EXERCISE 2.5
1. A learning platform charges a fixed monthly fee and an
additional cost per digital learning module accessed. A student
observes that when she accessed 10 modules, her bill was `400.
When she accessed 14 modules, her bill was `500. If the monthly
bill y depends on the number of modules accessed, x, according to
the relation y = ax + b, find the values of a and b.
2. A gym charges a fixed monthly fee and an additional cost per hour
for using the badminton court. A student using the gym observed
that when she used the badminton court for 10 hours, her bill
was `800. When she used it for 15 hours, her bill was `1100. If the
monthly bill y depends on the hours of the use of the badminton
court, x, according to the relation y = ax + b, find the values of a and b.
3. Consider the relationship between temperature measured in
degrees Celsius (°C) and degrees Fahrenheit (°F), which is given by
°C = a °F + b. Find a and b, given that ice melts at 0 degrees Celsius
and 32 degrees Fahrenheit, and water boils at 100 degrees Celsius
and 212 degrees Fahrenheit.
(Hint: When °C = 0, °F = 32 and when °C = 100, °F = 212. Use this
information to find a and b, and thus, the linear relationship
between °C and °F.)
2.6 Visualising linear relationships
As we have seen in many examples in this chapter, a linear pattern or
relationship can be expressed in the form of an equation y = ax + b.
Now we shall learn to plot such an equation as a straight line.
To plot a linear equation, we need to identify any two points on the
line. For example, to plot y = 2x + 1, we may identify two points as follows:
When x = 0, y = 1, so (0, 1) is a point on the line. We can call this point
A. We say that the x-coordinate of A is 0 and the y-coordinate is 1. Also,
when x = 3, y = 7. Thus, B (3, 7) is another point on the line. We plot these
points on the coordinate plane, join them and extend the line in both
directions as
Think and Reflect
Identify other points on the line by completing the following table.
x 1 2 5 7 9 12 20
y 3 15
Note that (1, 3) and (7, 15) are also the points on the line y = 2x + 1. Plot
the points on the line. You may use a graph paper for this task. Observe
that if a point lies on a line, its coordinates must satisfy the equation
of the line. Thus, we can verify that (7, 15) lies on the line y = 2x + 1 by
substituting x = 7 and y = 15 in the equation.
Example 12: Let us plot the points (–1, –3), (0, 0), (1, 3), (3, 9), (4, 12) in
the coordinate plane on a graph paper as shown in Fig. 2.6. Join the points
(–1, –3) and (4, 12) using a ruler. Doing so, observe that all five points lie
on a straight line. Can you guess the equation of this line by looking at the
relationship between the x and y coordinates of each point?
For each point, the value of the y-coordinate is three times that of
the x-coordinate. We can therefore say that y = 3x.
Example 13: Let us plot the points (– 3, 6), (– 2, 4), (0, 0), (1, – 2), (2, – 4),
(3, – 6) in the coordinate plane on a graph paper as shown in . Join
the points (– 3, 6) and (3, – 6) using a ruler. Doing so, observe that all five
points lie on a straight line. Can you guess the equation of this line by
looking at the relationship between the x and y coordinates of each point?
Example 14: Draw the graphs of y =
1
2 x, y = x, y = 2x by selecting
suitable points on these lines.
(Hint: In order to graph y =
1
2 x, we could take the points (0, 0)
and (4, 2). Can you verify that these lie on the line?)
shows all the three graphs on the same axes. Does this help
you to conclude anything about the linear equation y = ax, a > 0 as a
varies? What happens when a > 1 and when a < 1?
(Hint: You may also plot the equations y = 3x and y =
1
3 x on the
same axes.)
First of all, we observe that the straight lines representing an
equation of the form y = ax, always pass through the origin (0, 0). Further,
when a > 1, the line is steeper than the line y = x, which is equally
inclined to both axes. However, when a < 1, the line is less steep than
the line y = x. In fact, a is referred to as the slope of the line y = ax. We
will learn more about the concept of the slope of a line in the chapter
on linear equations.
For now, we focus on another important observation. A linear
growth is represented by a straight line with positive slope while a
linear decay is represented by a straight line with negative slope.
Note that the number of square tiles in the pattern in Fig. 2.4 leads to
the sequence 1, 3, 5, 7, …. in which the difference of consecutive terms
is the constant 2. The slope of the line representing the linear
relationship between the term number and the number of square tiles is
y = 2x – 1 is 2. Hence, the slope represents the constant difference
between consecutive terms of the sequence.
Example 15: Now let us draw the graphs of y =
–1
3 x, y = – x, y = – 3x
by selecting suitable points on these lines. Fig. 2.10 shows the graphs of
these linear equations without any points labelled on them.
shows all the three graphs on the same axes. Does this help
you to conclude anything about the linear equation y = – ax, a > 0, as a
varies? What will happen when a > 1 and when a < 1?
Think and Reflect
Differentiate between the graphs of the equations y = 3x + 1, and y = –3x + 1.
Example 16: Let us now draw the graphs of y = 2x – 1, y = 2x + 1,
y = 2x + 5, first individually (as shown in Fig. 2.12) and then on the same
axes (as shown in
Think and Reflect
Does this help you to conclude anything about the linear equation
y = ax + b when a is fixed but b varies?
(Hint: In these equations a = 2, and b takes the values –1, 1 and 5,
respectively.)
Now let us draw the graphs of the equations y = x + 3, y = 2x + 5 and
y = 3x – 2. See and observe where these lines cut the y-axis.
We see that,
y = 2x + 5 cuts the y-axis at A (0, 5),
y = x + 3 cuts the y-axis at B (0, 3), and
y = 3x – 2 cuts the y-axis at C (0, –2).
We observe that any straight line written in the form y = ax + b cuts
the y-axis at the point (0, b). The length b is referred to as the y-intercept
of the line. Thus, the y-intercept of the line y = x + 3 is 3. This means that
the line cuts the y-axis at a distance of 3 units from the origin in the
positive direction. Similarly, the y-intercept of the line y = 3x – 2 is – 2.
This means that the line cuts the y-axis at a distance of 2 units from the
origin in the negative direction.
After observing the graphs in Figures 2.8 to 2.14, we may conclude
the following:
(i) In the equation y = ax + b, a represents the slope of the line
and b represents the y-intercept.
(ii) If we change the values of a, keeping b fixed, the slope of the
line changes while the y-intercept remains fixed.
(iii) If we change the values of b keeping the value of a fixed, the
lines shift but remain parallel to the original line. Thus, lines
with equal slopes but different y-intercepts are parallel to
each other.