How to solve a system of equations of linear type
To fully understand how to solve the systemequations, we should consider what it is. As is clear from the term itself, a "system" is a collection of several equations related to each other. There are systems of algebraic and differential equations. In this article we will pay attention to how to solve a system of equations of the first type.
By definition, an equation is called algebraic,in which onlysimple mathematical operations; addition, division, subtraction, multiplication, exponentiation, and finding the root. The algorithm for solving an equation of this type is reduced to finding a structure equivalent to it by means of its transformations, but a simpler one.
Systems of algebraic equations are divided into linear and nonlinear.
System of linear equations (also widelythe abbreviation SLAU is used) differs from the system of nonlinear equations in that the unknown variables here are in the first degree. The general form of SLAE in matrix entries is as follows: Ax = b, where A is the set of known coefficients, x are variables, and b is the set of known free terms.
There are many ways of how to solve a system of equations of this type, theyare subdivided into direct and iterative methods. Direct methods allow us to find the values of variables for a certain number of mathematical transformations, and iterative algorithms use the algorithm of successive approximation and refinement.
Let's analyze by an example how to solve the system of linearequations, using a direct method of finding the value of variables. Direct methods include the methods of Gauss, Jordan-Gauss, Cramer, sweeps and some others. One of the simplest can be called the method of Cramer, usually it is with him in the curriculum begins acquaintance with matrices. This method is designed to solve square SLAU, i.e. Such systems, in which the number of equations is equal to the number of unknown variables in a row. Also, in order to solve the system of equations by the Cramer method, it is necessary to make sure that the free terms are not zeros (this is a necessary condition).
The solution algorithm is as follows: a matrix 1 consisting of the known coefficients of the a-system is constructed and its main determinant Δχ is found. The determinant is found by subtracting the product of the elements of the secondary diagonal from the product of elementsthe main one.
Further, a matrix 2 is compiled, where the values of the free elements b are substituted in the first column, similarly to the previous example, the determinant Δχ1.
We compose the matrix 3, the values of the free coefficients are substituted in the second column, we find the determinant of the matrix Δx2. And so on, until we compute the determinant of that matrix, where the coefficients b are in the last column.
To find the value of a particular variable, the determinants obtained by substituting the free coefficients must be divided into a principal determinant, i.e. x1= Δx1/ Δx, x2= Δx2/ Δx and so on.
If you have any questions on how to solve the system of equations in one way or another, I recommend referring to the reference and educational material, which details all the basic steps.