Question 902029
1,-1,0,1
0,1,-1,4
1,0,1,-1


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

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

divide row 2 by 1
1,-1,0,1
0,1,-1,4
0,1,1,-2

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

divide row 3 by 2
1,-1,0,1
0,1,-1,4
0,0,1,-3

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

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

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

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

final

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