Systems of equations have been widely used in the economic industry for mathematical modeling of various processes. For example, when solving management and production planning tasks, logistics routes (transport problem) or equipment placement.
Equation systems are used not only in the field of mathematics, but also in physics, chemistry and biology, in solving problems of finding the population size.
A system of linear equations is called two or more equations with several variables, for which it is necessary to find a common solution. Such a sequence of numbers under which all equations become true equalities or prove that the sequence does not exist.
Linear equation
Equations of the form ax + by = c are called linear. The notation x, y is unknown, the value of which must be found, b, a are the coefficients of the variables, c is the free term of the equation.
Solving an equation by constructing its graph will look like a straight line, all of whose points are a solution to the polynomial.
Types of systems of linear equations
The simplest examples are systems of linear equations with two variables X and Y.
F1 (x, y) = 0 and F2 (x, y) = 0, where F1,2 are functions, and (x, y) are function variables.
To solve a system of equations means to find such values (x, y) at which the system turns into the correct equality or to establish that there are no suitable values of x and y.
A pair of values (x, y), written in the form of coordinates of a point, is called a solution of a system of linear equations.
If the systems have one common solution or if there is no solution, they are called equivalent.
Homogeneous systems of linear equations are systems whose right-hand side is zero. If the right part after the equal sign has meaning or is expressed by a function, such a system is heterogeneous.
The number of variables can be much more than two, then we should talk about an example of a system of linear equations with three variables or more.
Faced with systems, students suggest that the number of equations must necessarily coincide with the number of unknowns, but this is not so. The number of equations in the system does not depend on variables; there can be arbitrarily many of them.
Simple and complex methods for solving systems of equations
There is no general analytical method for solving such systems; all methods are based on numerical solutions. In the school course of mathematics, methods such as permutation, algebraic addition, substitution, as well as the graphical and matrix method, are solved in detail by the Gauss method.
The main task in teaching methods of solution is to teach you to analyze the system correctly and find the optimal solution algorithm for each example. The main thing is not to learn the system of rules and actions for each method, but to understand the principles of applying this or that method
The solution of examples of systems of linear equations of the 7th grade of a program of a comprehensive school is quite simple and is explained in great detail. In any mathematics textbook, this section is given enough attention. The solution of examples of systems of linear equations by the Gauss and Cramer method is studied in more detail in the first courses of higher educational institutions.
Substitution systems solution
The actions of the substitution method are aimed at expressing the value of one variable through the second. The expression is substituted into the remaining equation, then it is brought to the form with one variable. The action is repeated depending on the number of unknowns in the system.
We give a solution to an example of a system of linear equations of class 7 by the substitution method:
As you can see from the example, the variable x was expressed in terms of F (X) = 7 + Y. The resulting expression, substituted in the 2nd equation of the system in place of X, helped to obtain one variable Y in the 2nd equation. The solution to this example does not cause difficulties and allows you to get the value of Y. The last step is to check the received values.
Solving an example of a system of linear equations by substitution is not always possible. The equations can be complex and the expression of a variable through the second unknown will be too cumbersome for further calculations. When there are more than 3 unknowns in the system, a substitution solution is also impractical.
Solution of an example of a system of linear inhomogeneous equations:
Solution Using Algebraic Addition
When searching for a solution to systems by the addition method, term-by-term addition and multiplication of equations by various numbers is performed. The ultimate goal of mathematical operations is a single variable equation.
For the application of this method, practice and observation are required. Solving a system of linear equations by the addition method with the number of variables 3 or more is not easy. Algebraic addition is convenient to use when fractions and decimal numbers are present in the equations.
Algorithm of action of the solution:
- Multiply both sides of the equation by a certain number. As a result of arithmetic, one of the coefficients of the variable should become equal to 1.
- Add the resulting expression term by term and find one of the unknowns.
- Substitute the obtained value in the 2nd equation of the system to search for the remaining variable.
Method of solution by introducing a new variable
A new variable can be introduced if it is required to find a solution in the system for no more than two equations, the number of unknowns should also be no more than two.
The method is used to simplify one of the equations by introducing a new variable. The new equation is solved with respect to the introduced unknown, and the obtained value is used to determine the initial variable.
It can be seen from the example that by introducing a new variable t, it was possible to reduce the first equation of the system to the standard square trinomial. You can solve the polynomial by finding the discriminant.
It is necessary to find the value of the discriminant by the well-known formula: D = b2 - 4 * a * c, where D is the desired discriminant, b, a, c are the factors of the polynomial. In the given example, a = 1, b = 16, c = 39, therefore, D = 100. If the discriminant is greater than zero, then there are two solutions: t = -b ± √ D / 2 * a, if the discriminant is less than zero, then the solution is one: x = -b / 2 * a.
The solution for the resulting systems is found by the addition method.
Visual method for solving systems
Suitable for systems with 3 equations. The method consists in plotting on the coordinate axis the graphs of each equation included in the system. The coordinates of the intersection points of the curves will be the general solution of the system.
The graphical method has a number of nuances. Consider several examples of solving systems of linear equations in a visual way.
As you can see from the example, two points were built for each line, the values of the variable x were chosen arbitrarily: 0 and 3. Based on the values of x, values were found for y: 3 and 0. Points with coordinates (0, 3) and (3, 0) were marked on the chart and connected by a line.
The steps must be repeated for the second equation. The intersection point of lines is a solution to the system.
In the following example, it is required to find a graphical solution to the system of linear equations: 0.5x-y + 2 = 0 and 0.5x-y-1 = 0.
As you can see from the example, the system does not have a solution, because the graphs are parallel and do not intersect along their entire length.
The systems from examples 2 and 3 are similar, but during the construction it becomes obvious that their solutions are different. It should be remembered that it is not always possible to say whether the system has a solution or not, it is always necessary to build a schedule.
Matrix and its varieties
Matrices are used to write down a system of linear equations. A matrix is a table of a special kind, filled with numbers. A matrix of the form n * m has n - rows and m - columns.
The matrix is square when the number of columns and rows is equal to each other. Matrix - a vector is a matrix of one column with an infinitely possible number of rows. A matrix with units on one of the diagonals and other zero elements is called unity.
The inverse matrix is such a matrix, when multiplied by which the original turns into the identity one, such a matrix exists only for the original square.
Rules for transforming a system of equations into a matrix
In relation to systems of equations, coefficients and free terms of equations are written as matrix numbers, one equation - one row of the matrix.
A row of a matrix is called nonzero if at least one element of the row is not equal to zero. Therefore, if the number of variables differs in any of the equations, then it is necessary to enter zero in the place of the missing unknown.
The columns of the matrix must strictly correspond to the variables. This means that the coefficients of the variable x can only be written in one column, for example, the first, the coefficient of the unknown y - only in the second.
When the matrix is multiplied, all elements of the matrix are sequentially multiplied by a number.
Options for finding the inverse matrix
The formula for finding the inverse matrix is quite simple: K -1 = 1 / | K |, where K -1 is the inverse matrix, and | K | - matrix determinant. | K | should not be zero, then the system has a solution.
The determinant is easily calculated for the “two by two” matrix, you only need to multiply the elements diagonally by each other. For the “three by three” option, there is a formula | K | = a 1 b 2 c 3 + a 1 b 3 c 2 + a 3 b 1 c 2 + a 2 b 3 c 1 + a 2 b 1 c 3 + a 3 b 2 c 1 . You can use the formula, but you can remember that you need to take one element from each row and each column so that the product does not repeat the numbers of columns and rows of elements.
Solving examples of linear systems of equations by the matrix method
The matrix method of finding a solution allows one to reduce cumbersome records in solving systems with a large number of variables and equations.
In the example, a nm are the coefficients of the equations, the matrix is the vector x n are variables, and b n are free terms.
Next, you need to find the inverse matrix and multiply the original by it. Finding the values of the variables in the resulting unit matrix is an easy task.
Gauss solution of systems
In higher mathematics, the Gauss method is studied together with the Cramer method, and the process of finding a solution to systems is called the Gauss-Cramer solution method. These methods are used when finding variable systems with a large number of linear equations.
The Gauss method is very similar to permutation and algebraic addition solutions, but more systematic. In a school course, a Gaussian solution is applied to systems of 3 and 4 equations. The purpose of the method is to bring the system to the form of an inverted trapezoid. By algebraic transformations and substitutions, the value of one variable is found in one of the equations of the system. The second equation is an expression with 2 unknowns, but 3 and 4, respectively, with 3 and 4 variables.
After reducing the system to the described form, the further solution is reduced to the sequential substitution of the known variables in the equations of the system.
In school textbooks for grade 7, an example of a Gauss solution is described as follows:
As can be seen from the example, in step (3), two equations 3x 3 -2x 4 = 11 and 3x 3 + 2x 4 = 7 were obtained. The solution to any of the equations will allow one of the variables x n to be known.
Theorem 5, which is mentioned in the text, states that if one of the equations of the system is replaced by equivalent, then the resulting system will also be equivalent to the original.
The Gauss method is difficult for secondary school students to perceive, but it is one of the most interesting ways for the development of ingenuity of children studying in the in-depth study program in mathematical and physical classes.
For simplicity, it is customary to write calculations as follows:
Coefficients of equations and free terms are written in the form of a matrix, where each row of the matrix corresponds to one of the equations of the system. A vertical bar separates the left side of the equation from the right. Roman numerals denote the numbers of equations in the system.
First, a matrix is written with which to work, then all the actions carried out with one of the lines. The resulting matrix is written after the arrow sign and continue to perform the necessary algebraic actions until the result is achieved.
As a result, you should get a matrix in which one of the diagonals is 1, and all the other coefficients are zero, that is, the matrix is reduced to a single form. Do not forget to make calculations with the numbers of both parts of the equation.
This recording method is less cumbersome and allows you not to be distracted by listing numerous unknowns.
The free application of any method of solution will require attentiveness and certain experience. Not all methods are applied in nature. Some ways of finding solutions are more preferable in that other area of activity of people, while others exist for educational purposes.