We will analyze two types of solving systems of equations:
1. Solution of the system by the substitution method.
2. Solution of the system by term-by-term addition (subtraction) of the equations of the system.
In order to solve the system of equations substitution method you need to follow a simple algorithm:
1. We express. From any equation, we express one variable.
2. Substitute. We substitute in another equation instead of the expressed variable, the resulting value.
3. We solve the resulting equation with one variable. We find a solution to the system.
To solve system by term-by-term addition (subtraction) need:
1. Select a variable for which we will make the same coefficients.
2. We add or subtract the equations, as a result we get an equation with one variable.
3. We solve the resulting linear equation. We find a solution to the system.
The solution of the system is the intersection points of the graphs of the function.
Let us consider in detail the solution of systems using examples.
Example #1:
Let's solve by the substitution method
Solving the system of equations by the substitution method2x+5y=1 (1 equation)
x-10y=3 (2nd equation)
1. Express
It can be seen that in the second equation there is a variable x with a coefficient of 1, hence it turns out that it is easiest to express the variable x from the second equation.
x=3+10y
2. After expressing, we substitute 3 + 10y in the first equation instead of the variable x.
2(3+10y)+5y=1
3. We solve the resulting equation with one variable.
2(3+10y)+5y=1 (open brackets)
6+20y+5y=1
25y=1-6
25y=-5 |: (25)
y=-5:25
y=-0.2
The solution of the system of equations is the intersection points of the graphs, therefore we need to find x and y, because the intersection point consists of x and y. Let's find x, in the first paragraph where we expressed we substitute y there.
x=3+10y
x=3+10*(-0.2)=1
It is customary to write points in the first place, we write the variable x, and in the second place the variable y.
Answer: (1; -0.2)
Example #2:
Let's solve by term-by-term addition (subtraction).
Solving a system of equations by the addition method3x-2y=1 (1 equation)
2x-3y=-10 (2nd equation)
1. Select a variable, let's say we select x. In the first equation, the variable x has a coefficient of 3, in the second - 2. We need to make the coefficients the same, for this we have the right to multiply the equations or divide by any number. We multiply the first equation by 2, and the second by 3 and get a total coefficient of 6.
3x-2y=1 |*2
6x-4y=2
2x-3y=-10 |*3
6x-9y=-30
2. From the first equation, subtract the second to get rid of the variable x. Solve the linear equation.
__6x-4y=2
5y=32 | :5
y=6.4
3. Find x. We substitute the found y in any of the equations, let's say in the first equation.
3x-2y=1
3x-2*6.4=1
3x-12.8=1
3x=1+12.8
3x=13.8 |:3
x=4.6
The point of intersection will be x=4.6; y=6.4
Answer: (4.6; 6.4)
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The article introduces such a concept as the definition of a system of equations and its solution. Frequently encountered cases of system solutions will be considered. The following examples will help explain the solution in detail.
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Definition of a system of equations
To proceed to the definition of a system of equations, it is necessary to pay attention to two points: the type of record and its meaning. To understand this, we need to dwell on each of the types in detail, then we can come to the definition of systems of equations.
For example, let's take two equations 2 x + y = - 3 and x = 5 , after which we combine with a curly bracket of such a plan:
2 x + y = - 3 , x = 5 .
Equations joined by a curly brace are considered to be records of systems of equations. They define sets of solutions to the equations of the given system. Each solution must be a solution to all given equations.
In other words, this means that any solutions to the first equation will be solutions to all equations united by the system.
Definition 1
Systems of equations is a number of equations, united by a curly bracket, having many solutions of equations that are simultaneously solutions for the entire system.
Main types of systems of equations
There are quite a lot of types of equations, like systems of equations. In order to make it convenient to solve and study them, they are divided into groups according to certain characteristics. This will help in considering systems of equations of certain types.
To begin with, the equations are classified by the number of equations. If there is one equation, then it is an ordinary equation, if there are more of them, then we are dealing with a system consisting of two or more equations.
Another classification affects the number of variables. When the number of variables is 1, we say that we are dealing with a system of equations with one unknown, when 2 - with two variables. Consider an example
x + y = 5 , 2 x - 3 y = 1
Obviously, the system of equations includes two variables x and y.
When writing such equations, the number of all variables in the record is considered. Their presence in each equation is optional. At least one equation must have one variable. Consider an example of a system of equations
2 x \u003d 11, x - 3 z 2 \u003d 0, 2 7 x + y - z \u003d - 3
This system has 3 variables x, y, z. The first equation has explicit x and implicit y and z. Implicit variables are variables that have 0 in the coefficient. The second equation has x and z and y is an implicit variable. Otherwise it can be written like this
2 x + 0 y + 0 z = 11
And the other equation is x + 0 · y − 3 · z = 0 .
The third classification of equations is the form. The school teaches simple equations and systems of equations, starting with systems of two linear equations with two variables . It means that the system includes 2 linear equations. For example, consider
2 x - y = 1 , x + 2 y = - 1 and - 3 x + y = 0 . 5 , x + 2 2 3 y = 0
These are basic simple linear equations. Further, you can encounter systems containing 3 or more unknowns.
In the 9th grade, they solve equations with two variables and non-linear ones. In integer equations, the exponent is increased to increase the complexity. Such systems are called systems of nonlinear equations with a certain number of equations and unknowns. Consider examples of such systems
x 2 - 4 x y = 1 , x - y = 2 and x = y 3 x y = - 5
Both systems are two-variable and both are non-linear.
When solving, you can meet fractional rational equations. For example
x + y = 3 , 1 x + 1 y = 2 5
They can simply call it a system of equations without specifying which ones. Rarely specify the type of system itself.
Senior classes move on to the study of irrational, trigonometric and exponential equations. For example,
x + y - x y = 5 , 2 x y = 3 , x + y = 5 π 2 , sin x + cos 2 y = - 1 , y - log 3 x = 1 , x y = 3 12 .
Higher educational institutions study and research solutions to systems of linear algebraic equations (SLAE). The left side of such equations contains polynomials with the first degree, and the right side contains some numbers. The difference from school ones is that the number of variables and the number of equations can be arbitrary, most often not the same.
Solving systems of equations
Definition 2Solving a system of equations with two variables is a pair of variables that, when substituted, turns each equation into a true numerical inequality, that is, it is a solution for each equation of this system.
For example, a pair of values \u200b\u200bx \u003d 5 and y \u003d 2 are the solution to the system of equations x + y \u003d 7, x - y \u003d 3. Because when substituting, the equations turn into true numerical inequalities 5 + 2 = 7 and 5 − 2 = 3. If we substitute the pair x = 3 and y = 0, then the system will not be solved, since the substitution will not give the correct equation, namely, we will get 3 + 0 = 7.
Let us formulate a definition for systems containing one or more variables.
Definition 3
Solving a system of equations with one variable- this is the value of the variable, which is the root of the equations of the system, which means that all equations will be converted into true numerical equalities.
Consider the example of a system of equations with one variable t
t 2 \u003d 4, 5 (t + 2) \u003d 0
The number - 2 is a solution to the equation, since (− 2) · 2 = 4 , and 5 · (− 2 + 2) = 0 are correct numerical equalities. For t = 1, the system is not solved, since when substituting, we get two incorrect equalities 12 = 4 and 5 · (1 + 2) = 0 .
Definition 4
Solving a system with three or more variables call a triple, a quadruple and further values, respectively, which turn all the equations of the system into true equalities.
If we have the values of the variables x = 1, y = 2, z = 0, then substituting them into the system of equations 2 x = 2, 5 y = 10, x + y + z = 3, we get 2 1 = 2, 5 2 = 10 and 1 + 2 + 0 = 3. So these numerical inequalities are true. And the values (1 , 0 , 5) will not be a solution, since, by substituting the values, the second of them will be wrong, as well as the third: 5 0 = 10 , 1 + 0 + 5 = 3 .
Systems of equations may have no solutions at all or have an infinite set. This can be seen with an in-depth study of this topic. It can be concluded that the system of equations is the intersection of the sets of solutions of all its equations. Let's break down a few definitions:
Definition 5
incompatible a system of equations is called when it has no solutions, otherwise it is called joint.
Definition 6
Uncertain a system is called when it has an infinite number of solutions, and certain with a finite number of solutions or in their absence.
Such terms are rarely used at school, as they are calculated for higher education programs. educational institutions. Acquaintance with equivalent systems will deepen the existing knowledge on solving systems of equations.
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Systems of linear equations.A system of equations is called linear if all the equations in the system are linear. It is customary to write a system of equations using curly brackets, for example:
Definition:A pair of values of variables that turns into a true equality each equation with two variables included in the system is called solution of a system of equations.
Solve the system means to find all its solutions or to prove that there are no solutions.
When solving a system of linear equations, the following three cases are possible:
the system has no solutions;
the system has exactly one solution;
The system has infinitely many solutions.
I . Solving a system of linear equations by the substitution method.
This method can also be called the "substitution method" or the method of eliminating unknowns.
Here we have a system of two equations with two unknowns. Note that the free terms (numbers -5 and -7) are located on the left side of the equation. We write the system in the usual form.
Do not forget that when transferring a term from part to part, you need to change its sign.
What does it mean to solve a system of linear equations? To solve a system of equations means to find such values of variables that turn each equation of the system into a true equality. This statement is true for any systems of equations with any number of unknowns.
We decide.
From the first equation of the system we express:
. This is the substitution. The resulting expression is substituted into the second equation of the system instead of the variable
Let's solve this equation for one variable.
We open the brackets, give like terms and find the value 
:
4) Next, we return to the substitution
to calculate the value 
.We already know the value, it remains to find:
5) Couple 
is the only solution to the given system.
Answer: (2.4; 2.2).
After any system of equations has been solved in any way, I strongly recommend that you check it on a draft. This is done easily and quickly.
1) Substitute the found answer in the first equation: 
- the correct equality is obtained.
2) We substitute the found answer into the second equation: 
- the correct equality is obtained.
The considered method of solution is not the only one; from the first equation it was possible to express , but not .
You can vice versa - express something from the second equation and substitute it into the first equation. However, it is necessary to evaluate the substitution so that it contains as few fractional expressions as possible. The most disadvantageous of the four ways is to express from the second or from the first equation:
or 
However, in some cases, fractions are still indispensable. Any task should strive to perform in the most rational way. This saves time and also reduces the chance of making a mistake.
Example 2
Solve a system of linear equations
II. Solution of the system by the method algebraic addition(subtraction) of system equations
In the course of solving systems of linear equations, one can use not the substitution method, but the method of algebraic addition (subtraction) of the equations of the system. This method saves time and simplifies calculations, however, now it will become more and more clear.
Solve the system of linear equations:
Let's take the same system as the first example.
1) Analyzing the system of equations, we notice that the coefficients of the variable y are identical in absolute value and opposite in sign (–1 and 1). In this situation, the equations can be added term by term:
2) Let's solve this equation for one variable.
As you can see, as a result of termwise addition, we have lost the variable . This, in fact, is the essence of the method - to get rid of one of the variables.
3) Now everything is simple: 
- substitute into the first equation of the system (you can also into the second):
AT finishing the solution should look something like this:
Answer: (2.4; 2.2).
Example 4
Solve the system of linear equations:
In this example, you can use the substitution method, but the big minus is that when we express any variable from any equation, we will get a solution in common fractions. Few people like actions with fractions, which means it is a waste of time, and there is a high probability of making a mistake.
Therefore, it is advisable to use term-by-term addition (subtraction) of equations. We analyze the coefficients for the corresponding variables:
As you can see, the numbers in pairs (14 and 7), (-9 and -2) are different, therefore, if we add (subtract) the equations right now, we will not get rid of the variable. Thus, I would like to see in one of the pairs the same modulo numbers, for example, 14 and -14 or 18 and -18.
We will consider the coefficients of the variable .
14x - 9y \u003d 24;
7x - 2y \u003d 17.
We select a number that would be divisible by both 14 and 7, and it should be as small as possible. In mathematics, such a number is called the least common multiple. If you are at a loss with the selection, then you can simply multiply the coefficients.
We multiply the second equation by 14: 7 \u003d 2.
As a result:
Now subtract the second from the first equation term by term.
It should be noted that it would be the other way around - subtract the first from the second equation, this does not change anything.
Now we substitute the found value into one of the equations of the system, for example, into the first one: 
Answer: (3:2)
Let's solve the system in a different way. Consider the coefficients for the variable .
14x - 9y \u003d 24;
7x - 2y \u003d 17.
Obviously, instead of a pair of coefficients (-9 and -3), we need to get 18 and -18.
To do this, multiply the first equation by (-2), multiply the second equation by 9:
We add the equations term by term and find the values of the variables:
Now we substitute the found value of x into one of the equations of the system, for example, into the first one:
Answer: (3:2)
The second method is somewhat more rational than the first, since adding is easier and more pleasant than subtracting. Most often, when solving systems, they tend to add and multiply, rather than subtract and divide.
Example 5
Solve the system of linear equations:
This is an example for an independent solution (answer at the end of the lecture).
Example 6
Solve a system of equations
Solution. The system has no solutions, since two equations of the system cannot be satisfied simultaneously (from the first equation 
and from the second 
Answer: There are no solutions.
Example 7
solve the system of equations
Solution. The system has infinitely many solutions, since the second equation is obtained from the first by multiplying by 2 (i.e., in fact, there is only one equation with two unknowns).
Answer: Infinitely many solutions.
III. Solution of the system using matrices.
The determinant of this system is the determinant composed of the coefficients of the unknowns. This determinant
Systems of equations received wide application in economic industry in mathematical modeling of various processes. For example, when solving problems of production management and planning, 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, when solving problems of finding the population size.
A system of linear equations is a term for 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 the sequence does not exist.
Linear Equation
Equations of the form ax+by=c are called linear. The designations x, y are the unknowns, the value of which must be found, b, a are the coefficients of the variables, c is the free term of the equation.
Solving the equation by plotting its graph will look like a straight line, all points of which are the solution of the polynomial.
Types of systems of linear equations
The simplest are examples of 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.
Solve a system of equations - it means to find such values (x, y) at which the system turns into a true equality or establish that suitable values x and y do not exist.
A pair of values (x, y), written as point coordinates, is called a solution to a system of linear equations.
If the systems have one common solution or there is no solution, they are called equivalent.
Homogeneous systems of linear equations are systems whose right side is equal to zero. If the right part after the "equal" sign has a value or is expressed by a function, such a system is not homogeneous.
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, schoolchildren assume 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 the variables, there can be an arbitrarily large number of them.
Simple and complex methods for solving systems of equations
There is no general analytical way to solve such systems, all methods are based on numerical solutions. The school mathematics course describes in detail such methods as permutation, algebraic addition, substitution, as well as the graphical and matrix method, the solution by the Gauss method.
The main task in teaching methods of solving is to teach how to correctly analyze the system and find the optimal solution algorithm for each example. The main thing is not to memorize a system of rules and actions for each method, but to understand the principles of applying a particular method.
The solution of examples of systems of linear equations of the 7th grade of the general education school program is quite simple and is explained in great detail. In any textbook on mathematics, this section is given enough attention. The solution of examples of systems of linear equations by the method of Gauss and Cramer is studied in more detail in the first courses of higher educational institutions.
Solution of systems by the substitution method
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 reduced to a single variable form. The action is repeated depending on the number of unknowns in the system
Let's give an example of a system of linear equations of the 7th class by the substitution method:
As can be seen from the example, the variable x was expressed through F(X) = 7 + Y. The resulting expression, substituted into the 2nd equation of the system in place of X, helped to obtain one variable Y in the 2nd equation. Solution this example does not cause difficulties and allows you to get the Y value. The last step is to check the received values.
It is not always possible to solve an example of a system of linear equations by substitution. The equations can be complex and the expression of the variable in terms of the second unknown will be too cumbersome for further calculations. When there are more than 3 unknowns in the system, the 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. ultimate goal mathematical operations is an equation with one variable.
For applications this method it takes practice and observation. It is not easy to solve a system of linear equations using the addition method with the number of variables 3 or more. Algebraic addition is useful when the equations contain fractions and decimal numbers.
Solution action algorithm:
- Multiply both sides of the equation by some number. As a result of the arithmetic operation, one of the coefficients of the variable must become equal to 1.
- Add the resulting expression term by term and find one of the unknowns.
- Substitute the resulting value into the 2nd equation of the system to find the remaining variable.
Solution method by introducing 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 of the equations by introducing a new variable. The new equation is solved with respect to the entered unknown, and the resulting value is used to determine the original variable.
It can be seen from the example that by introducing a new variable t, it was possible to reduce the 1st equation of the system to a standard square trinomial. You can solve a polynomial by finding the discriminant.
It is necessary to find the value of the discriminant using the well-known formula: D = b2 - 4*a*c, where D is the desired discriminant, b, a, c are the multipliers of the polynomial. In the given example, a=1, b=16, c=39, hence D=100. If the discriminant is greater than zero, then there are two solutions: t = -b±√D / 2*a, if the discriminant less than zero, then there is only one solution: x= -b / 2*a.
The solution for the resulting systems is found by the addition method.
A visual method for solving systems
Suitable for systems with 3 equations. The method is to build on coordinate axis graphs of each equation included in the system. The coordinates of the points of intersection of the curves and will be common solution systems.
The graphic method has a number of nuances. Consider several examples of solving systems of linear equations in a visual way.
As can be seen from the example, two points were constructed for each line, 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. Points with coordinates (0, 3) and (3, 0) were marked on the graph and connected by a line.
The steps 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 to the 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.
The systems from Examples 2 and 3 are similar, but when constructed, 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 graph.
Matrix and its varieties
Matrices are used to briefly write down a system of linear equations. A table is called a matrix. special kind filled with numbers. n*m has n - rows and m - columns.
A matrix is square when the number of columns and rows is equal. A matrix - a vector is a matrix of one column with infinitely possible number lines. A matrix with units along one of the diagonals and other zero elements is called identity.
An inverse matrix is such a matrix, when multiplied by which the original one turns into a unit one, such a matrix exists only for the original square one.
Rules for transforming a system of equations into a matrix
With regard to systems of equations, the coefficients and free members of the equations are written as numbers of the matrix, one equation is one row of the matrix.
A matrix row is called non-zero if at least one element of the row is not equal to zero. Therefore, if in any of the equations the number of variables differs, then it is necessary to enter zero in 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 multiplying a matrix, all matrix elements 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| must not be equal to zero, then the system has a solution.
The determinant is easily calculated for a two-by-two matrix, it is only necessary 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, or you can remember that you need to take one element from each row and each column so that the column and row numbers of the elements do not repeat in the product.
Solution of examples of systems of linear equations by the matrix method
The matrix method of finding a solution makes it possible to reduce cumbersome notations when solving systems with large quantity variables and equations.
In the example, a nm are the coefficients of the equations, the matrix is a vector x n are the variables, and b n are the free terms.
Solution of systems by the Gauss method
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 method of solving. These methods are used to find system variables with a lot of linear equations.
The Gaussian method is very similar to substitution and algebraic addition solutions, but is more systematic. In the school course, the Gaussian solution is used for 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, and 3 and 4 - with 3 and 4 variables, respectively.
After bringing the system to the described form, the further solution is reduced to the sequential substitution of known variables into the equations of the system.
In school textbooks for grade 7, an example of a Gaussian solution is described as follows:
As can be seen from the example, at step (3) two equations were obtained 3x 3 -2x 4 =11 and 3x 3 +2x 4 =7. The solution of any of the equations will allow you to find out one of the variables x n.
Theorem 5, which is mentioned in the text, says that if one of the equations of the system is replaced by an equivalent one, then the resulting system will also be equivalent to the original one.
The Gauss method is difficult for students to understand high school, but is one of the most interesting ways to develop the ingenuity of children enrolled in the advanced study program in math and physics classes.
For ease of recording calculations, it is customary to do the following:
Equation coefficients 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. separates the left side of the equation from the right side. Roman numerals denote the numbers of equations in the system.
First, they write down the matrix with which to work, then all the actions carried out with one of the rows. The resulting matrix is written after the "arrow" sign and continue to perform the necessary algebraic operations until the result is achieved.
As a result, a matrix should be obtained in which one of the diagonals is 1, and all other coefficients are equal to zero, that is, the matrix is reduced to a single form. We must not forget to make calculations with the numbers of both sides of the equation.
This notation is less cumbersome and allows you not to be distracted by listing numerous unknowns.
The free application of any method of solution will require care and a certain amount of experience. Not all methods are applied. Some ways of finding solutions are more preferable in a particular area of human activity, while others exist for the purpose of learning.
With this mathematical program, you can solve a system of two linear equations with two variables using the substitution method and the addition method.
The program not only gives the answer to the problem, but also leads detailed solution with explanations of the solution steps in two ways: the substitution method and the addition method.
This program May be useful for high school students general education schools in preparation for control work and exams, when testing knowledge before the exam, parents to control the solution of many problems in mathematics and algebra. Or maybe it's too expensive for you to hire a tutor or buy new textbooks? Or do you just want to get it done as soon as possible? homework math or algebra? In this case, you can also use our programs with a detailed solution.
In this way, you can conduct your own training and/or the training of your younger brothers or sisters, while the level of education in the field of tasks to be solved is increased.
Rules for Entering Equations
Any Latin letter can act as a variable.
For example: \(x, y, z, a, b, c, o, p, q \) etc.
When entering equations you can use brackets. In this case, the equations are first simplified. The equations after simplifications must be linear, i.e. of the form ax+by+c=0 with the accuracy of the order of the elements.
For example: 6x+1 = 5(x+y)+2
In equations, you can use not only integers, but also fractional numbers in the form of decimal and ordinary fractions.
Rules for entering decimal fractions.
Integer and fractional part decimal fractions can be separated by either a dot or a comma.
For example: 2.1n + 3.5m = 55
Rules for entering ordinary fractions.
Only a whole number can act as the numerator, denominator and integer part of a fraction.
The denominator cannot be negative.
When you enter numeric fraction The numerator is separated from the denominator by a division sign: /
The integer part is separated from the fraction by an ampersand: &
Examples.
-1&2/3y + 5/3x = 55
2.1p + 55 = -2/7(3.5p - 2&1/8q)
Solve a system of equations
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A bit of theory.
Solving systems of linear equations. Substitution method
The sequence of actions when solving a system of linear equations by the substitution method:
1) express one variable from some equation of the system in terms of another;
2) substitute the resulting expression in another equation of the system instead of this variable;
$$ \left\( \begin(array)(l) 3x+y=7 \\ -5x+2y=3 \end(array) \right. $$
Let's express from the first equation y through x: y = 7-3x. Substituting the expression 7-3x instead of y into the second equation, we get the system:
$$ \left\( \begin(array)(l) y = 7-3x \\ -5x+2(7-3x)=3 \end(array) \right. $$
It is easy to show that the first and second systems have the same solutions. In the second system, the second equation contains only one variable. Let's solve this equation:
$$ -5x+2(7-3x)=3 \Rightarrow -5x+14-6x=3 \Rightarrow -11x=-11 \Rightarrow x=1 $$
Substituting the number 1 instead of x into the equation y=7-3x, we find the corresponding value of y:
$$ y=7-3 \cdot 1 \Rightarrow y=4 $$
Pair (1;4) - solution of the system
Systems of equations in two variables that have the same solutions are called equivalent. Systems that do not have solutions are also considered equivalent.
Solving systems of linear equations by adding
Consider another way to solve systems of linear equations - the addition method. When solving systems in this way, as well as when solving by the substitution method, we pass from a given system to another system equivalent to it, in which one of the equations contains only one variable.
The sequence of actions when solving a system of linear equations by the addition method:
1) multiply the equations of the system term by term, choosing the factors so that the coefficients for one of the variables become opposite numbers;
2) add term by term the left and right parts of the equations of the system;
3) solve the resulting equation with one variable;
4) find the corresponding value of the second variable.
Example. Let's solve the system of equations:
$$ \left\( \begin(array)(l) 2x+3y=-5 \\ x-3y=38 \end(array) \right. $$
In the equations of this system, the coefficients of y are opposite numbers. Adding term by term the left and right parts of the equations, we obtain an equation with one variable 3x=33. Let's replace one of the equations of the system, for example the first one, with the equation 3x=33. Let's get the system
$$ \left\( \begin(array)(l) 3x=33 \\ x-3y=38 \end(array) \right. $$
From the equation 3x=33 we find that x=11. Substituting this x value into the equation \(x-3y=38 \) we get an equation with the variable y: \(11-3y=38 \). Let's solve this equation:
\(-3y=27 \Rightarrow y=-9 \)
Thus, we found the solution to the system of equations by adding: \(x=11; y=-9 \) or \((11; -9) \)
Taking advantage of the fact that in the equations of the system the coefficients of y are opposite numbers, we reduced its solution to the solution of an equivalent system (by summing both parts of each of the equations of the original symmeme), in which one of the equations contains only one variable.
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