In matrix form, the system can be written as follows and the solved using indicated row operations:
[(-3,5,1),(2,3,-1),(-4,2,3)][(x),(y),(z)] = [(10),(7),(-1)]
R_1 + R_2 to R_1
[(-1,8,0),(2,3,-1),(-4,2,3)][(x),(y),(z)] = [(17),(7),(-1)]
R_3 + 2R_3 to R_3
[(-1,8,0),(2,3,-1),(0,8,1)][(x),(y),(z)] = [(17),(7),(13)]
2R_1 + R_2 to R_2
[(-1,8,0),(0,19,-1),(0,8,1)][(x),(y),(z)] = [(17),(41),(13)]
R_2 - 2R_3 to R_2
[(-1,8,0),(0,3,-3),(0,8,1)][(x),(y),(z)] = [(17),(15),(13)]
R_2/3 to R_2
[(-1,8,0),(0,1,-1),(0,8,1)][(x),(y),(z)] = [(17),(5),(13)]
R_3 - 8R_2 to R_3
[(-1,8,0),(0,1,-1),(0,0,9)][(x),(y),(z)] = [(17),(5),(-27)]
R_3/9 to R_3
[(-1,8,0),(0,1,-1),(0,0,1)][(x),(y),(z)] = [(17),(5),(-3)]
-1R_1 to R_1
[(1,-8,0),(0,1,-1),(0,0,1)][(x),(y),(z)] = [(-17),(5),(-3)]
R_3 + R_2 to R_2
[(1,-8,0),(0,1,0),(0,0,1)][(x),(y),(z)] = [(-17),(2),(-3)]
8R_2 + R_1 to R_1
[(1,0,0),(0,1,0),(0,0,1)][(x),(y),(z)] = [(-1),(2),(-3)]