When solving systems of equations by using matrices, many teachers present a Gauss-Jordan elimination approach to row reducing matrices that can involve painfully tedious operations with fractions (which I will call the traditional method). In this essay, I present an alternative method to row reduce matrices that does not introduce additional fractions until the very last steps. The students in my classes seemed to appreciate the efficiency and accuracy that the alternative method offered. Freed from unnecessary computational demands, students were instead able to spend more time focusing on designing an appropriate system of equations for a given problem and interpreting the results of their calculations. I found that these students made relatively few arithmetic mistakes as compared to students I tutored in the traditional method, and many of these students who saw both approaches preferred the alternative method.