A method is developed using the variational principle to solve a two-region, monoenergetic, one-dimensional transport problem arising in the calculation of the neutron flux-distortion factor. The Roussopoulos functional for the problem is modified with the help of a Lagrangian multiplier accommodating the interface conditions. The complete set of Case eigen-functions in each region are selected as the trial functions. The stationarity condition of the modified functional leads to a decoupled set of equations for each set of unknown coefficients. These are solved using Gauss quadrature to approximate integrals. Two problems are solved using the formalism developed; the problem of computing the flux-depression due to a foil placed in a medium with a constant source, and the problem of evaluating the flux-distortion due to a foil placed in an exponentially varying flux. The results are compared with the previously reported values and excellent agreement is observed.