Students often instrumentally use variables and unknowns without considering the variational thinking behind them. Using parameters to modify the coefficients or unknowns in equations or systems of linear equations (without altering their structure) involves consciously incorporating variational thinking into problem-solving. We will test the scope of this approach to undergraduate students by using contextual problems modelled with systems of linear equations that have one solution, infinitely many solutions, or none. In this context, knowing the solution was not enough to decide; instead, modifications to the system were necessary, and incorporating parameters proved to be very useful for this purpose. The goal is for students, in addition to seeing variational thinking as a valuable strategy for determining the validity of a solution, to develop the ability to distinguish between unknowns, variables, and parameters.