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

We have studied how to write the coordinates of a given point in the Cartesian plane. Do We know where the points (2, 0), (−3, 0), (4, 0) and (n, 0), for any real number n, lie in the Cartesian plane? Yes, they all lie on the x-axis. But do We know why? Because on the x-axis, the y-coordinate of each point is 0. In fact, every point on the x-axis is of the form (x, 0). Can We now guess the equation of the x-axis? It is given by y = 0. Note that y = 0 can be expressed as 0.x + 1.y = 0. Similarly, observe that the equation of the y-axis is given by x = 0.

Now, consider the equation x − 2 = 0. If this is treated as an equation in one variable x only, then it has the unique solution x = 2, which is a point on the number line. However, when treated as an equation in two variables, it can be expressed as x + 0y − 2 = 0. This has infinitely many solutions. In fact, they are all of the form (2, r), where r is any real number. Also, We can check that every point of the form (2, r) is a solution of this equation. So as, an equation in two variables, x − 2 = 0 is represented by the line AB in the graph in Figure 4.8.

Figure 4.8

Example 9

Solve the equation
2x + 1 = x − 3, and represent the solution(s) on
(i) the number line,
(ii) the Cartesian plane.

Solution :

We solve 2x + 1 = x − 3, to get
2x − x = −3 − 1
i.e., x = −4

(i) The representation of the solution on the number line is shown in Figure 4.9, where
x = − 4 is treated as an equation in one variable.

Figure 4.9

(ii) We know that x = − 4 can be written as x + 0y = − 4
which is a linear equation in the variables x and y. This is represented by a straight line. Now all the values of y are permissible because 0y is always 0. However, x must satisfy the equation x = − 4. Hence, two solutions of the given equation are x = − 4, y = 0 and x = − 4, y = 2.
Note that the graph AB is a line parallel to the y-axis and at a distance of 4 units to the left of it (see Figure 4.10).

Figure 4.10

Similarly, We can obtain a line parallel to the x-axis corresponding to equations of the type
y = 3, or 0x + 1y = 3

EXERCISE 4.4

Q1. Give the geometric representations of y = 3 as an equation
(i) in one variable
(ii) in two variables

A1.
(i)

(ii)

Q2. Give the geometric representations of 2x + 9 = 0 as an equation
(i) in one variable
(ii) in two variables

A2
(i)

(ii)