In a superimposed unicast and multicast transmission system, one layer of Successive Interference Cancellation (SIC) is required at each receiver to remove the multicast stream before decoding the unicast stream. In this paper, we show that a linearly-precoded Rate-Splitting (RS) strategy at the transmitter can efficiently exploit this existing SIC receiver architecture. By splitting the unicast messages into common and private parts and encoding the common parts along with the multicast message into a super-common stream decoded by all users, the SIC is used for the dual purpose of separating the unicast and multicast streams as well as better managing the multi-user interference between the unicast streams. The precoders are designed with the objective of maximizing the Weighted Sum Rate (WSR) of the unicast messages subject to a Quality of Service (QoS) requirement of the multicast message and a sum power constraint. Numerical results show that RS outperforms existing Multi-User Linear-Precoding (MU-LP) and power-domain Non-Orthogonal Multiple Access (NOMA) in a wide range of user deployments (with a diversity of channel directions and channel strengths). Moreover, since one layer of SIC is required to separate the unicast and multicast streams, the performance gain of RS comes without any increase in the receiver complexity compared with MU-LP. Hence, in such non-orthogonal unicast and multicast transmissions, RS provides rate and QoS enhancements at no extra cost for the receivers.