The system
x-y=2
y-z=0
x+z=6
is the same as
1x-1y+0z=2
0x+1y-1z=0
1x+0y+1z=6
and that converts to this matrix
1 -1 0 2
0 1 -1 0
1 0 1 6
Now perform row reduction
1 -1 0 2
0 1 -1 0
1 0 1 6
1 -1 0 2
0 1 -1 0
0 1 1 4 R3 + (-1)*R1
1 0 -1 2 R1 + (1)*R2
0 1 -1 0
0 1 1 4
1 0 -1 2
0 1 -1 0
0 0 2 4 R3 + (-1)*R2
1 0 -1 2
0 1 -1 0
0 0 1 2 0.5*R3
1 0 0 4 R1 + (1)*R3
0 1 -1 0
0 0 1 2
1 0 0 4
0 1 0 2 R2 + (1)*R3
0 0 1 2
The right hand column has the values, 4, 2 and 2.
So x = 4, y = 2, and z = 2 are the three solutions
The solution as an ordered triple is therefore (4, 2, 2)