In this paper, by using the Lie symmetry analysis, all of the geometric vector fields of the $(3+1)$ -Burgers system are obtained. We find the 1, 2, and 3-dimensional optimal system of the Burger system and then by applying the 3-dimensional optimal system reduce the order of the system. Also the nonclassical symmetries of the $(3+1)$ -Burgers system will be found by employing nonclassical methods. Finally, the ansatz solutions of BSequations with the aid of the tanh method has been presented. The achieved solutions are investigated through two- and three-dimensional plots for different values of parameters. The analytical simulations are presented to ensure the efficiency of the considered technique. The behavior of the obtained results for multiple cases of symmetries is captured in the present framework. The outcomes of the present investigation show that the considered scheme is efficient and powerful to solve nonlinear differential equations that arise in the sciences and technology.