Short Notes: Linear Equation in two variables

Linear equation in two variables

An equation in the form of ax+by+c=0 is known as a linear equation in two variables.

The name is such because this equation exists in two variables, i.e, x and y. Hence, it contains two variables x and y along with some real numbers which are constant, i.e., a, b and c.

Note: a≠0 and b≠0, because if they become zero one of the variables either x or y will be eliminated from the                          equation.

Pair of linear equations in two variables

Here we consider a pair of linear equation which exist in two variables and then check their nature. Means, we will try to see their representation (on graph) and solution (algebraically).


a1x+b1y+c1=0   and   a2x+b2y+c2=0

This forms a pair of linear equation in two variables when neither of  a1 , b1 , c1 , a2 , b2 , c2 are zero.



The methods with which the pair of linear equations can be solved and representated includes:

  1. Graphical method
  2. Algebraic method


  1. Graphical method

  • Intersecting lines

  • Parallel lines

  • Coincident lines





Intersecting Unique Consistent
Parallel No solution Inconsistent
Coincident Infinitely many solutions Consistent

2. Algebraic method

Few methods to solve the pair of equations are:

  • Substitution method
  • Elimination method
  • Cross multiplication method

It may sometimes happen that the equations are not in linear form, in these cases it is possible to transform them into linear equations first then apply the above algebraic method.

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