M-theory, a generalization of string theory, motivates the search for examples of volume minimizing cycles in Riemannian manifolds of G2 holonomy. Methods of calibrated geometry lead to a system of four coupled nonlinear partial differential equations whose solutions correspond to associa- tive submanifolds of R7, which are 3-dimensional and minimize volume in their real homology classes. Several approaches to finding new solutions are investigated, the most interesting of which exploits the quaternionic structure of the PDE system. A number of examples of associative 3-planes are explicitly given; these may possibly be projected to nontrivial volume minimizing cycles in, for example, the G2-manifold R6 × S1.