Class 9 LINEAR EQUATIONS IN TWO VARIABLES FOR CBSE-NCERT

### Topic Covered

star Equations of Lines Parallel to the x-axis and y-axis

### Equations of Lines Parallel to the x-axis and y-axis

You have studied how to write the coordinates of a given point in the Cartesian plane.

Do you 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 you 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 you 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 + 0.y – 2 = 0. This has infinitely many solutions. In fact, they are all of the form (2, r), where r is any real number. Also, you 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 Fig. 4.8.

Q 3230367212

Solve the equation 2x + 1 = x – 3, and represent the solution(s) on
(i) the number line,
(ii) the Cartesian plane.
Class 9 Chapter 4 Example 9
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 Fig. 4.9, where
x = – 4 is treated as an equation in one variable.

(ii) We know that x = – 4 can be written as
x + 0.y = – 4

which is a linear equation in the variables x and y. This is represented by a line. Now all the values of y are permissible because 0.y 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 Fig. 4.10).

Similarly, you can obtain a line parallel to the x-axis corresponding to equations of the type

y = 3 or 0.x + 1.y = 3