This study utilizes two robust methodologies to examine the precise solutions of the Dirac integrable system. The Homogeneous Balance Method (HB) is initially employed to generate an accurate solution. The system of equations for the quasi-solution is solved, where all the equations are of the same nature. The quasi-solution of the traveling wave results in the solitary wave solution of the system. The singular manifold method (SMM) is utilized following the Lie reduction of the Dirac system in order to search for the traveling wave solutions of the system. Both approaches demonstrate the existence of traveling wave solutions inside the system. The precise solutions of the Dirac system are shown in three-dimensional graphs. We have created solutions to the examined problem, including bright solutions, periodic soliton solutions, and complicated solutions.