FOR A QUESTION LIKE FIND THE NUMBER OF SOLUTIONS FOR THE EQUATIONS ax+by+cz=p  AND THE OPTIONS GIVEN AS

                    dx+ey+fz=q

                    lx+my+nz=r

1)no sol

2)unique sol

3)infinite sol

4)none

 

HOW DO U GO FOR THESE GENERALY.

ONE METHOD IS BY APPLYIN ELEMENTARY ROW OPERATIONS

THE OTHER IS BY FINDIN THE RANK OF GIVEN MATRIX N BY EQUATIN IT TOTHAT OF THE AGGUMENTED MATRIX                         RITE!

 

 

I CAN GIVE U A EASIER METHOD THIS IS NOT A SHORT CUT THIS IS AL CONCEPTUAL.

 

FIRST OF AL ELEMINATE 'Z'  THEN U GET TWO EQUQTIONS IN 'X' AND 'Y' SOME THING LIKE

 

 

a₁x+b₁y+c₁=0

a₂x+b₂y+c₂=0

1)IF a₁/a₂b₁/b₂ THEN UNIQUE SOL EXISTS  N DET NOT EQUAL TO ZERO.

2)IFa₁/a₂=b₁/b₂ not=c₁/c₂ THEN NO SOL N DET=O

3)IF a₁/a₂=b₁/b₂=c₁/c₂ THEN THERE EXISTS INFINITE SOL N DET=O

 

THIS IS IN CASE OF NON HOMOGENOUS IF THE EQUATIONS R HOMOGENOUS THEN

AGAIN FIRST ELIMINATE 'Z'

 

U GET EQUATIONS IN THE FORM OFa₁x+b₁y=0

                                                         a₂x+b₂y=0

THEN IF DET=0 THEN IT HAS INFINITE SOL

             DET NOT EQUAL TO ZERO THEN IT HAS UNIQUE SOL i.e TRIVIAL SOL(AL ZEROS)

 

 

SEE WAT I HAVE WRITEN IS BIG BUT NOT THE PROCEDURE

TO ELEMINATE 'Z' IT TAKES MAX 10 TO 20SECS  N TO EQUATE a₁/a₂=..... TAKES NOT MORE THAN 40 SECS  SO THATS IT UR PROBLEM WILL BE SOLVED WITHIN 1MIN.!!!!!!!!!!!!!!!!!!!!!!!