Question 907712
1,1,1,-1,5
2,1,-1,1,6
1,-5,3,1,9
-1,-1,1,4,7


add  down (-2) *row 1 to row 2
1,1,1,-1,5
0,-1,-3,3,-4
1,-5,3,1,9
-1,-1,1,4,7

add  down (-1) *row 1 to row 3
1,1,1,-1,5
0,-1,-3,3,-4
0,-6,2,2,4
-1,-1,1,4,7

add  down (1) *row 1 to row 4
1,1,1,-1,5
0,-1,-3,3,-4
0,-6,2,2,4
0,0,2,3,12

divide row 2 by -1
1,1,1,-1,5
0,1,3,-3,4
0,-6,2,2,4
0,0,2,3,12

add  down (6) *row 2 to row 3
1,1,1,-1,5
0,1,3,-3,4
0,0,20,-16,28
0,0,2,3,12

add  down (0) *row 2 to row 4
1,1,1,-1,5
0,1,3,-3,4
0,0,20,-16,28
0,0,2,3,12

divide row 3 by 20
1,1,1,-1,5
0,1,3,-3,4
0,0,1,-16/20,28/20
0,0,2,3,12

add  down (-2) *row 3 to row 4
1,1,1,-1,5
0,1,3,-3,4
0,0,1,-4/5,7/5
0,0,0,23/5,46/5

divide row 4 by 23/5
1,1,1,-1,5
0,1,3,-3,4
0,0,1,-4/5,7/5
0,0,0,1,2

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

add up  (4/5) *row 4 to row 3
1,1,1,-1,5
0,1,3,-3,4
0,0,1,0,3
0,0,0,1,2

add up  (3) *row 4 to row 2
1,1,1,-1,5
0,1,3,0,10
0,0,1,0,3
0,0,0,1,2

add up  (1) *row 4 to row 1
1,1,1,0,7
0,1,3,0,10
0,0,1,0,3
0,0,0,1,2

add up  (-3) *row 3 to row 2
1,1,1,0,7
0,1,0,0,1
0,0,1,0,3
0,0,0,1,2

add up  (-1) *row 3 to row 1
1,1,0,0,4
0,1,0,0,1
0,0,1,0,3
0,0,0,1,2

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

final

1,0,0,0,3
0,1,0,0,1
0,0,1,0,3
0,0,0,1,2
"3","1","3","2"
(3,1,3,2)