the gradient and
the y-intercept of each of the linear equation in terms of a and b, where a and b are
constants.
2
1.
x 1 2 3 4 5 6
y 30.0 15.1 5.0 3.8 1.9 0.9
The above table shows the experimental values of two variables, x and y. It is known that
x and y are related by the equation , where p and k are constant. One of the
experimental values of y is incorrectly recorded.
a) Draw the graph of against x, using a scale of 2cm to 1 unit on the
x-axis and 2 cm to 0.2 units on the .
b) From your graph, find
i. The value of y that is incorrectly recorded and its correct value
ii. The value of p and of k.
2.
x 1.0 1.5 2.0 2.5 3.0 3.5
y 0.81 2.72 5.76 9.92 15.21 21.62
The table shows the experimental values of two variables, x and y. It is known that x and
y are related by an equation in the form , where p and m re constants.
a) Draw the graph of against x, using a scale of 2 cm to 0.5 units on both the x-
axis.
b) From your graph, find
i. The value of p,
ii. The value of m,
iii. Thevalue of y when x = 1.9
3
3. Two variables x and y are known to be connected in the relation bx = ay(x+b), where
a and b are constants. The following table shows experimental values of x and y.
x 1.0 2.0 3.0 4.0 5.0
y 0.200 0.286 0.333 0.364 0.385
a) By using a scale of 4 cm to represent 1 unit on the x-axis and 2 cm to represent 1
unit on the axis, plot a graph of against x.
b) From your graph, find the values of a and b.
c) By drawing a straight line on the same axes as part (a), solve the equation
, giving your answer correct to 1 decimal place.
Hint: y_intercept = 3, gradient = 2
4. In a laboratory experiment, the mass y, in grammes, of the culture of a particular
fungus increases with time t hours in a relation , where a and c are
constants. The result of the experiment is tabulated in table below.
t 2 4 6 8
y 2.50 2.70 2.85 2.95
Express the relation in a linear form, and hence , plot the graph of y
against , using a scale of 2 cm to 0.10 hour on the -axis and 2
cm to 0.05 g on the y-axis.
Hence, find
a) the value of a and n
b) the initial mass of the fungus in the culture.
Hint: y_intercept = 1.828, gradient = 1.122 , Initial mass….t=0
4
5. The variable x and y are related by the equation where a and b are
constants. Table below shows the corresponding values of x and y obtained from an
experiment.
t 2 4 6 8 10
y 1.98 20.1 54.4 118.4 169.7
It is known that one of the value of y had been wrongly recorded.
a) By using a scale of 2 cm to 1 unit on the horizontal axis and 2 cm to 2 units on the
vertical axis, plot a graph of against x.
b) Use the graph in (a) to find
i) the values of a and b
ii) the correct value of y that has been wrongly recorded.
c) By drawing a suitable straight line on the same axes as part (a), find the roots of
the equation
Hint: y_intercept = -3.0, gradient = 2.0 ,
6. The 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 2 4 6 8 10 12
y 4.50 10.00 22.78 51.26 115.33 259.49
a) Plot against x by using a scale of 2 cm to 2 units on the x-axis and 2 cm
to 0.2 units on the -axis.
b) Use your graph in (a) to find the values of
i) p
ii) k.
Answer: p=2, k= 1.5.
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7. The 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 q are constants.
x 2.5 3.0 3.5 4.0 4.5 5.0
y 1.0 2.7 4.1 6.5 6.8 8.0
One of the values is incorrectly recorded.
a) Using a scale of 2 cm to 5 units on both axes, plot the graph of xy against .
Hence, draw the line of best fit.
b) Use your graph in(a) to answer the following questions:
i) State the value of y which is incorrectly recorded and determine its actual value.
ii) Find the value of p and of q.
Answer: y incorrect = 6.5, y actual=5.5, p=-20, q= 2.
8. The table below shows the values of two variables, x and y, that are related by the
equation , where m and n are constants. One of the values of y is incorrectly
recorded.
x 1 2 3 4 5
y 8.1 14.6 17.0 47.2 85.0
a) Plot the graph of against (x + 1) using a scale of 2 cm to 1 unit on the
(x + 1)-axis and 2 cm to 0.2 units on the -axis. Hence, draw the line of best
fit.
b) From your graph, find
i) the value of y that is incorrectly recorded and state the actual value of y.
ii) the value of m and of n.
Answer: y incorrect = 17.0, y actual=26.3, m=2.5,n=1.8.
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9. The values of the variables x and y are related by the equation , where m
and n are constants. The values of x and y are recorded in the following table.
x 0.25 0.5 1 2 4
y 0.14 0.22 0.40 0.67 1.00
One of the values of y in the table is incorrectly recorded.
a) Reduce the equation to the linear form.
b) Plot the graph of against using a scale of 2 cm to 0.5 units on the - axis
and 2 cm to 1 unit on the -axis. Hence, draw the line of best fit.
c) From your graph,
i) determine the value of y that is incorrectly recorded and state the actual value
of y.
ii) estimate the value of m and n.
Answer: y incorrect = 0.14, y actual=0.12, m=2.0,n=0.5
10. The graph below shows the graph of against . Express
a) against
b) y in term of x.
log10y
(4,4)
(0,2)
0 log10x
Answer: a) , b)
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