An exact solution is derived for stabilizing a given but arbitrary, linear time-invariant discrete system by a first-order discrete-time feedback controller, which has received considerable attention in the past few years. An approach has been recently proposed to compute the first-order controllers, given in the form of C(z) = (zx(1) + x(2))/(z + x(3)). This approach derives the stabilizing set in the x(1)-x(2) plane by fixing x(3), and then repeat the procedure by sweeping over all possible values of x(3). In this note, from the geometrical point of view, we present an exact solution to the problem.