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

When the coefficient ratios are equal but differ from the constant ratio, $\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} \ne \dfrac{c_1}{c_2}$, the lines are parallel and never meet, giving no solution. Such a system is inconsistent.