Write down the equations as follows.
x + 2y + z = 2
x + z = 8
y - z = -5
Then you can create the matrix for this system
1 2 1 2
1 0 1 8
0 1 -1 -5
Subtract row 2 from row 1, then interchange these two rows.
1 0 1 8
0 2 0 -6
0 1 -1 -5
Divide row 2 by two, then subtract row 2 from row 3.
1 0 1 8
0 1 0 -3
0 0 -1 -2
Add row 3 to row 1 then multiply row 3 by -1.
1 0 0 6
0 1 0 -3
0 0 1 2
This matrix reads your solution.
x = 6
y = -3
z = 2
