we can derive an equi-satisfiable system that only comprises inequalities, with a mapping (if not a bijection) from the original system to the inequality system, and vice versa.
The most common reformulation I'm aware uses a QR decomposition of the transpose of the equality constraint matrix, \(E^\top\) \cite{Goldfarb_1982}. Chubanov's method exploits a different (never explicitly presented?) reformulation, based on \(P_E\), the orthogonal projector in the null space of (the rows of) \(E\). The original (homogeneous) system of linear equalities and inequalities becomes