Question 881485
1,1,1,1,4
1,-2,-1,-1,3
2,-1,1,-1,2
1,-1,2,-2,-7


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

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

add  down (-1) *row 1 to row 4
1,1,1,1,4
0,-3,-2,-2,-1
0,-3,-1,-3,-6
0,-2,1,-3,-11

divide row 2 by -3
1,1,1,1,4
0,1,-2/-3,-2/-3,-1/-3
0,-3,-1,-3,-6
0,-2,1,-3,-11

add  down (3) *row 2 to row 3
1,1,1,1,4
0,1,2/3,2/3,1/3
0,0,1,-1,-5
0,-2,1,-3,-11

add  down (2) *row 2 to row 4
1,1,1,1,4
0,1,2/3,2/3,1/3
0,0,1,-1,-5
0,0,7/3,-5/3,-31/3


add  down (-7/3) *row 3 to row 4
1,1,1,1,4
0,1,2/3,2/3,1/3
0,0,1,-1,-5
0,0,0,6/9,12/9

divide row 4 by 2/3
1,1,1,1,4
0,1,2/3,2/3,1/3
0,0,1,-1,-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  (1) *row 4 to row 3
1,1,1,1,4
0,1,2/3,2/3,1/3
0,0,1,0,-3
0,0,0,1,2

add up  (-2/3) *row 4 to row 2
1,1,1,1,4
0,1,6/9,0,-1
0,0,1,0,-3
0,0,0,1,2

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

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

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

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

final
1,0,0,0,4
0,1,0,0,1
0,0,1,0,-3
0,0,0,1,2
"4","1","-3","2"
(4,1,-3,2)
x,y,z,w