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In this paper mathematician K. M. Brown's method is used to solve load-flow problems. The method is particularly effective for solving ill-conditioned nonlinear algebraic equations. It is a variation of Newton's method incorporating Gaussian elimination in such a way that the most recent information is always used at each step of the algorithm; similar to what is done in the Gauss-Seidel process. The iteration converges locally and the convergence is quadratic in nature. A general discussion of ill-conditioning of a system of algebraic equations is given, and it is also shown by the fixed-point formulation that the proposed method falls in the general category of successive approximation methods. Digital computer solutions by the proposed method are given for cases for which the standard load-flow methods failed to converge, namely 11-, 13-and 43-bus ill-conditioned test systems. A comparison of this method with the standard load-flow methods is also presented for the well-conditioned AEP 30-, and 57-bus systems.