This paper joins discussions on normalized regression and decomposition equations in devising a simple and general algorithm for obtaining the normalized regression and applying it to the Oaxaca decomposition. This resolves the invariance problem in the detailed Oaxaca decomposition. An algorithm to calculate an asymptotic covariance matrix for estimates in the normalized regression for hypothesis testing is also derived. We extend these algorithms to non-linear equations where the underlying equation is linear and decompose differences in the first moment.