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.
Instruction
1. At first bring the system of the equations into such look when all unknown stand in strictly certain order. For example, all unknown of X will stand the first in every line, all Y – after X, all Z - after Y and so on. In the right part of each equation of unknown should not be. Mentally define the coefficients facing each unknown and also coefficients in the right part of each equation.
2. Write down the received coefficients in the form of an expanded matrix. The expanded matrix is the matrix made of coefficients at unknown and a column of free members. After that you pass to elementary transformations in a matrix. Begin to rearrange places of its line until you find proportional or identical. As soon as such lines appear, remove them everything, except one.
3. If in a matrix there is a zero line, remove also it. The zero line is a line in which all elements are equal to zero. Then try to divide or multiply lines of a matrix by any numbers, except zero. It will help you to simplify further transformations, having got rid of fractional coefficients.
4. Begin to add other lines increased by any number other than zero to lines of a matrix. Do it until you find zero elements in lines. The ultimate goal of all transformations is to transfer all matrix to step (a triangular look) when each following line has more and more zero elements. In registration of a task it is possible to emphasize with a simple pencil the received short flight of stairs and to circle with number circles, located on the steps of this ladder.
5. Then bring the received matrix back into an initial type of a system of the equations. In the lowermost equation the ready result will be already visible: what the unknown standing on the last place of each equation is equal to. Having substituted the received value of the unknown in the equation located above, receive value of the second unknown. And so on, do not calculate value of all unknown yet.