The aim of this work is the study of the MHD unsteady flow of an incompressible viscous and radiative, optically thin gray fluid past an infinitely vertical plate moving with constant velocity. The magnetic Reynolds number is not small, so that the induced magnetic field is present. The partial differential equations (PDEs) and their initial and boundary conditions, describing the problem under consideration, are dimensionalized and the numerical solution is obtained by using the finite volume method. The numerical results for the velocity, temperature and the induced magnetic field are shown in figures for different parameters of the problem under consideration, such as the magnetic parameter, M, the radiation parameter, R, the magnetic Prandtl parameter,  and the Grashof number, Gr. The analysis of these results showed that the flow, temperature and induced magnetic fields are noticeable influenced by these parameters. This study will bring out insight in the problem of an optically gray thin fluid flow under the influence of radiation in the presence of an induced magnetic field.