If $c_1 = c_2 = 0$, the system $a_1 x + b_1 y = 0,\; a_2 x + b_2 y = 0$ is —

A homogeneous system with $c_1 = c_2 = 0$ always has at least the trivial solution $(0,0)$, since $x=y=0$ satisfies both equations. Hence it can never be inconsistent.