NCERT Class 9 Mathematics Textbook for Blind Students made Screen Readable by Professor T K Bansal.

So far, We have obtained the solutions of a linear equation in two variables algebraically. Now, let us look at their geometric representation. We know that each such equation has infinitely many solutions. How can we show them in the coordinate plane? We may have some indication in which we write the solution as pairs of values. The solutions of the linear equation in Example 3, namely,
x + 2y = 6 equation (1)
can be expressed in the form of a table as follows by writing the values of y by the side of the corresponding values of x :

Table 1

SERIAL NO.	X =	Y =
1	0	3
2	2	2
3	4	1
4	6	0
5	-	-

In the previous chapter, We studied how to plot the points on a graph paper. Let us now plot the points (0, 3), (2, 2), (4, 1) and (6, 0) on a graph paper. Let us now join any two of these points and obtain a line. Let us call this as line AB (see Figure 4.2).

Figure 4.2

 

Do you see that the other two points also lie on the line AB? Now, let us pick another point on this line, say (8, −1). Is this a solution?
In fact, 8 + 2(−1) = 6.
So, (8, −1) is a solution of the equation. Pick any other point on this line AB and verify whether its coordinates satisfy the equation or not.

Now, take any point not lying on the line AB, say (2, 0). Do its coordinates satisfy the equation? Check, and see that they do not.

Let us list our observations:
1. Every point whose coordinates satisfy Equation (1) lies on the line AB.
2. Every point (a, b) that lies on the line AB gives a solution of the equation x + 2y = 6.
3. Any point, which does not lie on the line AB, is not a solution of Equation (1).

So, We can conclude that every point on the Line AB satisfies the equation of the line and every solution of the equation is a point on the line AB. In fact, a linear equation in two variables is represented geometrically by a line whose points make up the collection of solutions of the equation. This is called the graph of the linear equation. So, to obtain the graph of a linear equation in two variables, it is enough to plot two points corresponding to two solutions and join them by a line. However, it is advisable to plot more than two such points so that We can immediately check the correctness of the graph.

Remark :
The reason that a, degree one, polynomial equation a x + b y + c = 0 is called a linear equation is that its geometrical representation is a straight line.

Example 5

Given the point (1, 2), find the equation of a line on which it lies. How many such equations are there?

Solution :

Here (1, 2) is a solution of a linear equation We are looking for. So, We are looking for any line passing through the point (1, 2). One example of such a linear equation is
x + y = 3.
Others are y − x = 1, y = 2x, since they are also satisfied by the coordinates of the point (1, 2). In fact, there are infinitely many linear equations in two variables which are satisfied by the coordinates of the point (1, 2). Can We see this pictorially?

Example 6

Draw the graph of x + y = 7.

Solution :

To draw the graph, we need at least two solutions of the equation. We can check that x = 0, y = 7, and x = 7, y = 0 are solutions of the given equation. So, We can use the following table to draw the graph:

Table 2

 

X =	Y=
0	7
7	0

Draw the graph by plotting the two points from Table 2 and then by joining the two points by a line (see Figure 4.3).

Figure 4.3

 

Example 7

We know that the force applied on a body is directly proportional to the acceleration produced in the body. Write an equation to express this situation and plot the graph of the equation.

Solution :

Here the variables involved are force and acceleration. Let the force applied be y units and the acceleration produced be x units. From ratio and proportion, We can express this fact as
y = kx,
where k is a constant.
(From our study of science, We know that k is actually the mass of the body.)

Now, since we do not know what k is, we can not draw the precise graph of y = kx. However, if we give a certain value to k, then we can draw the graph. Let us take k = 3, i.e., we draw the line represented by y = 3x.

For this we find two of its solutions, say (0, 0) and (2, 6) (see Figure 4.4).

Figure 4.4

 

From the graph, We can see that when the force applied is 3 units, the acceleration produced is 1 unit. Also, note that (0, 0) lies on the graph which means the acceleration produced is 0 units, when the force applied is 0 units.

Remark :
The graph of the equation of the form y = kx is a line which always passes through the origin.

Example 8

For each of the graphs given in Figure 4.5 select the equation whose graph it is from the choices given below:

Figure 4.5 (i)

 

(a) For Figure 4.5 (i),
(i) x + y = 0
(ii) y = 2x
(iii) y = x
(iv) y = 2x + 1

Figure 4.5 (ii)

 

(b) For Figure 4.5 (ii),
(i) x + y = 0
(ii) y = 2x
(iii) y = 2x + 4
(iv) y = x − 4

Figure 4.5 (iii)

 

(c) For Figure 4.5 (iii),
(i) x + y = 0
(ii) y = 2x
(iii) y = 2x + 1 (
iv) y = 2x − 4

Solution :

(a) In Figure 4.5 (i), the points on the line are (−1, −2), (0, 0), (1, 2). By inspection, y = 2x is the equation corresponding to this graph. We can find that the y-coordinate in each case is double that of the x-coordinate.

(b) In Figure 4.5 (ii), the points on the line are (−2, 0), (0, 4), (1, 6). We know that the coordinates of the points of the graph (line) satisfy the equation y = 2x + 4. So,y = 2x + 4 is the equation corresponding to the graph in Figure 4.5 (ii).

(c) In Figure 4.5 (iii), the points on the line are (−1, −6), (0, −4), (1, −2), (2, 0). By inspection, We can see that y = 2x − 4 is the equation corresponding to the given graph (line).

EXERCISE 4.3

Q1. Draw the graph of each of the following linear equations in two variables:
(i) x + y = 4
(ii) x − y = 2
(iii) y = 3x
(iv) 3 = 2x + y

A1.
(i)

(ii)

(iii)

 

(iv)

Q2. Give the equations of two lines passing through (2, 14). How many more such lines are there, and why?

A2. 7x − y = 0 and x + y = 16; infintely many.
[Through a point infinitely many lines can be drawn]

Q3. If the point (3, 4) lies on the graph of the equation 3y = ax + 7, find the value of a.

A3. 5/3

Q4. The taxi fare in a city is as follows:
For the first kilometer, the fare is ₹ 8 and for the subsequent distance it is ₹ 5 per km. Taking the distance covered as x km and total fare as ₹ y, write a linear equation for this information, and draw its graph.

A4. 5x − y + 3 = 0

Q5. From the choices given below, choose the equation whose graphs are given in Figure 4.6 and Figure 4.7.

Figure 4.6

 

For Figure 4.6
(i) y = x
(ii) x + y = 0
(iii) y = 2x
(iv) 2 + 3y = 7x

Figure 4.7

 

For Figure 4.7
(i) y = x + 2 (ii) y = x − 2 (iii) y = −x + 2 (iv) x + 2y = 6

A5. For Figure 4.6, x + y = 0 and for Figure 4.7, y = −x + 2.

Q6. If the work done by a body on application of a constant force is directly proportional to the distance travelled by the body, express this in the form of an equation in two variables and draw the graph of the same by taking the constant force as 5 units. Also read from the graph the work done when the distance travelled by the body is (i) 2 units
(ii) 0 unit

A6. Supposing x is the distance and y is the work done. Therefore according to the problem the equation will be y = 5x.
(i) 10 units
(ii) 0 unit

 

Q7. Yamini and Fatima, two students of Class 9 of a school, together contributed Rs 100 towards the Prime Minister’s Relief Fund to help the earthquake victims. Write a linear equation which satisfies this data.
(We may take their contributions as ₹ x and ₹ y.) Draw the graph of the same.

A7. x + y = 100

 

Q8. In countries like USA and Canada, temperature is measured in Fahrenheit, whereas in countries like India, it is measured in Celsius. Here is a linear equation that converts Fahrenheit to Celsius: F = (9/5) C + 32
(i) Draw the graph of the linear equation above using Celsius for x-axis and Fahrenheit for y-axis.
(ii) If the temperature is 30°C, what is the temperature in Fahrenheit?
(iii) If the temperature is 95°F, what is the temperature in Celsius?
(iv) If the temperature is 0°C, what is the temperature in Fahrenheit and if the temperature is 0°F, what is the temperature in Celsius?
(v) Is there a temperature which is numerically the same in both Fahrenheit and Celsius? If yes, find it.

A8.
(i) See the figure below.

(ii) 86° F
(iii) 35° C
(iv) 32° F, −17.8° C (approximately)
(v) Yes, − 40° (both in F and C)