AN iteration process for the solution of the linear algebraic system Ax = ‖ with the singular matrix A, based on the normalized expansion of the generalized matrix B = ∥ A ⋮ − ‖∥, is presented. The process makes it possible to analyze the rounding errors directly, to calculate and correct the numerical solution, and to analyze the system for conditionality. A direct method of solving linear algebraic systems, based on the application of a normalized process, using elementary rotations or reflections with a specified order of permutation of the rows of the matrix and of the components of the free term is presented. The system enables the solution of the system to be found without working backwards and in this sense it is an orthogonal analogue of the well-known Jordan method. Under fairly general assumptions the process makes it possible, without having recourse to additional investigations, to analyze the system to be solved for conditionality and also to analyze the rounding errors directly. The process may be used to calculate the normal solution of an incompletely defined system. Two methods of refining the solution of the system are proposed: the first essentially uses the results of the calculations of the proposed process; the second may be recommended for refining the solution of a system obtained by other methods. It is recommended that in the first instance the proposed process be used to solve ill-conditioned systems, and also for the construction of a normal solution of incompletely defined systems whose dimensions permit a calculation to be carried out using only the operative memory of the computer.