The vibration of a sandwich plate is used, as an illustrative problem, to evaluate the effectiveness of some methods of obtaining one‐term approximations for the lowest eigenvalue of coupled systems. It is shown that, in using a method of weighted residuals, the boundary conditions satisfied by both the trial and the weighting function play significant roles in the accuracy of the eigenvalue. Moreover, the desired boundary conditions for the weighting function may differ from those of the trial function, so that the Galerkin method, in which the trial function and weighting function coincide, may not be the best method to use. In addition, it is shown that the values of some physical parameters may play a critical role in the accuracy obtained with a given trial function. It is demonstrated that good approximations of the fundamental frequency are obtainable upon uncoupling a set of differential equations, but uncoupling may lead to non‐self‐adjoint boundary conditions, in which case the weighting function should satisfy boundary conditions which are adjoint to those of the trial function. Attention is also called to a method, due to Lanczos, which does not require a choice of trial or weighting functions, is insensitive to the form in which the problem is stated, appears to require less effort for a given degree of accuracy, and is applicable to non‐self‐adjoint as well as self‐adjoint systems.