A linear programming-based method is presented for finding all solutions of nonlinear systems of equations with guaranteed accuracy. In this method, a new effective linear programming-based method is used to delete regions in which solutions do not exist. On the other hand, Krawczyk's method is used to find regions in which solutions exist. As an illustrative example, all solutions of the nonlinear system of equations describing equilibrium conditions of the "high polymer liquid system", which is a well-known ill-conditioned system of equations, are identified by the method.