The most straightforward way to check for consistency is to attempt to solve them. I will use basic algebra.
Label the equations:
A: x+2y+2z=2,
B: 3x-2y-z=5,
C: 2x-5y+3z=-4,
D: x+4y+6z=0.
From D, x=-4y-6z, so substitute for x in A and B (we can choose any two of A, B or C, so I chose A and B):
E: -4y-6z+2y+2z=-2y-4z=2,
F: -12y-18z-2y-z=5=-14y-19z=5.
Now we can solve for y and z.
-7E=14y+28z=-14.
-7E+F=28z-19z=-9, 9z=-9, z=-1.
So from E, -2y+4=2, 2y=2, y=1.
And from any of the equations A-D (for example, D):
x=-4+6=2.
So we have (x,y,z)=(2,1,-1) to check for consistency:
A: 2+2-2=2 OK
B: 6-2+1=5 OK
C: 4-5-3=-4 OK
D: 2+4-6=0 OK
So we have consistency x=2, y=1, z=-1.