This paper presents a numerical method for solving systems of singular integral equations. The method is based on Gauss-Chebyshev integration formulas for singular integrals. The authors develop two auxiliary formulas that are used to derive these integration formulas. These formulas are then applied to solve singular integral equations of the form: $$ \frac{1}{\pi}\int_{-1}^{1}\sum_{1}^{M}a_{i;\phi_{i}}(t)\frac{dt}{t-x}+\int_{-1}^{1}\sum_{1}^{M}k_{i;}(x,t)\phi_{i}(t)dt=g_{i}(x),\quad -1<x<1,\quad i=1,\cdots,M. $$ The unknown functions $\phi_1, \cdots, \phi_M$ are expressed in terms of fundamental functions $R_i(t)$, which are related to Chebyshev polynomials. The method involves approximating these functions using Chebyshev polynomials and then reducing the singular integral equations to systems of linear algebraic equations. The authors demonstrate the effectiveness of this method by solving a numerical example and comparing the results with an alternate method. The results show that the method is accurate and efficient, even for complex problems. The paper concludes that the method is a simple and effective way to solve systems of singular integral equations of the first kind.This paper presents a numerical method for solving systems of singular integral equations. The method is based on Gauss-Chebyshev integration formulas for singular integrals. The authors develop two auxiliary formulas that are used to derive these integration formulas. These formulas are then applied to solve singular integral equations of the form: $$ \frac{1}{\pi}\int_{-1}^{1}\sum_{1}^{M}a_{i;\phi_{i}}(t)\frac{dt}{t-x}+\int_{-1}^{1}\sum_{1}^{M}k_{i;}(x,t)\phi_{i}(t)dt=g_{i}(x),\quad -1<x<1,\quad i=1,\cdots,M. $$ The unknown functions $\phi_1, \cdots, \phi_M$ are expressed in terms of fundamental functions $R_i(t)$, which are related to Chebyshev polynomials. The method involves approximating these functions using Chebyshev polynomials and then reducing the singular integral equations to systems of linear algebraic equations. The authors demonstrate the effectiveness of this method by solving a numerical example and comparing the results with an alternate method. The results show that the method is accurate and efficient, even for complex problems. The paper concludes that the method is a simple and effective way to solve systems of singular integral equations of the first kind.