The direct method using reduced matrices, either formal or informal, is in a real sense not an iterative method at all like the simplex method. Rather a solution process is used in which the technique employed depends on the amount of reduction present. The reduced matrix can be computed in one step. The reduced grouped matrix can then be found in at least (m + n) steps (many of which can be integrated into a single step). With m > n, and eliminating the total deficiency of a linear form in one step, the linear forms can be integrated in at most n - 2 steps so a bound for the number of steps of the general solution is m + 2n - 1. No problem is known for which the actual number of steps is more than a small proportion of this.