The flow of a rarefied binary gas mixture is governed by transport equations which correspond to a coupled set of Fredholm integral equations of the second kind. By means of the Fourier transform technique and by expanding the transformed kernels in terms or orthogonal functions the problem is reduced to the solution of a system of linear algebraic equations. As a significant application, the case of a plane Poiseuille flow of a binary mixture following Sirovich-Hamel's model is considered. The convergence of the method is excellent.