We establish Lyapunov-type inequalities for systems of second order quasilinear differential equations with Dirichlet boundary conditions. By incorporating the positive parts of the coefficient functions, we correct and extend several known inequalities for quasilinear systems. Our results cover constant and variable coefficient cases and lead to explicit lower bounds for generalized eigenvalues. These inequalities generalize classical scalar results and improve the existing estimates within the literature. As an application, we obtain explicit and computable lower bounds for generalized eigenvalues of coupled quasilinear operators.