many solutions, as t can assume any real value. Case (iii) Consider the system of linear equations x = , ... ( ) = . ...
( ) Using equation ( ), we get x = − y ... ( ) Substituting ( ) in ( ) and simplifying, we get = . This is a contradicting result, which informs us that equation ( ) is inconsistent with equation ( ). So, a solution of ( ) is not a solution of ( ).
In other words, the system is inconsistent and has no solution. We note that the two equations represent two parallel straight lines (non-coincident) in two dimensional analytical geometry (see Fig. . ).
We know that two non-coincident parallel lines never meet in real points. Note ( ) Interchanging any two equations of a system of linear equations does not alter the solution of the system. ( ) Replacing an equation of a system of linear equations by a non-zero constant multiple of itself does not alter the solution of the system. ( ) Replacing an equation of a system of linear equations by addition of itself with a non-zero multiple of any other equation of the system does not alter the solution of the system.
– – – O – – x' y' ( , ) ( , ) ( , ( , ) - - Applications of Matrices and Determinants Definition . A system of linear equations having at least one solution is said to be consistent . A system of linear equations having no solution is said to be inconsistent . Remark If the number of the equations of a system of linear equations is equal to the number of unknowns of the system, then the coefficient matrix A of the system is a square matrix.
Further, if A is a non-singular matrix, then the solution of system of equations can be found by any one of the following methods : (i) matrix inversion method, (ii)