Enhanced mobile broadband (eMBB) and ultra-reliable and low-latency communications (URLLC) are the two main expected services in the next generation of wireless networks. Accommodation of these two services on the same wireless infrastructure leads to a challenging resource allocation problem due to their heterogeneous specifications. To address this problem, slicing has emerged as an architecture that enables a logical network with specific radio access functionality to each of the supported services on the same network infrastructure. The allocation of radio resources to each slice according to their requirements is a fundamental part of the network slicing that is usually executed at the radio access network (RAN). In this work, we formulate the RAN resource allocation problem as a sum-rate maximization problem subject to the orthogonality constraint (i.e., service isolation), latency-related constraint and minimum rate constraint while maintaining the reliability constraint with the incorporation of adaptive modulation and coding (AMC). However, the formulated problem is not mathematically tractable due to the presence of a step-wise function associated with the AMC and a binary assignment variable. Therefore, to solve the proposed optimization problem, first, we relax the mathematical intractability of AMC by using an approximation of the non-linear AMC achievable throughput, and next, the binary constraint is relaxed to a box constraint by using the penalized reformulation of the problem. The result of the above two-step procedure provides a close-to-optimal solution to the original optimization problem. Furthermore, to ease the complexity of the optimization-based scheduling algorithm, a low-complexity heuristic scheduling scheme is proposed for the efficient multiplexing of URLLC and eMBB services. Finally, the effectiveness of the proposed optimization and heuristic schemes is illustrated through extensive numerical simulations.