Solving systems of linear equations is a core concept in linear algebra and a wide variety of problems found in the sciences and engineering can be formulated as linear equations. This study sought to explore undergraduate students' development of the schema for solving systems of linear equations. The triad framework was used to describe the schema development in general and the system of linear equations was used as an example. A case study of an undergraduate class doing a linear algebra course in 2020 was considered in this study, where fifteen students participated in the study whereby they responded to a task with three question on solving a system of linear equations. The findings revealed that albeit minor manipulation errors, students were able to solve given systems of linear equations using the Cramer's rule and Gaussian elimination. However, students could not adequately go beyond the algorithmic computations to attain appropriate mathematical reasoning and establish underlying relations required to solve systems of linear equations, as no-one attained the trans-stage of conceptualization. It follows from this study that the identifications of the challenges that students encounter when solving systems of linear equations empowers course instructors on how to overcome the challenges.