In this paper, a self-backhauling radio access system is studied and analyzed. In particular, we consider a scenario where a full-duplex access node is serving mobile users simultaneously in uplink and downlink, while also maintaining a wireless backhaul connection. The full-duplex capability of the access node, together with large antenna arrays, allows it to do all of this using the same center frequency. The minimum transmit powers for such a system are solved in a closed form under the condition that certain Quality of Service (QoS) requirements, defined in terms of minimum uplink and downlink data rates, are fulfilled. It is demonstrated with numerical results that, by using the derived expressions for the optimal transmit powers, the probability of fulfilling the QoS requirements is greatly increased, while simultaneously the overall transmit power usage of the system is significantly reduced when compared to a benchmark scheme.