Solve the system of equations. 
2x + y = 0 
x - 3y + z = 0 
3x + y - z = 0 


Since z is already eliminated from eq. 1, we
only need to eliminate z from eq. 2 and eq. 3

x - 3y + z = 0 
3x + y - z = 0
--------------
4x - 2y   = 0

This can be divided thru by 2 to make it
simpler:

2x - y = 0

So we put that together with eq 1 above and we
have this system of 2 equations and 2 unknowns:

2x - y = 0
2x + y = 0
------------
4x     = 0

 x = 0

substitute x = 0 in 

2x - y = 0

2(0) - y = 0
      -y = 0
       y = 0 

Substitute x = 0 and y = 0 in

x - 3y  + z = 0
0 -3(0) + z = 0
          z = 0 

Solution: (x,y,z) = (0,0,0)

Edwin