Polydisperse solutions of polymers, formed by a reversible polymerization process, are studied by computer simulation. The system is described by a stepwise elongation and contraction process which is based on adding (or deleting) one link to a chain end. In such a process, cyclic polymers cannot be formed. The growth of a polymer is controlled by two chemical potentials that determine the number of polymers and their length and, hence, the concentration of the system. Such a reversible polymerization process can be described by a magnetic analog of the n-vector model. Thus, this system can be used to emphasize the correspondence between polymer models and critical magnetic models. The entire phase space of concentrations of a polydisperse solution is studied. We describe the long range of validity of the semidilute regime, in which the polymer chains are only slightly overlapping. Our results also indicate that, basically, a polydisperse solution shows the same behavior as a monodisperse one. In the limit of dense chains, we study the hypothesis that the system is controlled by one "infinite" chain that leads to a transition toward a collapsed phase. We performed several studies in this regime and found no evidence for an infinite chain nor for a collapsed phase.