A perturbation procedure is introduced to solve for the probability amplitudes of a system of coupled spins under the action of two rf magnetic fields; one a large saturating signal, the other a relatively weak one. The power absorbed when the saturating signal connects two states in the system is affected by the presence of the second signal connecting two other states especially if both signals have one state in common. The mixing of these states by the saturating signal can become large enough in some instances that it becomes meaningless to speak in terms of population differences and rate equations for these levels. It is further shown that the effect of the intermediate levels is to cause a shift in the resonance frequency for these levels. For widely spaced levels this ``intermediate'' effect may be neglected, but for closely spaced levels it can become appreciable. A four-level system is used to illustrate the method. Relaxation effects are neglected.