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)]#