Non-orthogonal multiple access (NOMA) can increase the spectral efficiency and could play an important role in improving the capacity of 5G networks. In this paper, an optimization problem is formulated to maximize the overall sum rate in a sub-carrier based NOMA system. The optimal transmission power of each user is obtained based on the users' instantaneous channel state information under the total power and the minimum rate constraints. Moreover, two closed-form suboptimal solutions are also proposed for a two-user scenario to reduce the complexity of the optimal solution. The suboptimal approaches are also extended to multiuser scenario by pairing users for subbands transmission. Simulation results show that the derived sub-optimal solutions provide better performance than the orthogonal frequency division multiple access (OFDMA) in terms of coverage probability and sum rate. Moreover, the results show that the proposed sub-optimal solutions achieve a comparable results to the optimal one with lower complexity.