The principal result of this paper is, that the linear complementarity problem in \(\mathbb R^n\): \[ Mz+q\geq 0,\;z\geq 0,\;z^T(Mz+q)=0\tag{1} \] has a solution, which can be obtained by solving the linear program \[ \{p^Tz\mid Mz+q\geq 0,\;z\geq 0,\;p=r+M^Ts\}=\min! \tag{2} \] where \(M\) satisfies \(MZ_1=Z_2\), \(r^TZ_1+s^TZ_2>0\) and \(Z_1,Z_2\) are \((n,n)\)-matrices of Z-type (real square matrix with non-positive off-diagonal elements). Some earlier papers in this direction with smaller classes of matrices are reviewed, further hints are given, that a number of free boundary problems of fluid mechanics can be solved by solving a problem like (1) with \(M\) a Z-matrix. The paper concludes with two corollaries on finding the least element of a polyhedral set and solving certain quadratic programming problems (in connection with problems of type (1) and (2)).