Solving a linear system of equations is not the same as solving a linear program. The article is about the former, not the latter. Linear programming is an optimization problem, subject to constraints that are described by inequalities. Linear systems of equations are finding values of relationships defined by equalities. There are lots of matrix multiplications involved in both, but other than that the solutions don't really resemble each other. For example, Linear Programming is not even known to be solvable in strongly polynomial time (simplex is exponential in the worst case, and the best method is weakly polynomial), while the algorithm in the paper improves from one strongly polynomial result ot another.