Abstract: The paper describes two iterative algorithms for solving general systems of M simultaneous linear algebraic equations (SLAE) with real matrices of coefficients. The system can be determined, underdetermined, and overdetermined. Linearly dependent equations are also allowed. Both algorithms use the method of Lagrange multipliers to transform the original SLAE into a positively determined function F of real original variables and Lagrange multipliers λm. Function F is differentiated with respect to variables xi and the obtained relationships are used to express F in terms of Lagrange multipliers λm. The obtained function is minimized with respect to variables λm with the help of one of two the following minimization techniques: (1) relaxation method or (2) conjugate gradient method by Fletcher and Reeves. Numerical examples are given
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