Summary

A pair of linear equations in two variables can be represented, and solved using twothree methods.

(i) graphical method

(ii) algebraic method

In graphical method : the graph of a pair of linear equations in two variables is represented by twomultiple lines.

(i) If the lines intersect at a point, then that point gives the unique solution of the two equations. In this case, the pair of equations is .

(ii) If the lines coincide, then there are infinitely many solutions: each point on the line being a solution. In this case, the pair of equations is dependentindependent and also consistentinconsistent.

(iii) If the lines are parallel, then the pair of equations has noone solution.

In this case, the pair of equations is inconsistentconsistent.

In the algebraic method, the following methods are used for finding the solution(s) of a pair of linear equations:

(i) Substitution Method

(ii) Elimination Method

If a pair of linear equations is given by a1x + b1y + c1= 0 and a2 x + b2y + c2= 0, then the following situations can arise :

(i) a1a2≠ b1b2

In this case, the pair of linear equations is consistentinconsistent.

(ii) a1a2 = b1b2 ≠ c1c2

In this case, the pair of linear equations is inconsistentconsistent.

(iii) a1a2 = b1b2 = c1c2

In this case, the pair of linear equations is dependentindependent and consistentinconsistent.

There are several situations which can be mathematically represented by two equations that are not linear to start with. But we alter them so that they are reduced to a pair of linear equations.