In everyday life, you will find many things that share a one-to-one relationship with each other, for example the quantity and cost of things, the age and the height, the altitude and the temperature. Such relationships are linear in nature and can be expressed mathematically as a pair of linear equations in two variables.

The general form of a pair of linear equations in two variables x and y  as
 
,Where  are all real numbers ,

We know that a linear equation in two variables when plotted on a graph defines a line. So, this means when a pair of linear equations is plotted, two lines are defined. Now, we know that two lines in a plane can intersect each other, be parallel to each other, or coincide with each other. The points where the two lines intersect are called the solutions of the pair of linear equations.

Condition 1: Intersecting Lines

If then the pair of linear equations
  has a unique solution.

Condition 2: Coincident Lines

If then the pair of linear equations
  has infinite solutions.
A pair of linear equations, which has a unique or infinite solutions are said to be a consistent pair of linear equations.

Condition 3: Parallel Lines

If then a pair of linear equations
  has no solution.

A pair of linear equations which has no solution is said to be an inconsistent pair of linear equations.