Systems of equations are widely used ineconomic branch in the mathematical modeling of various processes. For example, when solving problems of management and production planning, logistics routes (transportation problem) or equipment placement.
Systems of the equation 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 for which all equations become true equalities or prove that a sequence does not exist.
Linear equation
Equations of the form ax + by = c are called linear. The notation x, y is unknown, whose value must be found, b, a are the coefficients of the variables, c is the free term of the equation.
Solving an equation by plotting its graph will have the form of a line, all points of which are the solution of a polynomial.
Types of systems of linear equations
Examples of systems of linear equations with two variables X and Y are considered the simplest.
F1 (x, y) = 0 and F2 (x, y) = 0, where F1,2 are functions, and (x, y) are function variables.
Solve equation system - it means to find such values (x, y) for which the system turns into a true 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 the coordinates of a point, is called a solution of a system of linear equations.
If systems have one common solution or solutions do not exist, they are called equivalent.
Homogeneous systems of linear equations are systems whose right side is equal to zero. If the right after the sign "equality" part has a value or expressed function, such a system is heterogeneous.
The number of variables can be much more than two, then you should talk about an example of a system of linear equations with three variables or more.
Facing systems the schoolchildren assumethat the number of equations must necessarily coincide with the number of unknowns, but it is not. The number of equations in the system does not depend on variables; there can be as many of them as you like.
Simple and complex methods for solving systems of equations
There is no common analytical method.solutions of similar systems, all methods are based on numerical solutions. In the school course of mathematics, such methods as permutation, algebraic addition, substitution, as well as the graphic and matrix method, the solution by the Gauss method are described in detail.
The main task in learning how to solve isThis will teach you how to analyze the system correctly and find the optimal solution algorithm for each example. The main thing is not to memorize the system of rules and actions for each method, but to understand the principles of applying one method or another.
Solving examples of systems of linear equations 7The grade school program is fairly simple and explained in great detail. In any math textbook, this section is given enough attention. The solution of examples of systems of linear equations by the Gauss and Kramer methods is studied in more detail at the first courses of higher educational institutions.
Substitution system solution
The substitution method actions are directed toexpression of 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.
Let us give the solution of an example of a system of linear equations of the 7th class using the substitution method:
As can be seen from the example, the variable x was expressedthrough F (X) = 7 + Y. The resulting expression, substituted into the 2nd equation of the system in place of X, helped to get one variable Y in the 2nd equation. The solution of this example is not difficult and allows you to get the value of Y. The last step is to check the values obtained.
Solve an example of a system of linear equationssubstitution is not always possible. The equations can be complex and the expression of the variable through the second unknown will be too cumbersome for further calculations. When unknown in the system more than 3 solution substitution is also impractical.
Solving an example of a system of linear inhomogeneous equations:
Solution using algebraic addition
When searching for the solution of systems by the method of addition, they produce by term addition and multiplication of equations by different numbers. The ultimate goal of mathematical operations is an equation with one variable.
This method requires practice.and observation. Solving a system of linear equations by the addition method with the number of variables 3 and more is not easy Algebraic addition is convenient to apply when fractions and decimal numbers are present in the equations.
Algorithm of decision actions:
- Multiply both sides of the equation by some number. As a result of the arithmetic operation, one of the coefficients of the variable should become equal to 1.
- Add up the resulting expression and find one of the unknowns.
- Substitute the obtained value in the 2nd equation of the system to find the remaining variable.
The way to solve the introduction of a new variable
A new variable can be introduced if the system needs to find a solution for no more than two equations, the number of unknowns should also be no more than two.
The method is used to simplify one ofequations by entering a new variable. The new equation is solved relative to the entered unknown, and the resulting value is used to determine the original variable.
From the example it can be seen that by introducing a new variable t, it was possible to reduce the 1st equation of the system to the standard quadratic trinomial. The polynomial can be solved by finding the discriminant.
It is necessary to find the discriminant value bythe well-known formula: D = b2 - 4 * a * c, where D is the desired discriminant, b, a, c are 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 there is one solution: x = -b / 2 * a.
The solution for the resulting systems is found by the addition method.
Visual method of solving systems
Suitable for systems with 3 equations.The method consists in plotting on the coordinate axis the graphs of each equation entering the system. The coordinates of the points of intersection of the curves and will be a general solution of the system.
Graphic method has a number of nuances. Let us consider several examples of solving systems of linear equations in a visual manner.
As you can see from the example, for each straight wastwo points were constructed, the values of the variable x were chosen arbitrarily: 0 and 3. Based on the values of x, the values for y were found: 3 and 0. The points with coordinates (0, 3) and (3, 0) were marked on the graph and connected by a line .
Actions must be repeated for the second equation. The point of intersection of the lines is the solution of the system.
In the following example, it is required to find a graphical solution of a system of linear equations: 0.5x-y + 2 = 0 and 0.5x-y-1 = 0.
As can be seen from the example, the system has no solution, because the graphs are parallel and do not intersect along their entire length.
Системы из примеров 2 и 3 похожи, но при building it becomes obvious that their solutions are different. It should be remembered that it is not always possible to tell whether the system has a solution or not, it is always necessary to build a graph.
Matrix and its varieties
Matrices are used to briefly write a system of linear equations. A matrix is a special type of table, filled with numbers. A matrix of the form n * m has n - rows and m - columns.
The matrix is square when the numbercolumns and rows equals 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 the unit matrix.
The inverse matrix is such a matrix when multiplied by which the initial one becomes the identity, such a matrix exists only for the initial square.
Rules for converting a system of equations into a matrix
As applied to systems of equations, coefficients and free terms of equations are written as matrix numbers, one equation is one row of the matrix.
The row of the matrix is called nonzero if at leastone line element is not equal to zero. Therefore, if in any of the equations the number of variables varies, then it is necessary to enter a zero in the place of the missing unknown.
Matrix columns must strictly conformvariable. This means that the coefficients of the variable x can be written only in one column, for example, the first, the coefficient of the unknown y - only in the second.
When a matrix is multiplied, all elements of the matrix are sequentially multiplied by a number.
Variants of 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 | - the determinant of the matrix. | K | should not be zero, then the system has a solution.
The determinant is easily calculated for the matrix "two by two", you only need to multiply the elements along the diagonal. For the "three by three" option, there is a formula | K | = a1at2from3 + a1at3from2 + a3at1from2 + a2at3from1 + a2at1from3 + a3at2from1. You can use the formula, but you canremember that it is necessary 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 systems of linear equations by the matrix method
The matrix method of finding a solution allows one to reduce cumbersome records when solving systems with a large number of variables and equations.
In example anm - coefficients of equations, matrix - vector xMr. - variables, and bMr. - free members.
Next, you need to find the inverse matrix and multiply the original one by it. Find the values of variables in the resulting identity matrix is an easy task.
Gauss system solution
В высшей математике способ Гаусса изучают together with the Kramer method, and the process of finding a solution to the systems is called the Gauss – Kramer solution method. These methods are used when finding variable systems with a large number of linear equations.
The Gauss method is very similar to solutions usingsubstitutions and algebraic addition, but more systematic. In a school course, the Gauss 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 is the value of one variable in one of the equations of the system. The second equation is an expression with 2 unknowns, well, 3 and 4, respectively, with 3 and 4 variables.
After the system is reduced to the described form, the further solution reduces to the successive substitution of known variables into the equations of the system.
In school textbooks for grade 7, an example of a Gauss solution is described as follows:
As you can see from the example, two equations 3x were obtained in step (3)3-2x4= 11 and 3x3+ 2x4= 7. Solving any of the equations will let you know one of the variables xMr..
Theorem 5, which is mentioned in the text, says that if one of the equations of the system is replaced by equivalent, then the resulting system will also be equivalent to the original one.
Gauss method is difficult for students to understand.secondary school, but is one of the most interesting ways to develop the ingenuity of children enrolled in the program of in-depth study in the mathematical and physical classes.
For simplicity, the recording of calculations is usually done as follows:
Equation Coefficients and Free Memberswritten 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 number of equations in the system.
First write the matrix with which towork, then all actions carried out with one of the lines. The resulting matrix is written after the sign "arrow" and continue to perform the necessary algebraic actions to achieve the result.
The result should be a matrix in whichone of the diagonals are 1, and all other coefficients are zero, that is, the matrix is reduced to a single form. We must not forget to make calculations with the numbers on both sides of the equation.
This recording method is less cumbersome and allows you not to be distracted by the enumeration of numerous unknowns.
Free use of any solutionwill require attentiveness and some experience. Not all methods are applied. Some ways of finding solutions are preferable in a different area of human activity, while others exist for training purposes.
