Solve the linear system using Gaussian elimination method : {2x-3y-6z=-3 , x+2y-2z=1 3x+y+z=5}.

The most efficient method I know is a modification of Gauss:
The procedure is long so I will summarize:

r1 <-> r2
-2r1+r2 -> r2
-3r1+r3 -> r3
-5r2+r3 -> r3
yields:


Next using back substitution we arrive at

x=9/17
y=31/17
z=27/17

Here's the elimination part: