For a quasi-linear hyperbolic system, the method of vanishing viscosity is used to construct solutions with strong discontinuity (shock). The solution consists of two regular regions separated by a free boundary (shock). A system of differential equations that governs the free boundary and its boundary values is derived by Melnikov's method. If the system is a conservation law, the differential equation is the well know Rankine-Hugoniot condition. If the system is non-conservation, the differential equation is in a form of Melnikov type integral that generalizes the Rankin-Hugoniot condition. Solutions in the regular regions are then obtained by the method of characteristics.