AbstractMathematical programming techniques are designed to optimize a selected criterion in the context of constraints. Modelling activity is often directed at an exploration of alternatives, where measurement of solution merit involves the consideration of several criteria, some of which may not be expressed or expressable in formal terms and therefore outwith the model.In a descriptive rather than formal manner, a presentation is given of two methods which allow a decision-maker or analyst to explore the solutions to underdetermined linear systems through the use of control variables. Such variables need not be controllable in reality, but do provide a powerful‘what if?’facility. The two methods stem from the application of (i) the generalized inverse and (ii) partitioning techniques to linear equations. Some of the underlying algebra is displayed, but the key features of the methods (and their common elements) are developed in the context of a small example.In a section on implementation the superiority of the partitioning techniques is established.