# Pair of Linear Equations in Two Variables Class 10 Notes Maths: Chapter 3

## Pair Of Linearequations Intwo Variablesa

- A pair of linear equations in two variables x and y, can be algebraically represented as : a
_{1}x + b_{1}y + c_{1}= 0 and a_{2}x + b_{2}y + c_{2}= 0.Here a_{1}, b_{1}, c_{1}, a_{2}, b_{2}and c_{2}are real numbers and a_{1}2 + b_{1}2 ≠ 0 and a_{2}2 + b22 ≠ 0. - There are two methods of solving a pair of linear equations :

(i) Graphical method, and

(ii) Algebraic method. - A pair of linear equations a
_{1}x + b_{1}y + c_{1}= 0 and a_{2}x + b_{2}y + c_{2}= 0 graphically or geometrically represents a pair of straight lines which are :

(i) intersecting, if a1/a_{2}≠b_{1}/b_{2}

(ii) parallel, if a_{1}/a_{2}≠b1/b_{2}≠c_{1}/c_{2}

(iii) coincident, if a_{1}/a_{2}=b_{1}/b_{2}=c_{1}/c_{2} - While solving a pair of linear equations by the graphical method, we have to first draw the lines represented by them on the graph.

(i) If the resultant line intersect, then the pair is said to be consistent and the coordinates of the point provide an unique solution.

(ii) If the lines are parallel, then they do not have any solutions and are called inconsistent.

(iii) If the lines are coincident, then there are infinite numbers of solutions and they are called consistent but having infinite solutions. - A pair of linear equations can be solved by the algebraic method in three ways :

(i) By substitution method,

(ii) By elimination method - For a pair of linear equations in two variables

a_{1}x + b_{1}y + c_{1}= 0 and a_{2}x + b_{2}y + c_{2}= 0

(i) if,a_{1}/a_{2}≠b_{1}/b_{2}then the pair of linear equations represents a consistent and unique solution;

(ii)if, a_{1}/a_{2}=b_{1}/b_{2}≠c_{1}/c_{2}then the pair of linear equations represents an inconsistent solution; and

(iii) if,then the pair of linear equations represents consistent but infinite Solution