2x + 3y - 4z = 7
3x + 4y - 2z = 9
5x + 7y - 6z = 20




Use Row 1 to get a 0 where the 3 is in Row 2:
Multiply Row 1 by -3 and the Row 2 by 2



Add Row 1 to Row 2



Restore Row 1 as it was



Use Row 1 to get a 0 where the 5 is in Row 3
Multiply Row 1 by -5 and Row 3 by 2



Add Row 1 to Row 3



Restore Row 1 as it was:



Use Row 2 to get a 0 where the -1 is in Row 3
Multiply Row 2 by -1 and keep Row 3 as it is



Add Row 2 to Row 3



Oh oh!  We can see there is no solution because
the bottom Row is an abbreviation for the equation:

0x + 0y + 0z = 8

There are no values of x,y, or z that we can substitute
for any of them that will make the left side equal to 8,
because the left side will always be 0 no matter what
numbers you substitute for the letters.  Therefore there
is no solution and this is called an "inconsistent" system.

Edwin