When a chemical sample made of N elements is analyzed by using sequential selective excitation by monochromatic X-ray beams and selective measurement of the characteristic X-rays, the production of secondary fluorescence does not interfere with the measurements. This experimental situation leads to a particular simple case of the Sherman equations which can be written in this instance as linear equations. The linear equations thus obtained are shown to be very similar to the equations appearing in the classical models of Beattie and Brissey and of Lachance and Traill. The linear algebra proves the existence of N different sets of solutions, but the Perron Frobenius theorem ensures that there is one and only one physically feasible solution, and also leads to the method for obtaining it. This equation solution method can be extended to the equations appearing when standard samples of pure elements are also measured.The propagation of the errors in the measurements to the errors in the sample concentrations has been calculated and simulated, and the results have shown that the solution is well conditioned. © 2007 Springer Science+Business Media, LLC.