Get like letters, equal signs and constants lined up vertically:

(1)   2x +  y      = 6
(2)   -x - 2y + 8z = 2
(3)   -y + 4z      = 3

We start by observing that z is already eliminated in (1),
and that x is already eliminated from (3).

We pick one of those, say (3).  

(3)   -y + 4z      = 3

Since x is already eliminated from (3), we eliminate x from 
the other two, (1) and (2):

(1)   2x +  y      = 6
(2)   -x - 2y + 8z = 2

To do that we multiply (2) by 2 so that the x terms will cancel
when we add the equations term by term, getting (4):

(1)   2x +  y       =  6
     -2x - 4y + 16z =  4
------------------------
(4)       -3y + 16z = 10

Now we take (3) and (4) together as a system of 2 equations in
only 2 unknowns and line them up vertically:

(3)   -y + 4z  =  3
(4)  -3y + 16z = 10

To do that we multiply (3) by -3 so that the y terms will cancel
when we add the equations term by term, getting (5):

      3y - 12z = -9
(4)  -3y + 16z = 10
-------------------
            4z = 1
(5)          z = 1/4

Using (5), substitute 1/4 for z in (3) to find y:

(3)   -y + 4z  = 3
   -y + 4(1/4) = 3
        -y + 1 = 3
            -y = 2
(6)          y = -2

Using (6), substitute -2 for y in (1)

(1)   2x +  y  =  6
     2x + (-2) = 6
        2x - 2 = 6
            2x = 8
(7)          x = 4

From (7), (6), and (5), we have the solution 

(x,y,z) = (4,-2,1/4)

Edwin