The general plane elasticity problem of an isotropic elastic inclusion inside a finite or infinite isotropic elastic medium, with different elastic properties, may be reduced to a complex Cauchy-type singular integral equation along the boundary of the inclusion, and the outer boundary of the medium surrounding the inclusion, if it exists. A numerical solution of this integral equation by reduction to a system of linear equations is presented. The numerical results obtained for the case of an elliptical inclusion inside an infinite medium converge to the well-known theoretical results available for this special case.