You must have come across situations like the one given below :

Akhila went to a fair in her village. She wanted to enjoy rides on the Giant Wheel
and play Hoopla (a game in which you throw a ring on the items kept in a stall, and if
the ring covers any object completely, you get it).

Recall, from Class IX, that the following are examples of linear equations in two
variables:

You also know that an equation which can be put in the form ax + by + c = 0,
where a, b and c are real numbers, and a and b are not both zero, is called a linear
equation in two variables x and y.

In the previous section, you have seen how we can graphically represent a pair of
linear equations as two lines.

You have also seen that the lines may intersect, or may
be parallel, or may coincide.

Can we solve them in each case? And if so, how?

We
shall try and answer these questions from the geometrical point of view in this section.

In the previous section, we discussed how to solve a pair of linear equations graphically.
The graphical method is not convenient in cases when the point representing the
solution of the linear equations has non-integral coordinates like.

There is every possibility of making mistakes while reading such coordinates.

Is there any alternative method of finding the solution?

There are several algebraic methods, which we shall now discuss.

We shall explain the method of substitution by taking some examples.

Now let us consider another method of eliminating (i.e., removing) one variable.

This
is sometimes more convenient than the substitution method.

Let us see how this method
works.

So far, you have learnt how to solve a pair of linear equations in two variables by
graphical, substitution and elimination methods.

Here, we introduce one more algebraic
method to solve a pair of linear equations which for many reasons is a very useful
method of solving these equations.

Before we proceed further, let us consider the
following situation.

In this section, we shall discuss the solution of such pairs of equations which are not
linear but can be reduced to linear form by making some suitable substitutions.

We
now explain this process through some examples.

In this chapter, you have studied the following points:

1. Two linear equations in the same two variables are called a pair of linear equations in two
variables.

2. A pair of linear equations in two variables can be represented, and solved, by the: