  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 X(i) (i=1,...,N)
and Lagrange multipliers Lambda(i) (i=1,...,M). Function F is differentiated
with respect to variables X(i) and the obtained relationships are used to
express F in terms of Lagrange multipliers Lambda(i). The obtained function is
minimized with respect to variables Lambda(i) with the help of one of two the
following minimization techniques: (1) relaxation method or (2) method of
conjugate gradients by Fletcher and Reeves. Numerical examples are given.
