One of tasks of the higher mathematics – the proof of compatibility of a system of the linear equations. The proof needs to be carried out according to the theorem of Kronkera-Kapelli according to which a system in common if the rank of its main matrix is equal to a rank of an expanded matrix.

One of classical ways of the solution of systems of the linear equations is Gauss's method. It consists in a consecutive exception of variables when the system of the equations by means of simple transformations is transferred to a step system from which consistently there are all variables, starting with the last.

The asked question completely covers a main objective of the whole course "Linear Algebra". Therefore the answer can be given only in summary form, without detailed calculations and explanations. In general the linear equations are interesting what to solve by their perhaps purely algorithmic methods.

The linear system with three unknown has several ways of the decision. It is possible to find a solution of a system by means of Kremer's rule through determinants, by Gauss's method or using an easy way of substitution. The method of substitution is the basic for the solution of systems of the linear equations of a small order. It consists in serial expression from each equation of a system of one unknown variable, its substitution in the following equation and simplification of the received expressions.