This paper illustrates a comparative analysis of numerical solutions to a tridiagonal system of linear equations. It examines the effectiveness of the Recursive method described here, by comparing results with the LU decomposition method and the Thomas algorithm. Moreover, these approaches are utilized to tackle differential equations using the finite difference method since discretizing ODEs and PDEs often yields tridiagonal systems. Accurate numerical simulations in fields such as physics, engineering, and environmental research rely on solving these systems. It is important to note that the LU decomposition method and Thomas algorithm are both standard methods and require more calculation than the Recursive method. A Recursive method is given here, which is a sequence-based method. Solving the block tridiagonal system by this method is more efficient and easier, and the effectiveness is most pronounced when handling non-separable partial differential equations (PDEs), resulting in the formation of block tridiagonal systems. The current study demonstrates that the Recursive method is a direct approach. Numerical solutions for various example problems are compared across three methods, with results summarized in tables. MATLAB programming is used to obtain the solutions.