x - 2y + 3z = 11
2x      -  z =  3
     3y +  z = -8

The 2nd equation is the simplest
and it already has y eliminated in it.
so let's eliminate y also from the 1st 
and 3rd, so we'll have two equations 
with y eliminated in them:

To eliminate y from the 1st and 3rd equations,

 x - 2y + 3z = 11
     3y +  z = -8

we multiply the top one by 3 and the bottom one by 2
and add them, and the y's will cancel out:

3x - 6y +  9z =  33
     6y +  2z = -16
-------------------
3x      + 11z =  17

So we put that with the original second equation:

3x + 11z =  17
2x -   z =   3

To eliminate z we multiply the bottom one
through by 11, and add them so the z's will
cancel out:

 3x + 11z =  17
22x - 11z =  33
---------------
25x       =  50
        x =   2

Substitute that in 

  2x - z =  3
2(2) - z =  3 
   4 - z =  3
      -z = -1
       z =  1  

Substitute that in

     3y +  z = -8
      3y + 1 = -8
          3y = -9
           y = -3

(x,y,z) = (2,-3,1)

Edwin