How to solve a linear type of system of equations

For a complete understanding of how to solve a system of equations, you should consider what it is. As is clear from the term itself, a “system” is a collection of several equations that are interconnected. 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 only simple mathematical operations are performed on the variables, i.e. addition, division, subtraction, multiplication, exponentiation and finding the root. The algorithm for solving an equation of this type is reduced to finding equivalent to it, but simpler construction by means of its transformations.
Systems of algebraic equations are divided into linear and nonlinear.
The system of linear equations (the abbreviation SLAU is also widely used) differs from the system of nonlinear equations in that the unknown variables are in the first degree here. The general view of SLAE in the matrix notation looks like this: Ax = b, where A is the set of known coefficients, x are variables, b is the set of known free terms.

There are many ways to solve a system of equations of this type; they

are divided into direct and iterative methods. Direct methods allow you to find the values ​​of variables for a certain number of mathematical transformations, and iterative ones use the algorithm of sequential approximation and refinement.

Let us examine an example of how to solve a system of linear equations using the 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 Cramer method, usually it is with him in the curriculum that acquaintance with matrices begins. This method is intended to solve square SLAEs, i.e. such systems in which the number of equations is equal to the number of unknown variables in the 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: matrix 1 is composed, consisting of the known coefficients of the a-system, and its main determinant Δx is found. The determinant is found by subtracting the product of the elements of the secondary diagonal from the product of the elements

the main one.

Next, matrix 2 is compiled, where the values ​​of free elements b are substituted into the first column, and the determinant Δx _{1 is} found similarly to the previous example.

We compose the matrix 3, substitute the values ​​of the free coefficients into the second column, we find the determinant of the matrix Δx ₂ . And so on, until we calculate the determinant of the matrix where the coefficients b are in the last column.

To find the value of a variable, it is necessary to divide the determinants obtained by substituting the free coefficients into the main determinant, i.e. x ₁ = ∆x ₁ / ∆x, x ₂ = ∆x ₂ / ∆x, etc.
If you have questions about how to solve the system of equations in one way or another, I recommend that you turn to the reference and training material, which details all the basic steps.