Gauss method infinite set of solutions example. Gauss method for dummies: examples of solutions

Carl Friedrich Gauss, the greatest mathematician for a long time hesitated between philosophy and mathematics. Perhaps it was precisely such a mindset that allowed him to "leave" so noticeably in world science. In particular, by creating the "Gauss Method" ...

For almost 4 years, the articles of this site have dealt with school education, mainly from the side of philosophy, the principles of (mis)understanding, introduced into the minds of children. The time is coming for more specifics, examples and methods... I believe that this is exactly the approach to the familiar, confusing and important areas of life gives the best results.

We humans are so arranged that no matter how much you talk about abstract thinking, But understanding Always happens through examples. If there are no examples, then it is impossible to grasp the principles... Just as it is impossible to get to the top of a mountain except by walking the entire slope from the foot.

Same with school: for now living stories It is not enough that we instinctively continue to regard it as a place where children are taught to understand.

For example, teaching the Gauss method...

Gauss method in the 5th grade of the school

I’ll make a reservation right away: the Gauss method has a much wider application, for example, when solving systems of linear equations. What we are going to talk about takes place in the 5th grade. This start, having understood which, it is much easier to understand more "advanced options". In this article we are talking about method (method) of Gauss when finding the sum of a series

Here is an example that I brought from school younger son attending the 5th grade of the Moscow gymnasium.

School demonstration of the Gauss method

Math teacher using interactive whiteboard (modern methods training) showed the children a presentation of the history of the “creation of the method” by little Gauss.

The school teacher whipped little Karl (an outdated method, not used in schools these days) because he

instead of sequentially adding numbers from 1 to 100 to find their sum noticed that pairs of numbers equally spaced from the edges of an arithmetic progression add up to the same number. for example, 100 and 1, 99 and 2. Having counted the number of such pairs, little Gauss almost instantly solved the problem proposed by the teacher. For which he was executed in front of an astonished public. So that others would be discouraged from thinking.

What did little Gauss do? developed number sense? Noticed some feature number series with a constant step (arithmetic progression). AND exactly this later made him a great scientist, those who know how to notice, having feeling, instinct of understanding.

This is why mathematics is valuable, developing ability to see general in particular - abstract thinking. Therefore, most parents and employers instinctively consider mathematics an important discipline ...

“Then you need to learn mathematics, because it puts your mind in order.
M.V.Lomonosov".

However, the followers of those who flogged future geniuses with rods turned the Method into something the opposite. As my supervisor said 35 years ago: “The question has been learned.” Or as my youngest son said yesterday about Gauss’s method: “Maybe it’s not worth making a big science out of this, huh?”

The consequences of the creativity of the “scientists” are visible in the level of current school mathematics, the level of its teaching and the understanding of the “Queen of Sciences” by the majority.

However, let's continue...

Methods for explaining the Gauss method in 5th grade school

A mathematics teacher at a Moscow gymnasium, explaining the Gauss method according to Vilenkin, complicated the task.

What if the difference (step) of an arithmetic progression is not one, but another number? For example, 20.

The problem he gave to the fifth graders:

20+40+60+80+ ... +460+480+500

Before getting acquainted with the gymnasium method, let’s take a look at the Internet: how do school teachers and math tutors do it?..

Gaussian method: explanation No. 1

A well-known tutor on his YOUTUBE channel gives the following reasoning:

"Let's write the numbers from 1 to 100 as follows:

first a series of numbers from 1 to 50, and strictly below it another series of numbers from 50 to 100, but in the reverse order"

1, 2, 3, ... 48, 49, 50

100, 99, 98 ... 53, 52, 51

"Please note: the sum of each pair of numbers from the top and bottom rows is the same and equals 101! Let's count the number of pairs, it is 50 and multiply the sum of one pair by the number of pairs! Voila: The answer is ready!"

“If you couldn’t understand, don’t be upset!” the teacher repeated three times during the explanation. "You will take this method in 9th grade!"

Gaussian method: explanation No. 2

Another tutor, less well-known (judging by the number of views), takes a more scientific approach, offering a solution algorithm of 5 points that must be completed sequentially.

For the uninitiated, 5 is one of the Fibonacci numbers traditionally considered magical. A 5 step method is always more scientific than a 6 step method, for example. ...And this is hardly an accident, most likely, the Author is a hidden adherent of the Fibonacci theory

Given an arithmetic progression: 4, 10, 16 ... 244, 250, 256 .

Algorithm for finding the sum of numbers in a series using the Gauss method:

4, 10, 16 ... 244, 250, 256

256, 250, 244 ... 16, 10, 4

At the same time, you need to remember plus one rule : we must add one to the resulting quotient: otherwise we will get a result that is less by one than the true number of pairs: 42 + 1 = 43.

This is the required sum of the arithmetic progression from 4 to 256 with a difference of 6!

Gauss method: explanation in 5th grade at a Moscow gymnasium

Here's how to solve the problem of finding the sum of a series:

20+40+60+ ... +460+480+500

in the 5th grade of a Moscow gymnasium, Vilenkin’s textbook (according to my son).

After showing the presentation, the math teacher showed a couple of examples using the Gaussian method and gave the class a task of finding the sum of the numbers in a series in increments of 20.

This required the following:

As you can see, this is more compact and effective technique: number 3 is also a member of the Fibonacci sequence

My comments on the school version of the Gauss method

The great mathematician would definitely have chosen philosophy if he had foreseen what his “method” would be turned into by his followers German teacher, who flogged Karl with rods. He would have seen the symbolism, the dialectical spiral and the undying stupidity of the “teachers”, trying to measure the harmony of living mathematical thought with the algebra of misunderstanding ....

By the way: did you know. that our education system is rooted in the German school of the 18th and 19th centuries?

But Gauss chose mathematics.

What is the essence of his method?

IN simplification. IN observing and grasping simple patterns of numbers. IN turning dry school arithmetic into interesting and exciting activity , activating in the brain the desire to continue, rather than blocking high-cost mental activity.

Is it possible to use one of the given “modifications of Gauss’s method” to calculate the sum of the numbers of an arithmetic progression almost instantly? According to the “algorithms”, little Karl would be guaranteed to avoid spanking, develop an aversion to mathematics and suppress his creative impulses in the bud.

Why did the tutor so persistently advise fifth-graders “not to be afraid of misunderstanding” of the method, convincing them that they would solve “such” problems as early as 9th grade? Psychologically illiterate action. It was a good move to note: "See? You already in 5th grade you can solve problems that you will complete only in 4 years! What a great fellow you are!”

To use the Gaussian method, a level of class 3 is sufficient, when normal children already know how to add, multiply and divide 2-3 digit numbers. Problems arise due to the inability of adult teachers who are “out of touch” to explain the simplest things in normal human language, not to mention mathematical... They are unable to get people interested in mathematics and completely discourage even those who are “capable.”

Or, as my son commented: “making a big science out of it.”

Gauss method, my explanations

My wife and I explained this “method” to our child, it seems, even before school...

Simplicity instead of complexity or a game of questions and answers

"Look, here are the numbers from 1 to 100. What do you see?"

The point is not what exactly the child sees. The trick is to get him to look.

"How can you put them together?" The son realized that such questions are not asked “just like that” and you need to look at the question “somehow differently, differently than he usually does”

It doesn't matter if the child sees the solution right away, it's unlikely. It is important that he stopped being afraid to look, or as I say: “moved the task”. This is the beginning of the journey to understanding

“Which is easier: adding, for example, 5 and 6 or 5 and 95?” A leading question... But any training comes down to “guiding” a person to the “answer” - in any way acceptable to him.

At this stage, guesses may already arise about how to “save” on calculations.

All we did was hint: the “frontal, linear” method of counting is not the only possible one. If a child understands this, then later he will come up with many more such methods, because it's interesting!!! And he will definitely avoid “misunderstanding” mathematics and will not feel disgusted with it. He got the win!

If child discovered that adding pairs of numbers that add up to a hundred is a piece of cake, then "arithmetic progression with difference 1"- a rather dreary and uninteresting thing for a child - suddenly found life for him . Order emerged from chaos, and this always causes enthusiasm: that's how we are made!

A question to answer: why, after the insight a child has received, should he again be forced into the framework of dry algorithms, which are also functionally useless in this case?!

Why force stupid rewrites? sequence numbers in a notebook: so that even the capable do not have a single chance of understanding? Statistically, of course, but mass education is geared towards “statistics”...

Where did the zero go?

And yet, adding numbers that add up to 100 is much more acceptable to the mind than those that add up to 101...

The "Gauss School Method" requires exactly this: mindlessly fold pairs of numbers equidistant from the center of the progression, Despite everything.

What if you look?

Still zero - greatest invention humanity, which is more than 2,000 years old. And math teachers continue to ignore him.

It is much easier to transform a series of numbers starting with 1 into a series starting with 0. The sum will not change, will it? You need to stop “thinking in textbooks” and start looking... And see that pairs with a sum of 101 can be completely replaced by pairs with a sum of 100!

0 + 100, 1 + 99, 2 + 98 ... 49 + 51

How to abolish the "plus 1 rule"?

To be honest, I first heard about such a rule from that YouTube tutor...

What do I still do when I need to determine the number of members of a series?

I look at the sequence:

1, 2, 3, .. 8, 9, 10

and when you’re completely tired, then move on to a simpler row:

1, 2, 3, 4, 5

and I figure: if you subtract one from 5, you get 4, but I’m absolutely clear I see 5 numbers! Therefore, you need to add one! Number sense developed in primary school, suggests: even if there are a whole Google of members of the series (10 to the hundredth power), the pattern will remain the same.

What the hell are the rules?..

So that in a couple or three years you can fill all the space between your forehead and the back of your head and stop thinking? How to earn your bread and butter? After all, we are moving in even ranks into the era of the digital economy!

More about Gauss’s school method: “why make science out of this?..”

It was not for nothing that I posted a screenshot from my son’s notebook...

"What happened in class?"

“Well, I counted right away, raised my hand, but she didn’t ask. Therefore, while the others were counting, I began to do homework in Russian so as not to waste time. Then, when the others finished writing (???), she called me to the board. I said the answer."

“That’s right, show me how you solved it,” said the teacher. I showed it. She said: “Wrong, you need to count as I showed!”

“It’s good that she didn’t give a bad grade. And she made me write in their notebook “the course of the solution” in their own way. Why make a big science out of this?..”

The main crime of a math teacher

Hardly after that incident Carl Gauss experienced a high sense of respect for his school mathematics teacher. But if he knew how followers of that teacher will distort the very essence of the method...he would roar with indignation and through world organization intellectual property WIPO has achieved a ban on the use of its good name in school textbooks!..

In what main mistake school approach? Or, as I put it, a crime of school mathematics teachers against children?

Algorithm of misunderstanding

What do school methodologists do, the vast majority of whom don’t know how to think?

They create methods and algorithms (see). This a defensive reaction that protects teachers from criticism (“Everything is done according to...”) and children from understanding. And thus - from the desire to criticize teachers!(The second derivative of bureaucratic “wisdom”, a scientific approach to the problem). A person who does not grasp the meaning will rather blame his own misunderstanding, rather than the stupidity of the school system.

This is what happens: parents blame their children, and teachers... do the same for children who “don’t understand mathematics!”

Are you smart?

What did little Karl do?

A completely unconventional approach to a formulaic task. This is the essence of His approach. This the main thing that should be taught in school is to think not with textbooks, but with your head. Of course, there is also an instrumental component that can be used ... in search of simpler and effective methods accounts.

Gauss method according to Vilenkin

In school they teach that Gauss's method is to

What, if the number of elements of the series is odd, as in the problem that was assigned to my son?..

The "catch" is that in this case you should find an “extra” number in the series and add it to the sum of the pairs. In our example this number is 260.

How to detect? Copying all pairs of numbers into a notebook!(This is why the teacher made the kids do this stupid job of trying to teach "creativity" using the Gaussian method... And this is why such a "method" is practically inapplicable to large data series, AND this is why it is not the Gaussian method.)

A little creativity in the school routine...

The son acted differently.

(20 + 500, 40 + 480 ...).

0+500, 20+480, 40+460 ...

Not difficult, right?

But in practice it is made even easier, which allows you to carve out 2-3 minutes for remote sensing in Russian, while the rest are “counting”. In addition, it retains the number of steps of the method: 5, which does not allow the approach to be criticized for being unscientific.

Obviously this approach is simpler, faster and more universal, in the style of the Method. But... the teacher not only did not praise, but also forced me to rewrite it “in the correct way” (see screenshot). That is, she made a desperate attempt to stifle the creative impulse and the ability to understand mathematics at the root! Apparently, so that she could later be hired as a tutor... She attacked the wrong person...

Everything that I described so long and tediously can be explained normal child in half an hour maximum. Along with examples.

And in such a way that he will never forget it.

And it will step towards understanding...not just mathematicians.

Admit it: how many times in your life have you added using the Gaussian method? And I never did!

But instinct of understanding, which develops (or is extinguished) in the process of learning mathematical methods at school... Oh!.. This is truly an irreplaceable thing!

Especially in the age of universal digitalization, which we have quietly entered under the strict leadership of the Party and the Government.

A few words in defense of teachers...

It is unfair and wrong to place all responsibility for this style of teaching solely on school teachers. The system is in effect.

Some teachers understand the absurdity of what is happening, but what to do? Law on Education, Federal State Educational Standards, methods, technological maps lessons... Everything must be done “in accordance with and on the basis of” and everything must be documented. Step aside - stood in line to be fired. Let’s not be hypocrites: the salaries of Moscow teachers are very good... If they fire you, where to go?..

Therefore this site not about education. He is about individual education, only possible way get out of the crowd generation Z ...

Gauss method perfect for solving systems of linear algebraic equations (SLAEs). It has a number of advantages compared to other methods:

• firstly, there is no need to first examine the system of equations for consistency;
• secondly, the Gauss method can solve not only SLAEs in which the number of equations coincides with the number of unknown variables and the main matrix of the system is non-singular, but also systems of equations in which the number of equations does not coincide with the number of unknown variables or the determinant of the main matrix is ​​equal to zero;
• thirdly, the Gaussian method leads to results with a relatively small number of computational operations.

Brief overview of the article.

First, we give the necessary definitions and introduce notations.

Next, we will describe the algorithm of the Gauss method for the simplest case, that is, for systems of linear algebraic equations, the number of equations in which coincides with the number of unknown variables and the determinant of the main matrix of the system is not equal to zero. When solving such systems of equations, the essence of the Gauss method is most clearly visible, which is the sequential elimination of unknown variables. Therefore, the Gauss method is also called the method sequential elimination unknown. We will show detailed solutions of several examples.

In conclusion, we will consider the solution by the Gauss method of systems of linear algebraic equations, the main matrix of which is either rectangular or singular. The solution to such systems has some features, which we will examine in detail using examples.

Page navigation.

Basic definitions and notations.

Consider a system of p linear equations with n unknowns (p can be equal to n):

Where are unknown variables, are numbers (real or complex), and are free terms.

If , then the system of linear algebraic equations is called homogeneous, otherwise - heterogeneous.

The set of values ​​of unknown variables for which all equations of the system become identities is called decision of the SLAU.

If there is at least one solution to a system of linear algebraic equations, then it is called joint, otherwise - incompatible.

If a SLAE has a unique solution, then it is called certain. If there is more than one solution, then the system is called uncertain.

They say that the system is written in coordinate form, if it has the form
.

This system in matrix form records has the form , where - the main matrix of SLAE, - the matrix of the column of unknown variables, - the matrix of free members.

If we add to the matrix A as the (n + 1)-th column the matrix-column of free terms, then we get the so-called expanded matrix systems of linear equations. Usually, the augmented matrix is ​​denoted by the letter T, and the column of free members is separated by a vertical line from the rest of the columns, that is,

The square matrix A is called degenerate, if its determinant is zero. If , then matrix A is called non-degenerate.

The following point should be noted.

If the following actions are performed with a system of linear algebraic equations

• swap two equations,
• multiply both sides of any equation by an arbitrary and non-zero real (or complex) number k,
• to both parts of any equation add the corresponding parts of the other equation, multiplied by an arbitrary number k,

then we get an equivalent system that has the same solutions (or, like the original one, has no solutions).

For an extended matrix of a system of linear algebraic equations, these actions will mean elementary transformations with rows:

• swapping two lines,
• multiplication of all elements of any row of the matrix T by a non-zero number k ,
• adding to the elements of any row of the matrix the corresponding elements of another row, multiplied by an arbitrary number k .

Now we can proceed to the description of the Gauss method.

Solving systems of linear algebraic equations, in which the number of equations is equal to the number of unknowns and the main matrix of the system is nondegenerate, by the Gauss method.

What would we do at school if we were given the task of finding a solution to a system of equations .

Some would do that.

Note that by adding the left side of the first equation to the left side of the second equation, and the right side to the right side, you can get rid of the unknown variables x 2 and x 3 and immediately find x 1:

We substitute the found value x 1 =1 into the first and third equations of the system:

If we multiply both parts of the third equation of the system by -1 and add them to the corresponding parts of the first equation, then we get rid of the unknown variable x 3 and can find x 2:

We substitute the obtained value x 2 \u003d 2 into the third equation and find the remaining unknown variable x 3:

Others would have done differently.

Let us resolve the first equation of the system with respect to the unknown variable x 1 and substitute the resulting expression into the second and third equations of the system in order to exclude this variable from them:

Now let’s solve the second equation of the system for x 2 and substitute the result obtained into the third equation to eliminate the unknown variable x 2 from it:

From the third equation of the system it is clear that x 3 =3. From the second equation we find , and from the first equation we get .

Familiar solutions, right?

The most interesting thing here is that the second solution method is essentially the method of sequential elimination of unknowns, that is, the Gaussian method. When we expressed the unknown variables (first x 1, at the next stage x 2) and substituted them into the remaining equations of the system, we thereby excluded them. We carried out elimination until there was only one unknown variable left in the last equation. The process of sequentially eliminating unknowns is called direct Gauss method. After completing the forward move, we have the opportunity to calculate the unknown variable found in the last equation. With its help, we find the next unknown variable from the penultimate equation, and so on. The process of sequentially finding unknown variables while moving from the last equation to the first is called in reverse Gauss method.

It should be noted that when we express x 1 in terms of x 2 and x 3 in the first equation, and then substitute the resulting expression into the second and third equations, the following actions lead to the same result:

Indeed, such a procedure also makes it possible to eliminate the unknown variable x 1 from the second and third equations of the system:

Nuances with the elimination of unknown variables using the Gaussian method arise when the equations of the system do not contain some variables.

For example, in SLAU in the first equation there is no unknown variable x 1 (in other words, the coefficient in front of it is zero). Therefore, we cannot solve the first equation of the system for x 1 in order to eliminate this unknown variable from the remaining equations. The way out of this situation is to swap the equations of the system. Since we are considering systems of linear equations whose determinants of the main matrices are different from zero, there is always an equation in which the variable we need is present, and we can rearrange this equation to the position we need. For our example, it is enough to swap the first and second equations of the system , then you can resolve the first equation for x 1 and exclude it from the remaining equations of the system (although x 1 is no longer present in the second equation).

We hope you get the gist.

Let's describe Gaussian method algorithm.

Suppose we need to solve a system of n linear algebraic equations with n unknown variables of the form , and let the determinant of its main matrix be different from zero.

We will assume that , since we can always achieve this by rearranging the equations of the system. We exclude the unknown variable x 1 from all equations of the system, starting from the second one. To do this, add the first equation multiplied by to the second equation of the system, add the first multiplied by to the third equation, and so on, add the first multiplied by to the nth equation. The system of equations after such transformations will take the form

where , a .

We would come to the same result if we expressed x 1 in terms of other unknown variables in the first equation of the system and substituted the resulting expression into all other equations. Thus, the variable x 1 is excluded from all equations, starting from the second.

Next, we act similarly, but only with a part of the resulting system, which is marked in the figure

To do this, add the second multiplied by to the third equation of the system, add the second multiplied by to the fourth equation, and so on, add the second multiplied by to the nth equation. The system of equations after such transformations will take the form

where , a . Thus, the variable x 2 is excluded from all equations, starting from the third.

Next, we proceed to the elimination of the unknown x 3, while acting similarly with the part of the system marked in the figure

So we continue the direct course of the Gauss method until the system takes the form

From this moment, we begin the reverse course of the Gauss method: we calculate x n from the last equation as , using the obtained value x n we find x n-1 from the penultimate equation, and so on, we find x 1 from the first equation.

Let's look at the algorithm using an example.

Example.

Gaussian method.

Solution.

The coefficient a 11 is different from zero, so let’s proceed to the direct progression of the Gaussian method, that is, to the exclusion of the unknown variable x 1 from all equations of the system except the first. To do this, to the left and right sides of the second, third and fourth equations, add the left and right sides of the first equation, multiplied by , respectively. And :

The unknown variable x 1 has been eliminated, let's move on to eliminating x 2 . To the left and right sides of the third and fourth equations of the system we add the left and right sides of the second equation, multiplied by respectively And :

To complete the forward progression of the Gaussian method, we need to eliminate the unknown variable x 3 from the last equation of the system. Let us add to the left and right sides of the fourth equation, respectively, the left and right sides of the third equation, multiplied by :

You can begin the reverse of the Gaussian method.

From the last equation we have ,
from the third equation we get,
from the second,
from the first one.

To check, you can substitute the obtained values ​​of the unknown variables into the original system of equations. All equations turn into identities, which indicates that the solution using the Gauss method was found correctly.

Answer:

Now let’s give a solution to the same example using the Gaussian method in matrix notation.

Example.

Find the solution to the system of equations Gaussian method.

Solution.

The extended matrix of the system has the form . At the top of each column are the unknown variables that correspond to the elements of the matrix.

The direct approach of the Gaussian method here involves reducing the extended matrix of the system to a trapezoidal form using elementary transformations. This process is similar to the elimination of unknown variables that we did with the system in coordinate form. Now you will see this.

Let's transform the matrix so that all elements in the first column, starting from the second, become zero. To do this, to the elements of the second, third and fourth lines we add the corresponding elements of the first line multiplied by , and accordingly:

Next, we transform the resulting matrix so that in the second column all elements, starting from the third, become zero. This would correspond to eliminating the unknown variable x 2 . To do this, to the elements of the third and fourth rows we add the corresponding elements of the first row of the matrix, multiplied by respectively And :

It remains to exclude the unknown variable x 3 from the last equation of the system. To do this, to the elements of the last row of the resulting matrix we add the corresponding elements of the penultimate row, multiplied by :

It should be noted that this matrix corresponds to a system of linear equations

which was obtained earlier after a forward move.

It's time to turn back. In matrix notation, the inverse of the Gaussian method involves transforming the resulting matrix such that the matrix marked in the figure

became diagonal, that is, took the form

where are some numbers.

These transformations are similar to the forward transformations of the Gaussian method, but are performed not from the first line to the last, but from the last to the first.

Add to the elements of the third, second and first lines the corresponding elements of the last line, multiplied by , on and on respectively:

Now add to the elements of the second and first lines the corresponding elements of the third line, multiplied by and by, respectively:

At the last step of the reverse Gaussian method, to the elements of the first row we add the corresponding elements of the second row, multiplied by:

The resulting matrix corresponds to the system of equations , from where we find the unknown variables.

Answer:

NOTE.

When using the Gauss method to solve systems of linear algebraic equations, approximate calculations should be avoided, as this can lead to completely incorrect results. We recommend not rounding decimals. Better off decimals move on to ordinary fractions.

Example.

Solve a system of three equations using the Gauss method .

Solution.

Note that in this example the unknown variables have a different designation (not x 1, x 2, x 3, but x, y, z). Let's move on to ordinary fractions:

Let us exclude the unknown x from the second and third equations of the system:

In the resulting system, the unknown variable y is absent in the second equation, but y is present in the third equation, therefore, let’s swap the second and third equations:

This completes the direct progression of the Gauss method (there is no need to exclude y from the third equation, since this unknown variable no longer exists).

Let's start the reverse move.

From the last equation we find ,
from the penultimate


from the first equation we have

Answer:

X = 10, y = 5, z = -20.

Solving systems of linear algebraic equations in which the number of equations does not coincide with the number of unknowns or the main matrix of the system is singular, using the Gauss method.

Systems of equations, the main matrix of which is rectangular or square singular, may have no solutions, may have a single solution, or may have an infinite number of solutions.

Now we will understand how the Gauss method allows us to establish the compatibility or inconsistency of a system of linear equations, and in the case of its compatibility, determine all solutions (or one single solution).

In principle, the process of eliminating unknown variables in the case of such SLAEs remains the same. However, it is worth going into detail about some situations that may arise.

Let's move on to the most important stage.

So, let us assume that the system of linear algebraic equations, after completing the forward progression of the Gauss method, takes the form and not a single equation was reduced to (in this case we would conclude that the system is incompatible). A logical question arises: “What to do next”?

Let us write down the unknown variables that come first in all equations of the resulting system:

In our example these are x 1, x 4 and x 5. On the left sides of the equations of the system we leave only those terms that contain the written unknown variables x 1, x 4 and x 5, the remaining terms are transferred to the right side of the equations with the opposite sign:

Let's give the unknown variables that are on the right sides of the equations arbitrary values, where - arbitrary numbers:

After this, the right-hand sides of all equations of our SLAE contain numbers and we can proceed to the reverse of the Gaussian method.

From the last equation of the system we have, from the penultimate equation we find, from the first equation we get

The solution to a system of equations is a set of values ​​of unknown variables

Giving Numbers different meanings, we will obtain different solutions to the system of equations. That is, our system of equations has infinitely many solutions.

Answer:

Where - arbitrary numbers.

To consolidate the material, we will analyze in detail the solutions of several more examples.

Example.

Solve a homogeneous system of linear algebraic equations Gaussian method.

Solution.

Let us exclude the unknown variable x from the second and third equations of the system. To do this, to the left and right sides of the second equation, we add, respectively, the left and right sides of the first equation, multiplied by , and to the left and right sides of the third equation, we add the left and right sides of the first equation, multiplied by:

Now let’s exclude y from the third equation of the resulting system of equations:

The resulting SLAE is equivalent to the system .

We leave on the left side of the system equations only the terms containing the unknown variables x and y, and move the terms with the unknown variable z to the right side:

The online calculator finds a solution to a system of linear equations (SLE) using the Gaussian method. Given detailed solution. To calculate, select the number of variables and the number of equations. Then enter the data into the cells and click on the "Calculate" button.

Number representation:

Number of places after decimal separator

×

Warning

Clear all cells?

Close Clear

Data entry instruction. Numbers are entered as whole numbers (examples: 487, 5, -7623, etc.), decimal numbers (eg. 67., 102.54, etc.) or fractions. The fraction must be entered in the form a/b, where a and b (b>0) are integers or decimal numbers. Examples 45/5, 6.6/76.4, -7/6.7, etc.

Gauss method

The Gauss method is a method of transition from the original system of linear equations (using equivalent transformations) to a system that is easier to solve than the original system.

Equivalent transformations of a system of linear equations are:

• swapping two equations in the system,
• multiplying any equation in the system by a non-zero real number,
• adding to one equation another equation multiplied by an arbitrary number.

Consider a system of linear equations:

Let us write system (1) in matrix form:

(3)

A- called the coefficient matrix of the system, b− right side of the restrictions, x− vector of variables to be found. Let rank( A)=p.

Equivalent transformations do not change the rank of the coefficient matrix and the rank of the extended matrix of the system. The set of solutions of the system also does not change under equivalent transformations. The essence of the Gauss method is to reduce the matrix of coefficients A to diagonal or stepped.

Let's build an extended matrix of the system:

At the next stage, we reset all elements of column 2, below the element. If this element is zero, then this row is swapped with the row lying below this row and having a non-zero element in the second column. Next, reset all elements of column 2 below the leading element a 22. To do this, add lines 3, ... m with string 2 multiplied by − a 32 /a 22 , ..., −a m2/ a 22, respectively. Continuing the procedure, we obtain a matrix of diagonal or stepped form. Let the resulting extended matrix have the form:

Because rangA=rang(A|b), then the set of solutions (7) is ( n−p)− variety. Hence n−p the unknowns can be chosen arbitrarily. The remaining unknowns from system (7) are calculated as follows. From the last equation we express x p through the remaining variables and insert into the previous expressions. Next, from the penultimate equation we express x p−1 through the remaining variables and insert into the previous expressions, etc. Let's look at the Gauss method using specific examples.

Examples of solving a system of linear equations using the Gauss method

Example 1. Find a general solution to a system of linear equations using the Gauss method:

Denote by a ij elements i-th line and j th column.

a eleven . To do this, add lines 2,3 with line 1, multiplied by -2/3,-1/2, respectively:

Matrix recording type: Ax=b, Where

Denote by a ij elements i-th line and j th column.

Let's exclude the elements of the 1st column of the matrix below the element a eleven . To do this, add lines 2,3 with line 1, multiplied by -1/5,-6/5, respectively:

We divide each row of the matrix by the corresponding leading element (if the leading element exists):

Where x 3 , x

Substituting the upper expressions into the lower ones, we obtain the solution.

Then the vector solution can be represented as follows:

Where x 3 , x 4 are arbitrary real numbers.

The essence of the Gauss method is to transform (1) to a system with a triangular matrix, from which the values ​​of all unknowns are then obtained sequentially (in reverse). Let's consider one of the computational schemes. This circuit is called a single division circuit. So let's look at this diagram. Let a 11 ≠0 (leading element) divide the first equation by a 11. We get
(2)
Using equation (2), it is easy to eliminate the unknowns x 1 from the remaining equations of the system (to do this, it is enough to subtract equation (2) from each equation, previously multiplied by the corresponding coefficient for x 1), that is, in the first step we obtain
.
In other words, at step 1, each element of subsequent rows, starting from the second, is equal to the difference between the original element and the product of its “projection” onto the first column and the first (transformed) row.
Following this, leaving the first equation alone, we perform a similar transformation over the remaining equations of the system obtained in the first step: we select from among them the equation with the leading element and, with its help, exclude x 2 from the remaining equations (step 2).
After n steps, instead of (1), we obtain an equivalent system
(3)
Thus, at the first stage we obtain a triangular system (3). This stage is called forward stroke.
At the second stage (reverse), we find sequentially from (3) the values ​​x n, x n -1, ..., x 1.
Let us denote the resulting solution as x 0 . Then the difference ε=b-A x 0 called residual.
If ε=0, then the found solution x 0 is correct.

Calculations using the Gaussian method are performed in two stages:

1. The first stage is called the forward method. At the first stage, the original system is converted to a triangular form.
2. The second stage is called the reverse stroke. At the second stage, a triangular system equivalent to the original one is solved.

Purpose of the Gauss method

Types of Gaussian method

1. Classical Gaussian method;
2. Modifications of the Gauss method. One of the modifications of the Gaussian method is a scheme with the choice of the main element. A feature of the Gauss method with the choice of the main element is such a rearrangement of the equations so that at the kth step the leading element turns out to be the largest element in the kth column.
3. Jordano-Gauss method;

Example of a solution using the Gaussian method
Let's solve the system:

For ease of calculation, let's swap the lines:

Let's multiply the 2nd line by (2). Add the 3rd line to the 2nd

Multiply the 2nd line by (-1). Add the 2nd line to the 1st

From the 1st line we express x 3:
From the 2nd line we express x 2:
From the 3rd line we express x 1:

An example of a solution using the Jordano-Gauss method
Let us solve the same SLAE using the Jordano-Gauss method.

We will sequentially select the resolving element RE, which lies on the main diagonal of the matrix.
The resolution element is equal to (1).



NE = SE - (A*B)/RE
RE - resolving element (1), A and B - matrix elements forming a rectangle with elements STE and RE.
Let's present the calculation of each element in the form of a table:

Implementation of the Gaussian method

Using the Gaussian Method

Application of the Gauss method in game theory

Application of the Gauss method in solving differential equations

Application of the Jordano-Gauss method in linear programming

Two systems of linear equations are called equivalent if the set of all their solutions coincides.

Elementary transformations of a system of equations are:

1. Deleting trivial equations from the system, i.e. those for which all coefficients are equal to zero;
2. Multiplying any equation by a number other than zero;
3. Adding to any i-th equation any j-th equation multiplied by any number.

A variable x i is called free if this variable is not allowed, but the entire system of equations is allowed.

Theorem. Elementary transformations transform a system of equations into an equivalent one.

The meaning of the Gaussian method is to transform the original system of equations and obtain an equivalent resolved or equivalent inconsistent system.

So, the Gaussian method consists of the following steps:

1. Let's look at the first equation. Let's choose the first non-zero coefficient and divide the entire equation by it. We obtain an equation in which some variable x i enters with a coefficient of 1;
2. Let's subtract this equation from all the others, multiplying it by such numbers that the coefficients of the variable x i in the remaining equations are zeroed. We obtain a system resolved with respect to the variable x i and equivalent to the original one;
3. If trivial equations arise (rarely, but it happens; for example, 0 = 0), we cross them out of the system. As a result, there are one fewer equations;
4. We repeat the previous steps no more than n times, where n is the number of equations in the system. Each time we select a new variable for “processing”. If inconsistent equations arise (for example, 0 = 8), the system is inconsistent.

As a result, after a few steps we will obtain either a resolved system (possibly with free variables) or an inconsistent one. Allowed systems fall into two cases:

1. The number of variables is equal to the number of equations. This means that the system is defined;
2. Number of variables more number equations. We collect all the free variables on the right - we get formulas for the allowed variables. These formulas are written in the answer.

That's all! System of linear equations solved! This is a fairly simple algorithm, and to master it you do not have to contact a higher mathematics tutor. Consider an example:

Task. Solve the system of equations:

Description of steps:

1. Subtract the first equation from the second and third - we get the allowed variable x 1;
2. We multiply the second equation by (−1), and divide the third equation by (−3) - we get two equations in which the variable x 2 enters with a coefficient of 1;
3. We add the second equation to the first, and subtract from the third. We get the allowed variable x 2 ;
4. Finally, we subtract the third equation from the first - we get the allowed variable x 3;
5. We have received an approved system, write down the response.

The general solution to a simultaneous system of linear equations is new system, equivalent to the original one, in which all allowed variables are expressed in terms of free ones.

When might a general solution be needed? If you have to do fewer steps than k (k is how many equations there are). However, the reasons why the process ends at some step l< k , может быть две:

1. After the lth step, we obtained a system that does not contain an equation with number (l + 1). In fact, this is good, because... the authorized system is still obtained - even a few steps earlier.
2. After the lth step, we obtained an equation in which all coefficients of the variables are equal to zero, and the free coefficient is different from zero. This is a contradictory equation, and, therefore, the system is inconsistent.

It is important to understand that the emergence of an inconsistent equation using the Gaussian method is a sufficient basis for inconsistency. At the same time, we note that as a result of the lth step, no trivial equations can remain - all of them are crossed out right in the process.

Description of steps:

1. Subtract the first equation, multiplied by 4, from the second. We also add the first equation to the third - we get the allowed variable x 1;
2. Subtract the third equation, multiplied by 2, from the second - we get the contradictory equation 0 = −5.

So, the system is inconsistent because an inconsistent equation has been discovered.

Task. Explore compatibility and find a general solution to the system:

Description of steps:

1. We subtract the first equation from the second (after multiplying by two) and the third - we get the allowed variable x 1;
2. Subtract the second equation from the third. Since all the coefficients in these equations are the same, the third equation will become trivial. At the same time, multiply the second equation by (−1);
3. Subtract the second from the first equation - we get the allowed variable x 2. The entire system of equations is now also resolved;
4. Since the variables x 3 and x 4 are free, we move them to the right to express the allowed variables. This is the answer.

So, the system is consistent and indeterminate, since there are two allowed variables (x 1 and x 2) and two free ones (x 3 and x 4).