Linear system/Solution methods/Adaptivity/Remark

The methods described in fact, remark, and remark to solve a linear system differ with respect to speed, strategic conception, complexity of the coefficients, error-proneness. In the elimination method, the systematic reduction of the number of variables (reduction of dimension) is obvious, and it is unlikely to make mistakes (except for miscalculations). It is always clear how to continue. However, these advantages emerge starting with three variables. For two variables, it does not make a difference what method we choose.

The evaluation of the methods depends also on the features of the concrete system. Such features should be taken into account in order to find "short-cuts“ to the solution. The adequate choice of the solution method appropriate for the given problem is called adaptivity (a concept which is used in the didactic context with different meanings). If, for example, one row of the system has the form ${}x=3$, then one should recognize that a part of the solution can be read off immediately, and one should not add to this row any other row. Here, one should replace in the other rows everywhere ${}x$ by ${}3$, and continue then. Or: if four equations are given, where in two equations only the variables ${}x$ and ${}y$ appear, and where in the two other equations only the variables ${}z$ and ${}w$ appear, then one should realize that, in principle, two unrelated linear systems are given, each in two variables, and these should be solved independently. Or: it might be that a small subsystem of the system guarantees that there is no solution at all. Then only this has to be worked out, there is no need to consider the other equations. And: consider the exact question! If the question is whether a certain tuple is a solution, then we only have to plug this tuple into the equations, no manipulations are necessary.