Chapter I : Linear Equations in two variables in two variables

4. Solution by cross Multiplication:

Let us consider a system of equations:

From equation (1) we have

Putting this value of y in eq (2) we get

Or, 

Similarly by eliminating x, we get

Here we see that (3) and (4) are linear equation in one variable only.

Provided .

The above results can be written as:

and

and combining these two we get

The above result can be written in a picture form as:

When , then we can not divide equation’s (3) and (4) by to fine the value of x and y. if

then

and

If , then equation (2) reduces to

ka₁x + kb₁y + kc₁ = 0

Or,

Or,

Which is equation (1). Here we see that every solution of equation (1) is a solution of equation (2) hence the system has infinite solutions.

If , equation on (2) reduces to

Or,

Or, k (-c₁) + c₂ = 0

Or, C₂ = KC₁

But this is not true. Therefore, no solution exists.

The system of equations

has exactly one or, unique solution if

i. e; if

has no solution if

and has infinite solution if