The system $a_1 x + b_1 y = c_1,\; a_2 x + b_2 y = c_2$ is consistent and mutually dependent when —

When all three ratios are equal, $\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} = \dfrac{c_1}{c_2}$, the two equations represent the same line, so the system is consistent with infinitely many common solutions — i.e. mutually dependent.