Solving the slough by the Gauss method examples. Gauss method (successive exclusion of unknowns)

This online calculator finds a solution to a system of linear equations (SLE) using the Gaussian method. A detailed solution is given. To calculate, choose the number of variables and the number of equations. Then enter the data in the cells and click on the "Calculate."

x 1

+x2

+x 3

x 1

+x2

+x 3

x 1

+x2

+x 3

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Integers and/or Decimals

Number of digits after decimal separator

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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 typed in the form a/b, where a and b (b>0) are integer 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.

The equivalent transformations of the system of linear equations are:

swapping two equations in the system,
multiplication of 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:

(1)

We write system (1) in matrix form:

ax=b

(2)

(3)

A is called the coefficient matrix of the system, b− right side of constraints, 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 augmented 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 bring the matrix of coefficients A to diagonal or stepped.

Let's build the extended matrix of the system:

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

(7)

Because rankA=rank(A|b), then the set of solutions (7) is ( n−p) is a variety. Consequently n−p 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 rest of the variables and insert into the previous expressions. Next, from the penultimate equation, we express x p−1 through the rest of the variables and insert into the previous expressions, etc. Consider the Gauss method on specific examples.

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

Example 1. Find the general solution of 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 rows 2,3 with row 1, multiplied by -2/3, -1/2, respectively:

Matrix record type: ax=b, where

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

Exclude the elements of the 1st column of the matrix below the element a eleven . To do this, add rows 2,3 with row 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.

Two systems of linear equations are said to be equivalent if the set of all their solutions is the same.

Elementary transformations of the system of equations are:

Deletion from the system of trivial equations, i.e. those for which all coefficients are equal to zero;

Multiplying any equation by a non-zero number;

Addition to any i -th equation of any j -th equation, multiplied by any number.

The variable x i is called free if this variable is not allowed, and the whole system of equations is allowed.

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

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

So, the Gauss method consists of the following steps:

Consider the first equation. We choose the first non-zero coefficient and divide the whole equation by it. We obtain an equation in which some variable x i enters with a coefficient of 1;
Let us subtract this equation from all the others, multiplying it by numbers such that the coefficients for the variable x i in the remaining equations are set to zero. We get a system that is resolved with respect to the variable x i and is equivalent to the original one;
If trivial equations arise (rarely, but it happens; for example, 0 = 0), we delete them from the system. As a result, the equations become one less;
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 conflicting equations arise (for example, 0 = 8), the system is inconsistent.

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

The number of variables is equal to the number of equations. So the system is defined;
The number of variables is greater than the number of equations. We collect all free variables on the right - we get formulas for allowed variables. These formulas are written in the answer.

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

A task. Solve the system of equations:

Description of steps:

We subtract the first equation from the second and third - we get the allowed variable x 1;
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;
We add the second equation to the first, and subtract from the third. Let's get the allowed variable x 2 ;
Finally, we subtract the third equation from the first - we get the allowed variable x 3 ;
We have received an authorized system, we write down the answer.

The general solution of a joint system of linear equations is a 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 take fewer steps than k (k is how many equations in total). However, the reasons why the process ends at some step l< k , может быть две:

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

It is important to understand that the appearance of an inconsistent equation by the Gauss method is a sufficient reason for inconsistency. At the same time, we note that as a result of the l -th step, trivial equations cannot remain - all of them are deleted directly in the process.

Description of steps:

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

So, the system is inconsistent, since an inconsistent equation has been found.

A task. Investigate compatibility and find the general solution of the system:

Description of steps:

We subtract the first equation from the second (after multiplying by two) and the third - we get the allowed variable x 1;
Subtract the second equation from the third. Since all the coefficients in these equations are the same, the third equation becomes trivial. At the same time, we multiply the second equation by (−1);
We subtract the second equation from the first equation - we get the allowed variable x 2. The entire system of equations is now also resolved;
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 joint and indefinite, since there are two allowed variables (x 1 and x 2) and two free ones (x 3 and x 4).

Solving systems of linear equations by the Gauss method. Suppose we need to find a solution to the system from n linear equations with n unknown variables
the determinant of the main matrix of which is different from zero.

The essence of the Gauss method consists in the successive exclusion of unknown variables: first, the x 1 from all equations of the system, starting from the second, then x2 of all equations, starting with the third, and so on, until only the unknown variable remains in the last equation x n. Such a process of transforming the equations of the system for the successive elimination of unknown variables is called direct Gauss method. After the completion of the forward move of the Gauss method, from the last equation we find x n, using this value from the penultimate equation is calculated xn-1, and so on, from the first equation is found x 1. The process of calculating unknown variables when moving from the last equation of the system to the first is called reverse Gauss method.

Let us briefly describe the algorithm for eliminating unknown variables.

We will assume that , since we can always achieve this by rearranging the equations of the system. Eliminate the unknown variable x 1 from all equations of the system, starting from the second. 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, to n-th add the first equation, multiplied by . The system of equations after such transformations will take the form

where , a .

We would arrive at the same result if we expressed x 1 through other unknown variables in the first equation of the system and the resulting expression was substituted into all other equations. So the variable x 1 excluded from all equations, starting with 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, to n-th add the second equation, multiplied by . The system of equations after such transformations will take the form

where , a . So the variable x2 excluded from all equations, starting with the third.

Next, we proceed to the elimination of the unknown x 3, while we act 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 find xn-1 from the penultimate equation, and so on, we find x 1 from the first equation.

Example.

Solve System of Linear Equations Gaussian method.

Carl Friedrich Gauss, the greatest mathematician, hesitated for a long time, choosing 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 been concerned with school education, mainly from the point of view 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 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 catch the principles ... How impossible it is to be on the top of a mountain otherwise than by going through its entire slope from the foot.

Same with school: for now living stories not enough 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 will 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. it 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 my youngest son brought from school, attending the 5th grade of a Moscow gymnasium.

School demonstration of the Gauss method

A mathematics teacher using an interactive whiteboard (modern teaching methods) showed the children a presentation of the story of the "creation of the method" by little Gauss.

The school teacher whipped little Carl (an outdated method, now not used in schools) for being,

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 subjected to execution in front of an astonished public. To the rest to think was disrespectful.

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

This is the value of mathematics, which develops ability to see general in particular - abstract thinking. Therefore, most parents and employers instinctively consider mathematics an important discipline ...

“Mathematics should be taught later, so that it puts the mind in order.
M.V. Lomonosov".

However, the followers of those who flogged future geniuses turned the Method into something opposite. As my supervisor said 35 years ago: "They learned the question." Or, as my youngest son said yesterday about the Gauss 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 understanding of the "Queen of Sciences" by the majority.

However, let's continue...

Methods for explaining the Gauss method in the 5th grade of the school

A mathematics teacher at a Moscow gymnasium, explaining the Gauss method in Vilenkin's way, complicated the task.

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

The task he gave the fifth graders:

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

Before getting acquainted with the gymnasium method, let's look at the Web: how do school teachers - math tutors do it? ..

Gauss Method: Explanation #1

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

"let's write the numbers from 1 to 100 like this:

first a series of numbers from 1 to 50, and strictly below it another series of numbers from 50 to 100, but in 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 pass this method in the 9th grade!"

Gauss Method: Explanation #2

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

For the uninitiated: 5 is one of the Fibonacci numbers traditionally considered magical. The 5-step method is always more scientific than the 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:

Step 1: rewrite the given sequence of numbers in reverse, exactly under the first.

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

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

Step 2: calculate the sums of pairs of numbers arranged in vertical rows: 260.

Step 3: count how many such pairs are in the number series. To do this, subtract the minimum from the maximum number of the number series and divide by the step size: (256 - 4) / 6 = 42.

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

Step 4: multiply the sum of one pair of numbers by the number of pairs: 260 x 43 = 11,180

Step 5: since we calculated the amount pairs of numbers, then the amount received should be divided by two: 11 180 / 2 = 5590.

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

Gauss method: explanation in the 5th grade of the Moscow gymnasium

And here is how it was required to solve the problem of finding the sum of a series:

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

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

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

This required the following:

Step 1: be sure to write down all the numbers in the row in a notebook from 20 to 500 (in increments of 20).

Step 2: write consecutive terms - pairs of numbers: the first with the last, the second with the penultimate, etc. and calculate their sums.

Step 3: calculate the "sum of sums" and find the sum of the whole series.

As you can see, this is a more compact and efficient technique: the 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 followers would turn his "method" into. German teacher who flogged Karl with rods. He would have seen the symbolism and 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, do 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?

AT simplification. AT observation and capture simple patterns of numbers. AT turning dry school arithmetic into interesting and fun activity , activating the desire to continue in the brain, and not blocking high-cost mental activity.

Is it possible to calculate the sum of the numbers of an arithmetic progression with one of the above "modifications of the Gauss method" instantly? According to the "algorithms", little Karl would have been guaranteed to avoid spanking, cultivate an aversion to mathematics and suppress his creative impulses in the bud.

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

To use the Gaussian method, level 3 of the class 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 "do not enter" how to explain the simplest things in a normal human language, not just mathematical ... They are not able to interest mathematics and completely discourage even "able" ones.

Or, as my son commented, "make a big science out of it."

How (in the general case) to find out on which number the record of numbers in method No. 1 should be "unwrapped"?

What to do if the number of members of the series is odd?

Why turn into a "Rule Plus 1" what a child could just assimilate even in the first grade, if he had developed a "sense of number", and didn't remember"count in ten"?

And finally: where did ZERO disappear, a brilliant invention that is more than 2,000 years old and which modern mathematics teachers avoid using?!

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 - answers

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

It's not about what the child sees. The trick is to make him look.

"How can you put them together?" The son caught 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 ceased to be afraid to look, or as I say: "moved the task". This is the beginning of the path to understanding

"Which is easier: add, for example, 5 and 6 or 5 and 95?" A leading question... But after all, any training comes down to "guiding" a person to an "answer" - in any way acceptable to him.

At this stage, there may already be guesses about how to "save" on calculations.

All we have done is hint: the "frontal, linear" counting method is not the only one possible. If the child has truncated this, then later he will invent many more such methods, because it's interesting!!! And he will definitely avoid "misunderstanding" of mathematics, will not feel disgust for it. He got the win!

If a baby discovered that adding pairs of numbers that add up to a hundred is a trifling task, then "arithmetic progression with difference 1"- a rather dreary and uninteresting thing for a child - suddenly gave life to him . Out of chaos came order, and this is always enthusiastic: that's the way we are!

A quick question: why, after a child’s insight, should they again be driven into the framework of dry algorithms, which are also functionally useless in this case?!

Why make stupid rewrite sequence numbers in a notebook: so that even the capable would not have a single chance for understanding? Statistically, of course, but mass education is focused on "statistics" ...

Where did zero go?

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

The "school Gauss method" requires exactly this: mindlessly fold equidistant from the center of the progression of a pair of numbers, no matter what.

What if you look?

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

It's much easier to convert a series of numbers starting at 1 into a series starting at 0. The sum won't change, will it? You need to stop "thinking in textbooks" and start looking ... And to see that pairs with sum 101 can be completely replaced by pairs with sum 100!

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

How to abolish the "rule plus 1"?

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?

Looking at the sequence:

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

and when completely tired, then on a simpler row:

1, 2, 3, 4, 5

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

Fuck the rules?..

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

More about the school method of Gauss: "why make science out of this? .."

It was not in vain that I posted a screenshot from my son's notebook...

"What was there in the lesson?"

“Well, I immediately counted, raised my hand, but she didn’t ask. Therefore, while the others were counting, I began to do DZ 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. She said: "Wrong, you need to count as I showed!"

“It’s good that I didn’t put a deuce. And I made me write the “decision process” in their own way in a notebook. Why make a big science out of this? ..”

The main crime of a math teacher

hardly after that case Carl Gauss experienced a high sense of respect for the school teacher of mathematics. But if he knew how followers of that teacher pervert the essence of the method... he would have roared with indignation and, through the World Intellectual Property Organization WIPO, achieved a ban on the use of his good name in school textbooks! ..

What the main mistake of the school approach? Or, as I put it, the crime of school mathematics teachers against children?

Misunderstanding algorithm

What do school methodologists do, the vast majority of whom do not know how to think?

Create methods and algorithms (see). it 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, and not the stupidity of the school system.

What is happening: parents blame the children, and teachers ... the same for children who "do not understand mathematics! ..

Are you savvy?

What did little Carl do?

Absolutely unconventionally approached a template task. This is the quintessence of His approach. it the main thing that should be taught at 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 more efficient counting methods.

Gauss method according to Vilenkin

In school they teach that the Gauss method is to

in pairs find the sums of numbers equidistant from the edges of the number series, necessarily starting from the edges!

find the number of such pairs, and so on.

what, if the number of elements in the row is odd, as in the task that was assigned to the son? ..

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

How to discover? Rewriting all pairs of numbers in a notebook!(That's why the teacher made the kids do this stupid job, trying to teach "creativity" using the Gaussian method... And that's why such a "method" is practically inapplicable to large data series, And that's why it is not a Gaussian method).

A little creativity in the school routine...

The son acted differently.

At first he noted that it was easier to multiply the number 500, not 520.

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

Then he figured out: the number of steps turned out to be odd: 500 / 20 = 25.

Then he added ZERO to the beginning of the series (although it was possible to discard the last term of the series, which would also ensure parity) and added the numbers, giving a total of 500

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

26 steps are 13 pairs of "five hundred": 13 x 500 = 6500 ..

If we discarded the last member of the series, then there will be 12 pairs, but we should not forget to add the "discarded" five hundred to the result of the calculations. Then: (12 x 500) + 500 = 6500!

Easy, right?

But in practice it becomes 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 methodology: 5, which does not allow criticizing the approach for being unscientific.

Obviously this approach is simpler, faster and more versatile, in the style of the Method. But... the teacher not only didn't praise, but also forced me to rewrite it "in the right way" (see screenshot). That is, she made a desperate attempt to stifle the creative impulse and the ability to understand mathematics in the bud! Apparently, in order to later get hired as a tutor ... She attacked the wrong one ...

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

And so that he will never forget it.

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

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

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

Especially in the age of universal digitalization, which we quietly entered under the strict guidance 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 operation.

Some teachers understand the absurdity of what is happening, but what to do? The Law on Education, Federal State Educational Standards, methods, lesson cards... Everything should be done "in accordance and on the basis" and everything should be documented. Step aside - stood in line for dismissal. Let's not be hypocrites: the salary of Moscow teachers is very good... If they get fired, where should they go?..

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

Ever since the beginning of the 16th-18th centuries, mathematicians began to intensively study the functions, thanks to which so much has changed in our lives. Computer technology without this knowledge simply would not exist. To solve complex problems, linear equations and functions, various concepts, theorems and solution techniques have been created. One of such universal and rational methods and techniques for solving linear equations and their systems was the Gauss method. Matrices, their rank, determinant - everything can be calculated without using complex operations.

What is SLAU

In mathematics, there is the concept of SLAE - a system of linear algebraic equations. What does she represent? This is a set of m equations with the required n unknowns, usually denoted as x, y, z, or x 1 , x 2 ... x n, or other symbols. To solve this system by the Gaussian method means to find all unknown unknowns. If a system has the same number of unknowns and equations, then it is called an n-th order system.

The most popular methods for solving SLAE

In educational institutions of secondary education, various methods of solving such systems are being studied. Most often, these are simple equations consisting of two unknowns, so any existing method for finding the answer to them will not take much time. It can be like a substitution method, when another equation is derived from one equation and substituted into the original one. Or term by term subtraction and addition. But the Gauss method is considered the easiest and most universal. It makes it possible to solve equations with any number of unknowns. Why is this technique considered rational? Everything is simple. The matrix method is good because it does not require several times to rewrite unnecessary characters in the form of unknowns, it is enough to do arithmetic operations on the coefficients - and you will get a reliable result.

Where are SLAEs used in practice?

The solution of SLAE are the points of intersection of lines on the graphs of functions. In our high-tech computer age, people who are closely involved in the development of games and other programs need to know how to solve such systems, what they represent and how to check the correctness of the resulting result. Most often, programmers develop special linear algebra calculators, this includes a system of linear equations. The Gauss method allows you to calculate all existing solutions. Other simplified formulas and techniques are also used.

SLAE compatibility criterion

Such a system can only be solved if it is compatible. For clarity, we present the SLAE in the form Ax=b. It has a solution if rang(A) equals rang(A,b). In this case, (A,b) is an extended form matrix that can be obtained from matrix A by rewriting it with free terms. It turns out that solving linear equations using the Gaussian method is quite easy.

Perhaps some notation is not entirely clear, so it is necessary to consider everything with an example. Let's say there is a system: x+y=1; 2x-3y=6. It consists of only two equations in which there are 2 unknowns. The system will have a solution only if the rank of its matrix is equal to the rank of the augmented matrix. What is a rank? This is the number of independent lines of the system. In our case, the rank of the matrix is 2. Matrix A will consist of the coefficients located near the unknowns, and the coefficients behind the “=” sign will also fit into the expanded matrix.

Why SLAE can be represented in matrix form

Based on the compatibility criterion according to the proven Kronecker-Capelli theorem, the system of linear algebraic equations can be represented in matrix form. Using the Gaussian cascade method, you can solve the matrix and get the only reliable answer for the entire system. If the rank of an ordinary matrix is equal to the rank of its extended matrix, but less than the number of unknowns, then the system has an infinite number of answers.

Matrix transformations

Before moving on to solving matrices, it is necessary to know what actions can be performed on their elements. There are several elementary transformations:

By rewriting the system into a matrix form and carrying out its solution, it is possible to multiply all the elements of the series by the same coefficient.
In order to convert a matrix to canonical form, two parallel rows can be swapped. The canonical form implies that all elements of the matrix that are located along the main diagonal become ones, and the remaining ones become zeros.
The corresponding elements of the parallel rows of the matrix can be added one to the other.

Jordan-Gauss method

The essence of solving systems of linear homogeneous and inhomogeneous equations by the Gauss method is to gradually eliminate the unknowns. Let's say we have a system of two equations in which there are two unknowns. To find them, you need to check the system for compatibility. The Gaussian equation is solved very simply. It is necessary to write out the coefficients located near each unknown in a matrix form. To solve the system, you need to write out the augmented matrix. If one of the equations contains a smaller number of unknowns, then "0" must be put in place of the missing element. All known transformation methods are applied to the matrix: multiplication, division by a number, adding the corresponding elements of the rows to each other, and others. It turns out that in each row it is necessary to leave one variable with the value "1", the rest should be reduced to zero. For a more accurate understanding, it is necessary to consider the Gauss method with examples.

A simple example of solving a 2x2 system

To begin with, let's take a simple system of algebraic equations, in which there will be 2 unknowns.

Let's rewrite it in an augmented matrix.

To solve this system of linear equations, only two operations are required. We need to bring the matrix to the canonical form so that there are units along the main diagonal. So, translating from the matrix form back into the system, we get the equations: 1x+0y=b1 and 0x+1y=b2, where b1 and b2 are the answers obtained in the process of solving.

The first step in solving the augmented matrix will be as follows: the first row must be multiplied by -7 and the corresponding elements added to the second row, respectively, in order to get rid of one unknown in the second equation.
Since the solution of equations by the Gauss method implies bringing the matrix to the canonical form, then it is necessary to do the same operations with the first equation and remove the second variable. To do this, we subtract the second line from the first and get the necessary answer - the solution of the SLAE. Or, as shown in the figure, we multiply the second row by a factor of -1 and add the elements of the second row to the first row. This is the same.

As you can see, our system is solved by the Jordan-Gauss method. We rewrite it in the required form: x=-5, y=7.

An example of solving SLAE 3x3

Suppose we have a more complex system of linear equations. The Gauss method makes it possible to calculate the answer even for the most seemingly confusing system. Therefore, in order to delve deeper into the calculation methodology, we can move on to a more complex example with three unknowns.

As in the previous example, we rewrite the system in the form of an expanded matrix and begin to bring it to the canonical form.

To solve this system, you will need to perform much more actions than in the previous example.

First you need to make in the first column one single element and the rest zeros. To do this, multiply the first equation by -1 and add the second equation to it. It is important to remember that we rewrite the first line in its original form, and the second - already in a modified form.
Next, we remove the same first unknown from the third equation. To do this, we multiply the elements of the first row by -2 and add them to the third row. Now the first and second lines are rewritten in their original form, and the third - already with changes. As you can see from the result, we got the first one at the beginning of the main diagonal of the matrix and the rest are zeros. A few more actions, and the system of equations by the Gauss method will be reliably solved.
Now you need to do operations on other elements of the rows. The third and fourth steps can be combined into one. We need to divide the second and third lines by -1 to get rid of the negative ones on the diagonal. We have already brought the third line to the required form.
Next, we canonicalize the second line. To do this, we multiply the elements of the third row by -3 and add them to the second line of the matrix. It can be seen from the result that the second line is also reduced to the form we need. It remains to do a few more operations and remove the coefficients of the unknowns from the first row.
In order to make 0 from the second element of the row, you need to multiply the third row by -3 and add it to the first row.
The next decisive step is to add the necessary elements of the second row to the first row. So we get the canonical form of the matrix, and, accordingly, the answer.

As you can see, the solution of equations by the Gauss method is quite simple.

An example of solving a 4x4 system of equations

Some more complex systems of equations can be solved by the Gaussian method using computer programs. It is necessary to drive coefficients for unknowns into existing empty cells, and the program will calculate the required result step by step, describing each action in detail.

The step-by-step instructions for solving such an example are described below.

In the first step, free coefficients and numbers for unknowns are entered into empty cells. Thus, we get the same augmented matrix that we write by hand.

And all the necessary arithmetic operations are performed to bring the extended matrix to the canonical form. It must be understood that the answer to a system of equations is not always integers. Sometimes the solution can be from fractional numbers.

Checking the correctness of the solution

The Jordan-Gauss method provides for checking the correctness of the result. In order to find out whether the coefficients are calculated correctly, you just need to substitute the result into the original system of equations. The left side of the equation must match the right side, which is behind the equals sign. If the answers do not match, then you need to recalculate the system or try to apply another method of solving SLAE known to you, such as substitution or term-by-term subtraction and addition. After all, mathematics is a science that has a huge number of different methods of solving. But remember: the result should always be the same, no matter what solution method you used.

Gauss method: the most common errors in solving SLAE

During the solution of linear systems of equations, errors most often occur, such as incorrect transfer of coefficients to a matrix form. There are systems in which some unknowns are missing in one of the equations, then, transferring the data to the expanded matrix, they can be lost. As a result, when solving this system, the result may not correspond to the real one.

Another of the main mistakes can be incorrect writing out the final result. It must be clearly understood that the first coefficient will correspond to the first unknown from the system, the second - to the second, and so on.

The Gauss method describes in detail the solution of linear equations. Thanks to him, it is easy to perform the necessary operations and find the right result. In addition, this is a universal tool for finding a reliable answer to equations of any complexity. Maybe that is why it is so often used in solving SLAE.