Methods for solving systems of equations. Systems of equations - initial information

A system of linear equations is a union of n linear equations, each containing k variables. It is written like this:

Many, when faced with higher algebra for the first time, mistakenly believe that the number of equations must necessarily coincide with the number of variables. In school algebra this is usually the case, but for higher algebra this is, generally speaking, not true.

The solution of a system of equations is a sequence of numbers (k 1 , k 2 , ..., k n ), which is the solution to each equation of the system, i.e. when substituting into this equation instead of variables x 1 , x 2 , ..., x n gives the correct numerical equality.

Accordingly, to solve a system of equations means to find the set of all its solutions or to prove that this set is empty. Since the number of equations and the number of unknowns may not be the same, three cases are possible:

The system is inconsistent, i.e. the set of all solutions is empty. Enough rare case, which is easily detected no matter how the system is solved.
The system is consistent and defined, i.e. has exactly one solution. Classic variant, well known since the school bench.
The system is consistent and undefined, i.e. has infinitely many solutions. This is the hardest option. It is not enough to state that "the system has an infinite set of solutions" - it is necessary to describe how this set is arranged.

The variable x i is called allowed if it is included in only one equation of the system, and with a coefficient of 1. In other words, in the remaining equations, the coefficient for the variable x i must be equal to zero.

If we select one allowed variable in each equation, we get a set of allowed variables for the entire system of equations. The system itself, written in this form, will also be called allowed. Generally speaking, one and the same initial system can be reduced to different allowed systems, but this does not concern us now. Here are examples of allowed systems:

Both systems are allowed with respect to the variables x 1 , x 3 and x 4 . However, with the same success it can be argued that the second system is allowed with respect to x 1 , x 3 and x 5 . It is enough to rewrite the latest equation in the form x 5 = x 4 .

Now consider a more general case. Suppose we have k variables in total, of which r are allowed. Then two cases are possible:

The number of allowed variables r is equal to the total number of variables k : r = k . We get a system of k equations in which r = k allowed variables. Such a system is collaborative and definite, because x 1 \u003d b 1, x 2 \u003d b 2, ..., x k \u003d b k;
The number of allowed variables r is less than the total number of variables k : r< k . Остальные (k − r ) переменных называются свободными - они могут принимать любые значения, из которых легко вычисляются разрешенные переменные.

So, in the above systems, the variables x 2 , x 5 , x 6 (for the first system) and x 2 , x 5 (for the second) are free. The case when there are free variables is better formulated as a theorem:

Please note: this is very important point! Depending on how you write the final system, the same variable can be both allowed and free. Most of the tutors higher mathematics it is recommended to write out variables in lexicographic order, i.e. ascending index. However, you don't have to follow this advice at all.

Theorem. If in a system of n equations the variables x 1 , x 2 , ..., x r are allowed, and x r + 1 , x r + 2 , ..., x k are free, then:

If we assign values to free variables (xr + 1 = tr + 1 , xr + 2 = tr + 2 , ..., xk = tk ) and then find the values x 1 , x 2 , ..., xr , we get one of solutions.

If the values of the free variables in two solutions are the same, then the values of the allowed variables are also the same, i.e. solutions are equal.

What is the meaning of this theorem? To obtain all solutions of the allowed system of equations, it suffices to single out the free variables. Then, assigning to free variables different meanings, we will receive turnkey solutions. That's all - in this way you can get all the solutions of the system. There are no other solutions.

Conclusion: the allowed system of equations is always compatible. If the number of equations in the allowed system is equal to the number of variables, the system will be definite; if less, it will be indefinite.

And everything would be fine, but the question arises: how to get the resolved one from the original system of equations? For this there is

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Solve the system with two unknowns - this means finding all pairs of variable values that satisfy each of the given equations. Each such pair is called system solution.

Example:
The pair of values \(x=3\);\(y=-1\) is a solution to the first system, because by substituting these triples and minus ones into the system instead of \(x\) and \(y\), both equations become into valid equalities \(\begin(cases)3-2\cdot (-1)=5 \\3 \cdot 3+2 \cdot (-1)=7 \end(cases)\)

But \(x=1\); \(y=-2\) - is not a solution to the first system, because after substitution the second equation "does not converge" \(\begin(cases)1-2\cdot(-2)=5 \\3\cdot1+2 \cdot(-2)≠7 \end(cases)\)

Note that such pairs are often written shorter: instead of "\(x=3\); \(y=-1\)" they write like this: \((3;-1)\).

How to solve a system of linear equations?

There are three main ways to solve systems of linear equations:

Substitution method.

\(\begin(cases)x-2y=5\\3x+2y=7 \end(cases)\)\(\Leftrightarrow\) \(\begin(cases)x=5+2y\\3x+2y= 7\end(cases)\)\(\Leftrightarrow\)

Substitute the resulting expression instead of this variable into another equation of the system.

\(\Leftrightarrow\) \(\begin(cases)x=5+2y\\3(5+2y)+2y=7\end(cases)\)\(\Leftrightarrow\)

\(\begin(cases)13x+9y=17\\12x-2y=26\end(cases)\)

In the second equation, each term is even, so we simplify the equation by dividing it by \(2\).

\(\begin(cases)13x+9y=17\\6x-y=13\end(cases)\)

This system can be solved in any of the ways, but it seems to me that the substitution method is the most convenient here. Let's express y from the second equation.

\(\begin(cases)13x+9y=17\\y=6x-13\end(cases)\)

Substitute \(6x-13\) for \(y\) in the first equation.

\(\begin(cases)13x+9(6x-13)=17\\y=6x-13\end(cases)\)

The first equation has become normal. We solve it.

Let's open the parentheses first.

\(\begin(cases)13x+54x-117=17\\y=6x-13\end(cases)\)

Let's move \(117\) to the right and give like terms.

\(\begin(cases)67x=134\\y=6x-13\end(cases)\)

Divide both sides of the first equation by \(67\).

\(\begin(cases)x=2\\y=6x-13\end(cases)\)

Hooray, we found \(x\)! Substitute its value into the second equation and find \(y\).

\(\begin(cases)x=2\\y=12-13\end(cases)\)\(\Leftrightarrow\)\(\begin(cases)x=2\\y=-1\end(cases )\)

Let's write down the answer.

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. Every decision must be the solution of 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. At the school are simple equations and systems of equations, starting with systems of two linear equations in 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 the basic protozoa 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(SLAU). 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 2

Solving 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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