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).
Example:
a_{1}x+b_{1}y+c_{1}=0 and a_{2}x+b_{2}y+c_{2}=0
This forms a pair of linear equation in two variables when neither of a_{1 ,} b_{1 , }c_{1 }, a_{2 }, b_{2 , }c_{2 }are zero.
Solution
The methods with which the pair of linear equations can be solved and representated includes:
 Graphical method
 Algebraic method

Graphical method

Intersecting lines

Parallel lines

Coincident lines
LINES 
SOLUTION 
NATURE OF PAIR 
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.
Keep practicing…
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