This is lengthy.......be careful......even one mistake is deadly!!!
2 1 -3 -19
1 2 1 -4
1 -1 5 21 multiply R1 by (-1/2) add to R2
2 1 -3 -19
0 3/2 5/2 11/2 multiply R2 by 2
1 -1 5 21
2 1 - 3 -19
0 3 5 11
1 -1 5 21 multiply R1 by (-1/2) add to R3
2 1 - 3 -19
0 3 5 11
0 -3/2 13/2 61/2 multiply R3 by 2
2 1 -3 -19
0 3 5 11
0 -3 13 61 multiply R2 by (-1/3) add to R1
2 0 -14/3 -68/3
0 3 5 11
0 -3 13 61 add R2 to R3 put result in R3.....multiply R1 by 3
6 0 -14 -68
0 3 5 11
0 0 18 72 divide R1 by 2 , R3 by 18
3 0 - 7 -34
0 3 5 11
0 0 1 4 multiply R3 by 7.....add to R1
3 0 0 -6
0 3 5 11
0 0 1 4 multiply R3 by -5....add to R2
3 0 0 -6
0 3 0 - 9
0 0 1 4 divide R1, R2 by 3
1 0 0 - 2
0 1 0 - 3
0 0 1 4
Read across and the solutions are (x, y , z} = {-2, -3, 4 }