The authors consider a nonlinear system of differential equations in the form \[ \begin{aligned} & x'(t) = \alpha _1 (t)x(t) + \beta _1 (t)\left| {u(t)} \right| ^{\gamma - 2}u(t), \\ & u'(t) = - \beta _2 (t)\left| {x(t)} \right| ^{\beta - 2}x(t) - \alpha _1 (t)u(t),\end{aligned} \tag{1} \] containing as special cases the well-known equations of Emden-Fowler-type and half-linear equations. They prove some new Lyapunov-type inequalities for the nonlinear system (1) and some special cases of (1). The inequalities obtained here can be used for the study of qualitative nature of solutions. They also give some applications to show the importance of their results.