Comprensión del conjunto solución de un sistema de ecuaciones lineales de dos incógnitas: un estudio de casos

: This article reports the Chilean high school and university students’ understanding of the Solution Set of a System of Linear Equations of Three Equations and Two Unknowns (CSSEL 3x2). The study is based on the theory of Modes of Thought, with a Case Study methodology and the use of Implicative Statistics. As a strategy, equivalent equations were used to investigate the influence of the concept of linear dependence on the path to the different ways of conceiv-ing its set of solutions. The use of the concept of slope and the idea of linear combination as mechanisms to transit between the synthetic-geometric and ana-lytical modes associated with the CSSEL3x2 stand out in the students’ responses. A disarticulation between the different ways of thinking the whole solution when parameters are incorporated to the coefficients of the equations was observed, which is particularly noticeable for students who had studied Linear Algebra. On the other hand, we show that the concept of linear dependence that is brought into play in the analysis is restricted to the case of scalar multiples.