More reliable than the graphical method discussed in the previous paragraph.

Substitution method

We used this method in 7th grade to solve systems of linear equations. The algorithm that was developed in the 7th grade is quite suitable for solving systems of any two equations (not necessarily linear) with two variables x and y (of course, the variables can be designated by other letters, which does not matter). In fact, we used this algorithm in the previous paragraph, when the problem of a two-digit number led to a mathematical model, which is a system of equations. We solved this system of equations above using the substitution method (see example 1 from § 4).

An algorithm for using the substitution method when solving a system of two equations with two variables x, y.

1. Express y in terms of x from one equation of the system.
2. Substitute the resulting expression instead of y into another equation of the system.
3. Solve the resulting equation for x.
4. Substitute in turn each of the roots of the equation found in the third step instead of x into the expression y through x obtained in the first step.
5. Write the answer in the form of pairs of values ​​(x; y), which were found in the third and fourth steps, respectively.

4) Substitute one by one each of the found values ​​of y into the formula x = 5 - 3. If then
5) Pairs (2; 1) and solutions to a given system of equations.

Answer: (2; 1);

Algebraic addition method

This method, like the substitution method, is familiar to you from the 7th grade algebra course, where it was used to solve systems of linear equations. Let us recall the essence of the method using the following example.

Example 2. Solve system of equations

Let's multiply all terms of the first equation of the system by 3, and leave the second equation unchanged:
Subtract the second equation of the system from its first equation:

As a result of the algebraic addition of two equations of the original system, an equation was obtained that was simpler than the first and second equations of the given system. With this simpler equation we have the right to replace any equation of a given system, for example the second one. Then the given system of equations will be replaced by a simpler system:

This system can be solved using the substitution method. From the second equation we find. Substituting this expression instead of y into the first equation of the system, we get

It remains to substitute the found values ​​of x into the formula

If x = 2 then

Thus, we found two solutions to the system:

Method for introducing new variables

You were introduced to the method of introducing a new variable when solving rational equations with one variable in the 8th grade algebra course. The essence of this method for solving systems of equations is the same, but from a technical point of view there are some features that we will discuss in the following examples.

Example 3. Solve system of equations

Let's introduce a new variable. Then the first equation of the system can be rewritten in a simpler form: Let's solve this equation with respect to the variable t:

Both of these values ​​satisfy the condition and therefore are the roots of a rational equation with variable t. But that means either where we find that x = 2y, or
Thus, using the method of introducing a new variable, we managed to “stratify” the first equation of the system, which was quite complex in appearance, into two simpler equations:

x = 2 y; y - 2x.

What's next? And then each of the two simple equations obtained must be considered in turn in a system with the equation x 2 - y 2 = 3, which we have not yet remembered. In other words, the problem comes down to solving two systems of equations:

We need to find solutions to the first system, the second system and include all the resulting pairs of values ​​in the answer. Let's solve the first system of equations:

Let's use the substitution method, especially since everything is ready for it here: let's substitute the expression 2y instead of x into the second equation of the system. We get

Since x = 2y, we find, respectively, x 1 = 2, x 2 = 2. Thus, two solutions of the given system are obtained: (2; 1) and (-2; -1). Let's solve the second system of equations:

Let's use the substitution method again: substitute the expression 2x instead of y into the second equation of the system. We get

This equation has no roots, which means the system of equations has no solutions. Thus, only the solutions of the first system need to be included in the answer.

Answer: (2; 1); (-2;-1).

The method of introducing new variables when solving systems of two equations with two variables is used in two versions. First option: one new variable is introduced and used in only one equation of the system. This is exactly what happened in example 3. Second option: two new variables are introduced and used simultaneously in both equations of the system. This will be the case in example 4.

Example 4. Solve system of equations

Let's introduce two new variables:

Let's take into account that then

This will allow the given system to be rewritten in a much simpler form, but with respect to the new variables a and b:

Since a = 1, then from the equation a + 6 = 2 we find: 1 + 6 = 2; 6=1. Thus, regarding the variables a and b, we got one solution:

Returning to the variables x and y, we obtain a system of equations

Let us apply the method of algebraic addition to solve this system:

Since then from the equation 2x + y = 3 we find:
Thus, regarding the variables x and y, we got one solution:

Let us conclude this paragraph with a brief but rather serious theoretical discussion. You have already gained some experience in solving various equations: linear, quadratic, rational, irrational. You know that the main idea of ​​solving an equation is to gradually move from one equation to another, simpler, but equivalent to the given one. In the previous paragraph we introduced the concept of equivalence for equations with two variables. This concept is also used for systems of equations.

Definition.

Two systems of equations with variables x and y are called equivalent if they have the same solutions or if both systems have no solutions.

All three methods (substitution, algebraic addition and introducing new variables) that we discussed in this section are absolutely correct from the point of view of equivalence. In other words, using these methods, we replace one system of equations with another, simpler, but equivalent to the original system.

Graphical method for solving systems of equations

We have already learned how to solve systems of equations in such common and reliable ways as the method of substitution, algebraic addition and the introduction of new variables. Now let's remember the method that you already studied in the previous lesson. That is, let's repeat what you know about the graphical solution method.

The method of solving systems of equations graphically involves constructing a graph for each of the specific equations that are included in a given system and are located in the same coordinate plane, as well as where it is necessary to find the intersections of the points of these graphs. To solve this system of equations are the coordinates of this point (x; y).

It should be remembered that it is common for a graphical system of equations to have either one single correct solution, or an infinite number of solutions, or to have no solutions at all.

Now let’s look at each of these solutions in more detail. And so, a system of equations can have a unique solution if the lines that are the graphs of the system’s equations intersect. If these lines are parallel, then such a system of equations has absolutely no solutions. If the direct graphs of the equations of the system coincide, then such a system allows one to find many solutions.

Well, now let’s look at the algorithm for solving a system of two equations with 2 unknowns using a graphical method:

Firstly, first we build a graph of the 1st equation;
The second step will be to construct a graph that relates to the second equation;
Thirdly, we need to find the intersection points of the graphs.
And as a result, we get the coordinates of each intersection point, which will be the solution to the system of equations.

Let's look at this method in more detail using an example. We are given a system of equations that needs to be solved:

Solving equations

1. First, we will build a graph of this equation: x2+y2=9.

But it should be noted that this graph of the equations will be a circle with a center at the origin, and its radius will be equal to three.

2. Our next step will be to graph an equation such as: y = x – 3.

In this case, we must construct a straight line and find the points (0;−3) and (3;0).

3. Let's see what we got. We see that the straight line intersects the circle at two of its points A and B.

Now we are looking for the coordinates of these points. We see that the coordinates (3;0) correspond to point A, and the coordinates (0;−3) correspond to point B.

And what do we get as a result?

The numbers (3;0) and (0;−3) obtained when the line intersects the circle are precisely the solutions to both equations of the system. And from this it follows that these numbers are also solutions to this system of equations.

That is, the answer to this solution is the numbers: (3;0) and (0;−3).

We are already familiar with the concept of a linear equation in two unknowns. Equations can be present in one problem either individually or several equations at once. In such cases, the equations are combined into a system of equations.

What is a system of linear equations

System of equations- these are two or more equations for which it is necessary to find all their common solutions. Usually, to write a system of equations, they are written in a column and one common curly bracket is drawn. A recording of the system of linear equations is presented below.

( 4x + 3y = 6
( 2x + y = 4

This entry means that a system of two equations with two variables is given. If there were three equations in the system, then we would be talking about a system of three equations. And so on for any number of equations.

If all the equations present in a system are linear, then we say that a system of linear equations is given. In the example above, a system of two linear equations is presented. As noted above, the system may have general solutions. We will talk about the term “general solution” below.

What is the solution?

A solution to a system of two equations with two unknowns is a pair of numbers (x,y) such that if we substitute these numbers into the equations of the system, then each of the equations of the system turns into a true equality.

For example, we have a system of two linear equations. The solution to the first equation will be all pairs of numbers that satisfy this equation.

For the second equation, the solution will be pairs of numbers that satisfy this equation. If there is a pair of numbers that satisfies both the first and second equations, then this pair of numbers will be the solution to a system of two linear equations in two unknowns.

Graphic solution

Graphically, the solution to a linear equation is all the points of a certain line on the plane.

For a system of linear equations, we will have several straight lines (according to the number of equations). And the solution to the system of equations will be the point at which ALL lines intersect. If there is no such point, then the system will have no solutions. The point at which all lines intersect belongs to each of these lines, therefore the solution is called general.

By the way, plotting the equations of a system and finding their common point is one of the ways to solve a system of equations. This method is called graphical.

Other ways to solve linear equations

There are other ways to solve systems of linear equations in two variables. Basic methods for solving systems of linear equations with two unknowns.

Instructions

Addition method.
You need to write two strictly below each other:

549+45y+4y=-7, 45y+4y=549-7, 49y=542, y=542:49, y≈11.
In an arbitrarily chosen (from the system) equation, insert the number 11 instead of the already found “game” and calculate the second unknown:

X=61+5*11, x=61+55, x=116.
The answer to this system of equations is x=116, y=11.

Graphic method.
It consists of practically finding the coordinates of the point at which the lines are mathematically written in a system of equations. The graphs of both lines should be drawn separately in the same coordinate system. General view: – y=khx+b. To construct a straight line, it is enough to find the coordinates of two points, and x is chosen arbitrarily.
Let the system be given: 2x – y=4

Y=-3x+1.
A straight line is constructed using the first one, for convenience it should be written down: y=2x-4. Come up with (easier) values ​​for x, substituting it into the equation, solving it, and finding y. We get two points along which a straight line is constructed. (see picture)
x 0 1

y -4 -2
A straight line is constructed using the second equation: y=-3x+1.
Also construct a straight line. (see picture)

y 1 -5
Find the coordinates of the intersection point of two constructed lines on the graph (if the lines do not intersect, then the system of equations does not have - so).

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Helpful advice

If you solve the same system of equations in three different ways, the answer will be the same (if the solution is correct).

Sources:

• 8th grade algebra
• solve an equation with two unknowns online
• Examples of solving systems of linear equations with two

System equations is a collection of mathematical records, each of which contains a number of variables. There are several ways to solve them.

You will need

• -Ruler and pencil;
• -calculator.

Instructions

Let's consider the sequence of solving the system, which consists of linear equations having the form: a1x + b1y = c1 and a2x + b2y = c2. Where x and y are unknown variables, and b,c are free terms. When applying this method, each system represents the coordinates of points corresponding to each equation. To begin, in each case, express one variable in terms of another. Then set the variable x to any number of values. Two is enough. Substitute into the equation and find y. Construct a coordinate system, mark the resulting points on it and draw a line through them. Similar calculations must be carried out for other parts of the system.

The system has a unique solution if the constructed lines intersect and have one common point. It is incompatible if parallel to each other. And it has infinitely many solutions when the lines merge with each other.

This method is considered very visual. The main disadvantage is that the calculated unknowns have approximate values. More accurate results are provided by the so-called algebraic methods.

Any solution to a system of equations is worth checking. To do this, substitute the resulting values ​​for the variables. You can also find its solution using several methods. If the solution of the system is correct, then everyone should turn out the same.

Often there are equations in which one of the terms is unknown. To solve an equation, you need to remember and perform a certain set of actions with these numbers.

You will need

• - paper;
• - pen or pencil.

Instructions

Imagine that there are 8 rabbits in front of you, and you only have 5 carrots. Think about it, you still need to buy more carrots so that each rabbit gets one.

Let's present this problem in the form of an equation: 5 + x = 8. Let's substitute the number 3 in place of x. Indeed, 5 + 3 = 8.

When you substituted a number for x, you did the same thing as when you subtracted 5 from 8. So, to find unknown term, subtract the known term from the sum.

Let's say you have 20 rabbits and only 5 carrots. Let's make it up. An equation is an equality that holds only for certain values ​​of the letters included in it. The letters whose meanings need to be found are called . Write an equation with one unknown, call it x. When solving our rabbit problem, we get the following equation: 5 + x = 20.

Let's find the difference between 20 and 5. When subtracting, the number from which it is subtracted is the one being reduced. The number that is subtracted is called , and the final result is called the difference. So, x = 20 – 5; x = 15. You need to buy 15 carrots for the rabbits.

Check: 5 + 15 = 20. The equation is solved correctly. Of course, when it comes to such simple ones, checking is not necessary. However, when you have equations with three-digit, four-digit, etc. numbers, you definitely need to check to be absolutely sure of the result of your work.

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Helpful advice

To find the unknown minuend, you need to add the subtrahend to the difference.

To find the unknown subtrahend, you need to subtract the difference from the minuend.

Tip 4: How to solve a system of three equations with three unknowns

A system of three equations with three unknowns may not have solutions, despite a sufficient number of equations. You can try to solve it using the substitution method or using Cramer's method. Cramer's method, in addition to solving the system, allows you to evaluate whether the system is solvable before finding the values ​​of the unknowns.

Instructions

The substitution method consists of sequentially sequentially one unknown through two others and substituting the resulting result into the equations of the system. Let a system of three equations be given in general form:

a1x + b1y + c1z = d1

a2x + b2y + c2z = d2

a3x + b3y + c3z = d3

Express x from the first equation: x = (d1 - b1y - c1z)/a1 - and substitute into the second and third equations, then express y from the second equation and substitute into the third. You will obtain a linear expression for z through the coefficients of the system equations. Now go “backward”: substitute z into the second equation and find y, and then substitute z and y into the first and solve for x. The process is generally shown in the figure before finding z. Further writing in general form will be too cumbersome; in practice, by substituting , you can quite easily find all three unknowns.

Cramer's method consists of constructing a system matrix and calculating the determinant of this matrix, as well as three more auxiliary matrices. The system matrix is ​​composed of coefficients for the unknown terms of the equations. A column containing the numbers on the right-hand sides of equations, a column of right-hand sides. It is not used in the system, but is used when solving the system.

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note

All equations in the system must provide additional information independent of other equations. Otherwise, the system will be underdetermined and it will not be possible to find an unambiguous solution.

Helpful advice

After solving the system of equations, substitute the found values ​​into the original system and check that they satisfy all the equations.

By itself the equation with three unknown has many solutions, so most often it is supplemented by two more equations or conditions. Depending on what the initial data are, the course of the decision will largely depend.

You will need

• - a system of three equations with three unknowns.

Instructions

If two of the three systems have only two of the three unknowns, try to express some variables in terms of the others and substitute them into the equation with three unknown. Your goal in this case is to turn it into normal the equation with an unknown person. If this is , the further solution is quite simple - substitute the found value into other equations and find all the other unknowns.

Some systems of equations can be subtracted from one equation by another. See if it is possible to multiply one of or a variable so that two unknowns are canceled at once. If there is such an opportunity, take advantage of it; most likely, the subsequent solution will not be difficult. Remember that when multiplying by a number, you must multiply both the left side and the right side. Likewise, when subtracting equations, you must remember that the right-hand side must also be subtracted.

If the previous methods did not help, use the general method of solving any equations with three unknown. To do this, rewrite the equations in the form a11x1+a12x2+a13x3=b1, a21x1+a22x2+a23x3=b2, a31x1+a32x2+a33x3=b3. Now create a matrix of coefficients for x (A), a matrix of unknowns (X) and a matrix of free variables (B). Please note that by multiplying the matrix of coefficients by the matrix of unknowns, you will get a matrix of free terms, that is, A*X=B.

Find matrix A to the power (-1) by first finding , note that it should not be equal to zero. After this, multiply the resulting matrix by matrix B, as a result you will receive the desired matrix X, indicating all the values.

You can also find a solution to a system of three equations using Cramer's method. To do this, find the third-order determinant ∆ corresponding to the system matrix. Then successively find three more determinants ∆1, ∆2 and ∆3, substituting the values ​​of the free terms instead of the values ​​of the corresponding columns. Now find x: x1=∆1/∆, x2=∆2/∆, x3=∆3/∆.

Sources:

• solutions to equations with three unknowns

When starting to solve a system of equations, figure out what kind of equations they are. Methods for solving linear equations have been studied quite well. Nonlinear equations are most often not solved. There are only one special cases, each of which is practically individual. Therefore, the study of solution techniques should begin with linear equations. Such equations can even be solved purely algorithmically.

Instructions

Begin your learning process by learning how to solve a system of two linear equations with two unknowns X and Y by elimination. a11*X+a12*Y=b1 (1); a21*X+a22*Y=b2 (2). The coefficients of the equations are indicated by indices indicating their locations. Thus, the coefficient a21 emphasizes the fact that it is written in the first place in the second equation. In generally accepted notation, the system is written by equations located one below the other and jointly denoted by a curly bracket on the right or left (for more details, see Fig. 1a).

The numbering of equations is arbitrary. Choose the simplest one, such as one in which one of the variables is preceded by a coefficient of 1 or at least an integer. If this is equation (1), then next express, say, the unknown Y in terms of X (the case of excluding Y). To do this, transform (1) to the form a12*Y=b1-a11*X (or a11*X=b1-a12*Y when excluding X)), and then Y=(b1-a11*X)/a12. Substituting the latter into equation (2) write a21*X+a22*(b1-a11*X)/a12=b2. Solve this equation for X.
a21*X+a22*b1/a12-a11*a22*X/a12=b2; (a21-a11*a22/a12)*X=b2-a22*b1/a12;
X=(a12* b2-a22*b1)/(a12*a21-a11*a22) or X=(a22* b1-a12*b2)/(a11*a22-a12*a21).
Using the found connection between Y and X, you will finally obtain the second unknown Y=(a11* b2-a21*b1)/(a11*a22-a12*a21).

If the system were specified with specific numerical coefficients, then the calculations would be less cumbersome. But the general solution makes it possible to consider the fact that the unknowns found are exactly the same. Yes, and the numerators show some patterns in their construction. If the dimension of the system of equations were greater than two, then the elimination method would lead to very cumbersome calculations. To avoid them, purely algorithmic solutions have been developed. The simplest of them is Cramer's algorithm (Cramer's formulas). For you should find out the general system of equations of n equations.

A system of n linear algebraic equations with n unknowns has the form (see Fig. 1a). In it, aij are the coefficients of the system,
xj – unknowns, bi – free terms (i=1, 2, ... , n; j=1, 2, ... , n). Such a system can be written compactly in matrix form AX=B. Here A is the matrix of system coefficients, X is the column matrix of unknowns, B is the column matrix of free terms (see Figure 1b). According to Cramer's method, each unknown xi =∆i/∆ (i=1,2…,n). The determinant ∆ of the coefficient matrix is ​​called the main one, and ∆i the auxiliary one. For each unknown, the auxiliary determinant is found by replacing the i-th column of the main determinant with a column of free terms. The Cramer method for the case of second- and third-order systems is presented in detail in Fig. 2.

The system is a combination of two or more equalities, each of which contains two or more unknowns. There are two main ways to solve systems of linear equations that are used in the school curriculum. One of them is called the method, the other - the addition method.

Standard form of a system of two equations

Solving the system using the addition method

For example, suppose a system consisting of two equations is given. The first of them has the form 2x+4y=8, the second has the form 6x+2y=6. One of the options for completing the task is to multiply the second equation by a coefficient of -2, which will lead it to the form -12x-4y=-12. The correct choice of coefficient is one of the key tasks in the process of solving a system using the addition method, since it determines the entire further course of the procedure for finding unknowns.

Now it is necessary to add the two equations of the system. Obviously, the mutual destruction of variables with coefficients equal in value but opposite in sign will lead to the form -10x=-4. After this, it is necessary to solve this simple equation, from which it clearly follows that x = 0.4.

The last step in the solution process is to substitute the found value of one of the variables into any of the original equalities available in the system. For example, substituting x=0.4 into the first equation, you can get the expression 2*0.4+4y=8, from which y=1.8. Thus, x=0.4 and y=1.8 are the roots of the example system.

In order to make sure that the roots were found correctly, it is useful to check by substituting the found values ​​into the second equation of the system. For example, in this case we get an equality of the form 0.4*6+1.8*2=6, which is correct.

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A linear equation with two variables has the general form ax + by + c = 0. In it, a, b and c are coefficients - some numbers; and x and y are variables - unknown numbers that need to be found.

The solution to a linear equation with two variables is a pair of numbers x and y, for which ax + by + c = 0 is a true equality.

A given linear equation in two variables (for example, 3x + 2y – 1 = 0) has a set of solutions, that is, a set of pairs of numbers for which the equation is true. A linear equation with two variables is transformed into a linear function of the form y = kx + m, which is a straight line on the coordinate plane. The coordinates of all points lying on this line are solutions to a linear equation in two variables.

If two linear equations of the form ax + by + c = 0 are given and it is required to find values ​​of x and y for which both of them will have solutions, then we say that we must solve system of equations. A system of equations is written under a common curly brace. Example:

A system of equations can have no solution if the lines that are the graphs of the corresponding linear functions do not intersect (that is, parallel to each other). To conclude that there is no solution, it is enough to transform both linear equations with two variables to the form y = kx + m. If k is the same number in both equations, then the system has no solutions.

If a system of equations turns out to consist of two identical equations (which may not be obvious immediately, but after transformations), then it has an infinite number of solutions. In this case we talk about uncertainty.

In all other cases, the system has one solution. This conclusion can be drawn from the fact that any two non-parallel lines can intersect only at one point. It is this intersection point that will lie on both the first line and the second, that is, it will be a solution to both the first equation and the second. Therefore, it is a solution to a system of equations. However, it is necessary to stipulate situations when certain restrictions are imposed on the values ​​of x and y (usually according to the conditions of the problem). For example, x > 0, y > 0. In this case, even if the system of equations has a solution, but it does not satisfy the condition, then the conclusion is drawn that the system of equations has no solutions under the given conditions.

There are three ways to solve a system of equations:

1. By selection method. Most often this is very difficult to do.
2. Graphic method. When two straight lines (graphs of functions of the corresponding equations) are drawn on the coordinate plane and their point of intersection is found. This method may not give accurate results if the coordinates of the intersection point are fractional numbers.
3. Algebraic methods. They are versatile and reliable.