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This study investigates the coupling of the finite and boundary element methods in elastostatics where each method is used to model a different portion of the domain. The principal interest is in applying a boundary element method (BEM) to model the infinite domain (assumed to be isotropic linear elastic) while using the finite element method (FEM) to model regions with more complex constitutive relations. The approach taken in this study, referred to as a FEM-hosted coupling, treats each BEM subdomain as a single finite element. Two derivations for an IBEM stiffness matrix are given; the first is a physically intuitive direct derivation while the second is the corresponding variational derivation. Though the emphasis is on the IBEM, the DBEM is also addressed. The inherent incompatibility between the BEM and FEM methods is discussed and explained in terms of the shape function fallacy.