The equations that appear in the formulation of finite-position synthesis problems tend to increase their complexity and become quite large when solving for multiple-joint systems. In this work, we create a new set of equations that are suited to find all solutions for very large synthesis problems, targeting interval branch-and-cut algorithms. We obtain a system of equations in which all the variables are naturally bounded. Here we present the formulation and preliminary results, while its application to interval analysis is still a work in progress.