First, let us consider this system of equations

x + y = 2,       (1)
y + z = 2,       (2)
x + z = 2        (3)

which is a sub-system of the given one.

This system (1)-(3) has a unique solution, and we can easily find it. 
Add all three equations 1, (2) and (3) (both sides). You will get

2x + 2y + 2z = 6,   or   

x + y + z = 3.   (4)

Now, distract (1) from (4). You will get z = 1.
     
     Distract (2) from (4). You will get x = 1.

     Distract (3) from (4). You will get y = 1.

So, (x,y,z) = (1,1,1) is the unique solution of the system (1), (2) and (3).

Now we have the requirement that these values should satisfy the equation

ax + by + cz = 0.

Substitute here x=1, y=1 and z=1, and you will get

a*1 + b*1 + c*1 = 0,   or

a + b + c = 0.

So, the condition 

a + b + c = 0

is the necessary and sufficient condition for the system

x+y=2, y+z=2, x+z=2, ax+by+cz=0 

to be consistent.