In the analysis of logic programs, abstract domains for detecting sharing and linearity information are widely used. Devising abstract unification algorithms for such domains has proved to be rather hard. At the moment, the available algorithms are correct but not optimal, i.e., they cannot fully exploit the information conveyed by the abstract domains. In this paper, we define a new (infinite) domain ShLin-w which can be thought of as a general framework from which other domains can be easily derived by abstraction. ShLin-w makes the interaction between sharing and linearity explicit. We provide a constructive characterization of the optimal abstract unification operator on ShLin-w and we lift it to two well-known abstractions of ShLin-w. Namely, to the classical Sharing X Lin abstract domain and to the more precise ShLin-2 abstract domain by Andy King. In the case of single binding substitutions, we obtain optimal abstract unification algorithms for such domains. To appear in Theory and Practice of Logic Programming (TPLP).