Question 891010
x-y+3z=13
4x+y+2z=17
3x+2y+2z=1 

1,-1,3,13
4,1,2,17
3,2,2,1

add  down (-4/1) *row 1 to row 2
1,-1,3,13
0,5,-10,-35
3,2,2,1

add  down (-3/1) *row 1 to row 3
1,-1,3,13
0,5,-10,-35
0,5,-7,-38

divide row 2 by 5/1
1,-1,3,13
0,1,-2,-7
0,5,-7,-38

add  down (-5/1) *row 2 to row 3
1,-1,3,13
0,1,-2,-7
0,0,3,-3

divide row 3 by 3/1
1,-1,3,13
0,1,-2,-7
0,0,1,-1

We now have the value for the last variable.
We will work our way up and get the other solutions.

add up  (2/1) *row 3 to row 2
1,-1,3,13
0,1,0,-9
0,0,1,-1

add up  (-3/1) *row 3 to row 1
1,-1,0,16
0,1,0,-9
0,0,1,-1

add up  (1/1) *row 2 to row 1
1,0,0,7
0,1,0,-9
0,0,1,-1

final
1,0,0,7
0,1,0,-9
0,0,1,-1


(7,-9,-1)