2x+2y+2z=0
-2x+5y+2z=1
 8x+y+4z=-1

Get rid of the -2x by adding the 1st equation to the 2nd equation:

 2x+2y+2z=0
   +7y+4z=1
 8x+y+4z=-1

Get rid of the 8x by multiplying the 1st equation by -4 then 
adding it to the 3rd equation:

-8x-8y-8z=0
   +7y+4z=1
 8x+y+4z=-1

-8x-8y-8z=0
   +7y+4z=1
   -7y-4z=-1

Then divide the 1st equation through by -8

  x+ y+ z=0
   +7y+4z=1
   -7y-4z=-1

Get rid of the -7y by adding the 2nd equation to the 3rd equation

  x+ y+ z=0
   +7y+4z=1

The whole 3rd equation vanished when I did that.

Solve the second equation for y

  7y+4z=1
     7y=1-4z
      y=

Substitute in the first equation:

  x +  + z=0
  x + 
  x + 
  x = 
 
General solution:



Edwin