There are various ways to solve this. Here is an algebraic way.
(1) 4x+3y+2z=1
(2) 5x-2y-z=-1
(3) x-4y+3z=-7
(4)=(1)+2(2): 14x-y=-1, y=14x+1
(5)=(3)+3(2): 16x-10y=-10 simplifies to 8x-5y=-5=8x-5(14x+1)=-5 after substituting for y.
8x-70x-5=-5, -62x=0, x=0, so y=14x+1=1. Substitute for x and y in any of the original equations:
(1) 0+3+2z=1, 2z=-2, z=-1.
CHECK
(1) 3-2=1 OK
(2) 0-2+1=-1 OK
(3) 0-4-3=-7 OK
SOLUTION: x=0, y=1, z=-1.