This paper presents a numerical method for solving systems of singular integral equations, which are commonly encountered in physics and engineering. The authors develop Gauss-Chebyshev integration formulas for singular integrals and use these formulas to create a simple numerical method. The effectiveness of the method is demonstrated through a numerical example, which is also solved using an alternate method for comparison. The singular behavior of the unknown functions is characterized by fundamental functions derived from the integral equations. The method involves approximating the unknown functions using truncated series expansions of Chebyshev polynomials and applying collocation techniques at specific points. The paper includes detailed proofs of auxiliary formulas and examples to illustrate the application of the method. The results show that the proposed method is simple, effective, and converges well, making it a valuable tool for solving complex singular integral equations.This paper presents a numerical method for solving systems of singular integral equations, which are commonly encountered in physics and engineering. The authors develop Gauss-Chebyshev integration formulas for singular integrals and use these formulas to create a simple numerical method. The effectiveness of the method is demonstrated through a numerical example, which is also solved using an alternate method for comparison. The singular behavior of the unknown functions is characterized by fundamental functions derived from the integral equations. The method involves approximating the unknown functions using truncated series expansions of Chebyshev polynomials and applying collocation techniques at specific points. The paper includes detailed proofs of auxiliary formulas and examples to illustrate the application of the method. The results show that the proposed method is simple, effective, and converges well, making it a valuable tool for solving complex singular integral equations.