In this paper, a new algorithm is proposed to find the solutions of a general (square or nonsquare) Fully Fuzzy Linear Equation System (FFLS) with arbitrary trapezoidal fuzzy numbers, i.e. there are no sign restrictions on the variables or the parameters of the system. We introduce the “feasible (strong) fuzzy solution” and “approximate fuzzy solution” concepts, then accordingly “no solution” case of a general FFLS is defined. And a model is proposed by means of a mixed integer programming modelling of “min” and “max” concepts in the multiplication of two arbitrary trapezoidal fuzzy numbers. With the logic of goal programming, the objective function of this model is constructed within the deviation variables. Based on the proposed model, also an algorithm is presented to determine the nature of solutions of a general FFLS. The method is illustrated with some numerical examples. Our numerical results for the examples from the literature are analyzed within some distance functions.