In many areas, like geometric reasoning, computer modeling, implicitization, computer vision, robotics and kinematics, polynomial equations are used to model various behaviors and/or configurations. Frequently, it is necessary to derive direct dependence between parameters of the problem. Variable elimination helps to derive these dependencies and get direct correspondence between the parameters of the problem. In general, we are seeking to derive algebraic conditions on the parameters of a polynomial system so that solutions to the system exists. For some time now the Dixon resultant formulation has proved to be the most efficient tool to compute such a condition, called resultant. This work builds on the success of the Dixon resultant formulation to develop new method as well as provide number of insights, so that the Dixon-based methods can be applied more readily. The proposed method is able to better adapt to input polynomial systems with ad hoc structures with various optimization techniques. Besides the practical significance, the new construction bridges the gap between the Dixon-Bezout matrix formulations and Sylvester-type constructions.