In a real-time system, tasks are required to be completed before their deadlines. Under normal workload conditions, a scheduler with a proper scheduling policy can make all the tasks meet their deadlines. However, in practical environment, system workload may vary widely. Once system workload becomes too heavy, so that there does not exist a feasible schedule can make all the tasks meet their deadlines, we say the system is overloaded under which some tasks will miss their deadlines. To alleviate the degrees of system performance degradation caused by the missed deadline tasks, the design of scheduling is crucial. Many design objectives can be considered. In this paper, we first focus on maximizing the total number of tasks that can be completed before their deadlines. A scheduling method based on satisfiability modulo theories (SMT) is proposed. In the method, the problem of scheduling is treated as a satisfiability problem. The key work is to formalize the satisfiability problem using first-order language. After the formalization, a SMT solver (e.g., Z3, Yices) is employed to solve this satisfiability problem. An optimal schedule can be generated based on the solution model returned by the SMT solver. The correctness of this method and the optimality of the generated schedule can be verified in a straightforward manner. The time efficiency of the proposed method is demonstrated through various simulations. Moreover, in the proposed SMT-based scheduling method, we define the scheduling constraints as system constraints and target constraints. This means if we want to design scheduling to achieve other objectives, only the target constraints need to be modified. To demonstrate this advantage, we adapt the SMT-based scheduling method to other design objectives: maximizing effective processor utilization and maximizing obtained values of completed tasks. Only very little changes are needed in the adaption procedure, which means the proposed SMT-based scheduling method is flexible and sufficiently general.