Reordering method for solving linear system

Abstract

This paper introduces a method for solving systems of linear equations, applicable to both non-homogeneous and homogeneous scenarios. The method involves converting the system into homogeneous equations by incorporating the constant terms into the variables. We employ matrix analysis to classify the solutions, distinguishing between unique solutions, infinite solutions, and cases with no solutions. This approach is particularly effective for linear systems where the number of equations matches the number of unknowns, utilizing determinants as a key analytical tool. We provide illustrative examples for each scenario, along with a general solution for cases exhibiting infinite solutions.