The concept of systems of linear equations (SLEs) is fundamental and core in linear algebra, a subject, which has many applications in a number of disciplines. Gaussian elimination is a versatile method, which can be used to solve almost all types of SLEs by using row-reductions. This study focused on exploring undergraduate students’ conceptualizations of elementary row operations (EROs) as a means to solve SLEs. The purpose of this study was to explore undergraduate students’ conceptualizations of row reductions and their applications to the solutions of systems of equations. The perspectives of the action-process-object-schema theoretical framework were used in analyzing data and discussing the findings. To explore the students’ conceptualization of EROs, a descriptive research approach was followed. I considered a case study of 131 students registered for a mathematics for educators course, where linear algebra was one of the topics. The findings revealed that students attained the action conception of reducing a system with unique solutions but had challenges reducing and interpreting solutions to a system with non-unique solutions. The latter row-reduction implored process and object conceptions especially when variable elements in the augmented matrix were involved. As students find the learning of linear algebra difficult, this study contributes to the debate in literature on how to improve its teaching and make suggestions on the ways make more effective the learning of linear algebra.