. . Solution to a System of Linear equations The meaning of solution to a system of linear equations can be understood by considering the following cases : Case (i) Consider the system of linear equations x = , ... ( ) + = .
... ( ) These two equations represent a pair of straight lines in two dimensional analytical geometry (see the Fig. . ).
Using ( ), we get x = + y . ... ( ) Substituting ( ) in ( ) and simplifying, we get y = . Substituting y = in ( ) and simplifying, we get x = .
Both equations ( ) and ( ) are satisfied by x = and y = . That is, a solution of ( ) is also a solution of ( ) . So, we say that the system is consistent and has unique solution ( , ) . The point ( , ) is the point of intersection of the two lines and x .
Fig. . O ( , ) ( , ) ( , ) ( , ) x' y' ( , − - - Case (ii) Consider the system of linear equations = , ... ( ) = ...
( ) Using equation ( ), we get x = − y ... ( ) Substituting ( ) in ( ) and simplifying, we get0 = . This informs us that equation ( ) is an elementary transformation of equation ( ). In fact, by dividing equation ( ) by , we get equation ( ).
It is not possible to find uniquely x and y with just a single equation ( ). So we are forced to assume the value of one unknown, say y = , where t is any real number. Then, x . The two equations ( ) and ( ) represent one and only one straight line (coincident lines) in two dimensional analytical geometry (see Fig.
. ) . In other words, the system is consistent (a solution of ( ) is also a solution of ( )) and has infinitely