First level

Linear equations. Complete Guide (2019)

What are "linear equations"

or verbally - three friends were given apples each, based on the fact that Vasya has all the apples.

And now you have decided linear equation
Now let's give this term a mathematical definition.

Linear Equation - is an algebraic equation whose total degree of its constituent polynomials is. It looks like this:

Where and are any numbers and

For our case with Vasya and apples, we will write:

- “if Vasya gives all three friends the same number of apples, he will have no apples left”

"Hidden" linear equations, or the importance of identical transformations

Despite the fact that at first glance everything is extremely simple, when solving equations, you need to be careful, because linear equations are called not only equations of the form, but also any equations that are reduced to this form by transformations and simplifications. For example:

We see that it is on the right, which, in theory, already indicates that the equation is not linear. Moreover, if we open the brackets, we will get two more terms in which it will be, but don't jump to conclusions! Before judging whether the equation is linear, it is necessary to make all the transformations and thus simplify the original example. In this case, transformations can change the appearance, but not the very essence of the equation.

In other words, these transformations must be identical or equivalent. There are only two such transformations, but they play a very, VERY important role in solving problems. Let's consider both transformations on concrete examples.

Move left - right.

Let's say we need to solve the following equation:

Back in elementary school, we were told: "with X - to the left, without X - to the right." What expression with x is on the right? Right, not how not. And this is important, because if this seemingly simple question is misunderstood, the wrong answer comes out. And what is the expression with x on the left? Correctly, .

Now that we have dealt with this, we transfer all terms with unknowns to the left, and everything that is known to the right, remembering that if there is no sign in front of the number, for example, then the number is positive, that is, it is preceded by the sign " ".

Moved? What did you get?

All that remains to be done is to bring like terms. We present:

So, we have successfully parsed the first identical transformation, although I am sure that you already knew it and actively used it without me. The main thing - do not forget about the signs for numbers and change them to the opposite when transferring through the equal sign!

Multiplication-division.

Let's start right away with an example

We look and think: what do we not like in this example? The unknown is all in one part, the known is in the other, but something is stopping us ... And this is something - a four, because if it were not there, everything would be perfect - x is equal to a number - exactly as we need !

How can you get rid of it? We cannot transfer to the right, because then we need to transfer the entire multiplier (we cannot take it and tear it away from it), and transferring the entire multiplier also does not make sense ...

It's time to remember about the division, in connection with which we will divide everything just into! All - this means both the left and the right side. So and only so! What do we get?

Here is the answer.

Let's now look at another example:

Guess what to do in this case? That's right, multiply the left and right sides by! What answer did you get? Correctly. .

Surely you already knew everything about identical transformations. Consider that we just refreshed this knowledge in your memory and it's time for something more - For example, to solve our big example:

As we said earlier, looking at it, you cannot say that this equation is linear, but we need to open the brackets and perform identical transformations. So let's get started!

To begin with, we recall the formulas for abbreviated multiplication, in particular, the square of the sum and the square of the difference. If you don’t remember what it is and how brackets are opened, I strongly recommend reading the topic, as these skills will be useful to you when solving almost all the examples found on the exam.
Revealed? Compare:

Now it's time to bring like terms. Do you remember how we were told in the same primary classes “we don’t put flies with cutlets”? Here I am reminding you of this. We add everything separately - factors that have, factors that have, and other factors that do not have unknowns. As you bring like terms, move all unknowns to the left, and everything that is known to the right. What did you get?

As you can see, the x-square has disappeared, and we see a completely ordinary linear equation. It remains only to find!

And finally, I will say one more very important thing about identical transformations - identical transformations are applicable not only for linear equations, but also for square, fractional rational and others. You just need to remember that when transferring factors through the equal sign, we change the sign to the opposite, and when dividing or multiplying by some number, we multiply / divide both sides of the equation by the same number.

What else did you take away from this example? That looking at an equation it is not always possible to directly and accurately determine whether it is linear or not. You must first completely simplify the expression, and only then judge what it is.

Linear equations. Examples.

Here are a couple more examples for you to practice on your own - determine if the equation is linear and if so, find its roots:

Answers:

1. Is an.

2. Is not.

Let's open the brackets and give like terms:

Let's make an identical transformation - we divide the left and right parts into:

We see that the equation is not linear, so there is no need to look for its roots.

3. Is an.

Let's make an identical transformation - multiply the left and right parts by to get rid of the denominator.

Think why is it so important to? If you know the answer to this question, we move on to further solving the equation, if not, be sure to look into the topic so as not to make mistakes in more complex examples. By the way, as you can see, a situation where it is impossible. Why?
So let's go ahead and rearrange the equation:

If you coped with everything without difficulty, let's talk about linear equations with two variables.

Linear Equations with Two Variables

Now let's move on to a slightly more complicated one - linear equations with two variables.

Linear equations with two variables look like:

Where, and are any numbers and.

As you can see, the only difference is that one more variable is added to the equation. And so everything is the same - there are no x squared, there is no division by a variable, etc. etc.

What a life example to give you ... Let's take the same Vasya. Suppose he decides that he will give each of his 3 friends the same number of apples, and keep the apples for himself. How many apples does Vasya need to buy if he gives each friend an apple? What about? What if by?

The dependence of the number of apples that each person will receive on the total number of apples that need to be purchased will be expressed by the equation:

• - the number of apples that a person will receive (, or, or);
• - the number of apples that Vasya will take for himself;
• - how many apples Vasya needs to buy, taking into account the number of apples per person.

Solving this problem, we get that if Vasya gives one friend an apple, then he needs to buy pieces, if he gives apples - and so on.

And generally speaking. We have two variables. Why not plot this dependence on a graph? We build and mark the value of ours, that is, points, with coordinates, and!

As you can see, and depend on each other linearly, hence the name of the equations - “ linear».

We abstract from apples and consider graphically different equations. Look carefully at the two constructed graphs - a straight line and a parabola, given by arbitrary functions:

Find and mark the corresponding points on both figures.
What did you get?

You can see that on the graph of the first function alone corresponds one, that is, and linearly depend on each other, which cannot be said about the second function. Of course, you can object that on the second graph, x also corresponds to - , but this is only one point, that is, a special case, since you can still find one that corresponds to more than one. And the constructed graph does not resemble a line in any way, but is a parabola.

I repeat, one more time: the graph of a linear equation must be a STRAIGHT line.

With the fact that the equation will not be linear, if we go to any extent - this is understandable using the example of a parabola, although for yourself you can build a few more simple graphs, for example or. But I assure you - none of them will be a STRAIGHT LINE.

Do not believe? Build and then compare with what I got:

And what happens if we divide something by, for example, some number? Will there be a linear dependence and? We will not argue, but we will build! For example, let's plot a function graph.

Somehow it doesn’t look like a straight line built ... accordingly, the equation is not linear.
Let's summarize:

1. Linear Equation - is an algebraic equation in which the total degree of its constituent polynomials is equal.
2. Linear Equation with one variable looks like:
   , where and are any numbers;
   Linear Equation with two variables:
   , where, and are any numbers.
3. It is not always immediately possible to determine whether an equation is linear or not. Sometimes, in order to understand this, it is necessary to perform identical transformations, move similar terms to the left / right, not forgetting to change the sign, or multiply / divide both parts of the equation by the same number.

LINEAR EQUATIONS. BRIEFLY ABOUT THE MAIN

1. Linear equation

This is an algebraic equation in which the total degree of its constituent polynomials is equal.

2. Linear equation with one variable looks like:

Where and are any numbers;

3. Linear equation with two variables looks like:

Where, and are any numbers.

4. Identity transformations

To determine whether the equation is linear or not, it is necessary to make identical transformations:

• move left/right like terms, not forgetting to change the sign;
• multiply/divide both sides of the equation by the same number.

Learning to solve equations is one of the main tasks that algebra poses to students. Starting with the simplest, when it consists of one unknown, and moving on to more and more complex ones. If you have not mastered the actions to be performed with the equations from the first group, it will be difficult to deal with others.

To continue the conversation, we need to agree on notation.

General form of a linear equation with one unknown and the principle of its solution

Any equation that can be written like this:

a * x = in,

called linear. This is the general formula. But often in assignments, linear equations are written in an implicit form. Then it is required to perform identical transformations in order to obtain a generally accepted notation. These actions include:

• opening brackets;
• moving all terms with a variable value to the left side of the equality, and the rest to the right;
• reduction of like terms.

In the case when an unknown value is in the denominator of a fraction, it is necessary to determine its values ​​for which the expression will not make sense. In other words, it is supposed to know the domain of the equation.

The principle by which all linear equations are solved is to divide the value on the right side of the equation by the coefficient in front of the variable. That is, "x" will be equal to / a.

Particular cases of a linear equation and their solutions

During reasoning, there may be moments when linear equations take on one of the special forms. Each of them has a specific solution.

In the first situation:

a * x = 0, and a ≠ 0.

The solution to this equation will always be x = 0.

In the second case, "a" takes the value equal to zero:

0 * x = 0.

The answer to this equation is any number. That is, it has an infinite number of roots.

The third situation looks like this:

0*x=in, where in ≠ 0.

This equation doesn't make sense. Because there are no roots that satisfy him.

General form of a linear equation with two variables

From its name it becomes clear that there are already two unknown quantities in it. Linear Equations with Two Variables look like this:

a * x + b * y = c.

Since there are two unknowns in the entry, the answer will look like a pair of numbers. That is, it is not enough to specify only one value. This will be an incomplete answer. The pair of quantities at which the equation becomes an identity is a solution to the equation. Moreover, in the answer, the variable that comes first in the alphabet is always written first. It is sometimes said that these numbers satisfy him. Moreover, there can be an infinite number of such pairs.

How to solve a linear equation with two unknowns?

To do this, you just need to pick up any pair of numbers that turns out to be correct. For simplicity, you can take one of the unknowns equal to some prime number, and then find the second.

When solving, you often have to perform actions to simplify the equation. They are called identical transformations. Moreover, the following properties are always true for equations:

• each term can be transferred to the opposite part of the equality by replacing its sign with the opposite one;
• the left and right sides of any equation are allowed to be divided by the same number, if it is not equal to zero.

Examples of tasks with linear equations

First task. Solve linear equations: 4x \u003d 20, 8 (x - 1) + 2x \u003d 2 (4 - 2x); (5x + 15) / (x + 4) = 4; (5x + 15) / (x + 3) = 4.

In the equation that comes first in this list, it is enough to simply divide 20 by 4. The result will be 5. This is the answer: x \u003d 5.

The third equation requires that the identity transformation be performed. It will consist in opening brackets and bringing like terms. After the first action, the equation will take the form: 8x - 8 + 2x \u003d 8 - 4x. Then you need to transfer all the unknowns to the left side of the equality, and the rest to the right. The equation will look like this: 8x + 2x + 4x \u003d 8 + 8. After bringing like terms: 14x \u003d 16. Now it looks the same as the first one, and its solution is easy. The answer is x=8/7. But in mathematics it is supposed to isolate the whole part from an improper fraction. Then the result will be transformed, and "x" will be equal to one whole and one seventh.

In the remaining examples, the variables are in the denominator. This means that you first need to find out for what values ​​the equations are defined. To do this, you need to exclude numbers at which the denominators turn to zero. In the first of the examples it is "-4", in the second it is "-3". That is, these values ​​should be excluded from the answer. After that, you need to multiply both sides of the equality by the expressions in the denominator.

Opening the brackets and bringing like terms, in the first of these equations it turns out: 5x + 15 = 4x + 16, and in the second 5x + 15 = 4x + 12. After transformations, the solution to the first equation will be x = -1. The second turns out to be equal to "-3", which means that the last one has no solutions.

Second task. Solve the equation: -7x + 2y = 5.

Suppose that the first unknown x \u003d 1, then the equation will take the form -7 * 1 + 2y \u003d 5. Transferring the multiplier "-7" to the right side of the equality and changing its sign to plus, it turns out that 2y \u003d 12. So, y =6. Answer: one of the solutions of the equation x = 1, y = 6.

General form of inequality with one variable

All possible situations for inequalities are presented here:

• a * x > b;
• a*x< в;
• a*x ≥v;
• a * x ≤c.

In general, it looks like the simplest linear equation, only the equal sign is replaced by an inequality.

Rules for identical transformations of inequality

Just like linear equations, inequalities can be modified according to certain laws. They come down to this:

1. any literal or numeric expression can be added to the left and right parts of the inequality, and the inequality sign will remain the same;
2. it is also possible to multiply or divide by the same positive number, from this again the sign does not change;
3. when multiplying or dividing by the same negative number, the equality will remain true, provided that the inequality sign is reversed.

General form of double inequalities

In tasks, the following variants of inequalities can be presented:

• in< а * х < с;
• c ≤ a * x< с;
• in< а * х ≤ с;
• c ≤ a * x ≤ c.

It is called double because it is limited by inequality signs on both sides. It is solved using the same rules as the usual inequalities. And finding the answer comes down to a series of identical transformations. Until the simplest is obtained.

Features of solving double inequalities

The first of these is its image on the coordinate axis. There is no need to use this method for simple inequalities. But in difficult cases, it may be simply necessary.

To depict the inequality, it is necessary to mark on the axis all the points that were obtained during the reasoning. These are both invalid values, which are denoted by dots, and values ​​from inequalities obtained after transformations. Here, too, it is important to draw the points correctly. If the inequality is strict, then< или >, then these values ​​are punctured. In non-strict inequalities, the points must be painted over.

Then it is necessary to indicate the meaning of inequalities. This can be done with hatching or arcs. Their intersection will indicate the answer.

The second feature is related to its recording. Two options are offered here. The first is ultimate inequality. The second is in the form of gaps. This is where he gets into trouble. The answer in gaps always looks like a variable with an ownership sign and parentheses with numbers. Sometimes there are several gaps, then you need to write the “and” symbol between the brackets. These signs look like this: ∈ and ∩. The spacing brackets also play a role. Round is placed when the point is excluded from the answer, and rectangular includes this value. The infinity sign is always in parentheses.

Examples of solving inequalities

1. Solve the inequality 7 - 5x ≥ 37.

After simple transformations, it turns out: -5x ≥ 30. Dividing by “-5”, you can get the following expression: x ≤ -6. This is already an answer, but it can be written in another way: x ∈ (-∞; -6].

2. Solve the double inequality -4< 2x + 6 ≤ 8.

First you need to subtract 6 everywhere. It turns out: -10< 2x ≤ 2. Теперь нужно разделить на 2. Неравенство примет вид: -5 < x ≤ 1. Изобразив ответ на числовой оси, сразу можно понять, что результатом будет промежуток от -5 до 1. Причем первая точка исключена, а вторая включена. То есть ответ у неравенства такой: х ∈ (-5; 1].

Systems of equations are widely used in the economic industry in the mathematical modeling of various processes. For example, when solving problems of production management and planning, logistics routes (transport problem) or equipment placement.

Equation systems are used not only in the field of mathematics, but also in physics, chemistry and biology, when solving problems of finding the population size.

A system of linear equations is a term for two or more equations with several variables for which it is necessary to find a common solution. Such a sequence of numbers for which all equations become true equalities or prove that the sequence does not exist.

Linear Equation

Equations of the form ax+by=c are called linear. The designations x, y are the unknowns, the value of which must be found, b, a are the coefficients of the variables, c is the free term of the equation.
Solving the equation by plotting its graph will look like a straight line, all points of which are the solution of the polynomial.

Types of systems of linear equations

The simplest are examples of systems of linear equations with two variables X and Y.

F1(x, y) = 0 and F2(x, y) = 0, where F1,2 are functions and (x, y) are function variables.

Solve a system of equations - it means to find such values ​​(x, y) for which the system becomes a true equality, or to establish that there are no suitable values ​​of x and y.

A pair of values ​​(x, y), written as point coordinates, is called a solution to a system of linear equations.

If the systems have one common solution or there is no solution, they are called equivalent.

Homogeneous systems of linear equations are systems whose right side is equal to zero. If the right part after the "equal" sign has a value or is expressed by a function, such a system is not homogeneous.

The number of variables can be much more than two, then we should talk about an example of a system of linear equations with three variables or more.

Faced with systems, schoolchildren assume that the number of equations must necessarily coincide with the number of unknowns, but this is not so. The number of equations in the system does not depend on the variables, there can be an arbitrarily large number of them.

Simple and complex methods for solving systems of equations

There is no general analytical way to solve such systems, all methods are based on numerical solutions. The school mathematics course describes in detail such methods as permutation, algebraic addition, substitution, as well as the graphical and matrix method, the solution by the Gauss method.

The main task in teaching methods of solving is to teach how to correctly analyze the system and find the optimal solution algorithm for each example. The main thing is not to memorize a system of rules and actions for each method, but to understand the principles of applying a particular method.

The solution of examples of systems of linear equations of the 7th grade of the general education school program is quite simple and is explained in great detail. In any textbook on mathematics, this section is given enough attention. The solution of examples of systems of linear equations by the method of Gauss and Cramer is studied in more detail in the first courses of higher educational institutions.

Solution of systems by the substitution method

The actions of the substitution method are aimed at expressing the value of one variable through the second. The expression is substituted into the remaining equation, then it is reduced to a single variable form. The action is repeated depending on the number of unknowns in the system

Let's give an example of a system of linear equations of the 7th class by the substitution method:

As can be seen from the example, the variable x was expressed through F(X) = 7 + Y. The resulting expression, substituted into the 2nd equation of the system in place of X, helped to obtain one variable Y in the 2nd equation. The solution of this example does not cause difficulties and allows you to get the Y value. The last step is to check the obtained values.

It is not always possible to solve an example of a system of linear equations by substitution. The equations can be complex and the expression of the variable in terms of the second unknown will be too cumbersome for further calculations. When there are more than 3 unknowns in the system, the substitution solution is also impractical.

Solution of an example of a system of linear inhomogeneous equations:

Solution using algebraic addition

When searching for a solution to systems by the addition method, term-by-term addition and multiplication of equations by various numbers are performed. The ultimate goal of mathematical operations is an equation with one variable.

Applications of this method require practice and observation. It is not easy to solve a system of linear equations using the addition method with the number of variables 3 or more. Algebraic addition is useful when the equations contain fractions and decimal numbers.

Solution action algorithm:

1. Multiply both sides of the equation by some number. As a result of the arithmetic operation, one of the coefficients of the variable must become equal to 1.
2. Add the resulting expression term by term and find one of the unknowns.
3. Substitute the resulting value into the 2nd equation of the system to find the remaining variable.

Solution method by introducing a new variable

A new variable can be introduced if the system needs to find a solution for no more than two equations, the number of unknowns should also be no more than two.

The method is used to simplify one of the equations by introducing a new variable. The new equation is solved with respect to the entered unknown, and the resulting value is used to determine the original variable.

It can be seen from the example that by introducing a new variable t, it was possible to reduce the 1st equation of the system to a standard square trinomial. You can solve a polynomial by finding the discriminant.

It is necessary to find the value of the discriminant using the well-known formula: D = b2 - 4*a*c, where D is the desired discriminant, b, a, c are the multipliers of the polynomial. In the given example, a=1, b=16, c=39, hence D=100. If the discriminant is greater than zero, then there are two solutions: t = -b±√D / 2*a, if the discriminant is less than zero, then there is only one solution: x= -b / 2*a.

The solution for the resulting systems is found by the addition method.

A visual method for solving systems

Suitable for systems with 3 equations. The method consists in plotting graphs of each equation included in the system on the coordinate axis. The coordinates of the points of intersection of the curves will be the general solution of the system.

The graphic method has a number of nuances. Consider several examples of solving systems of linear equations in a visual way.

As can be seen from the example, two points were constructed for each line, the values ​​of the variable x were chosen arbitrarily: 0 and 3. Based on the values ​​of x, the values ​​for y were found: 3 and 0. Points with coordinates (0, 3) and (3, 0) were marked on the graph and connected by a line.

The steps must be repeated for the second equation. The point of intersection of the lines is the solution of the system.

In the following example, it is required to find a graphical solution to the system of linear equations: 0.5x-y+2=0 and 0.5x-y-1=0.

As can be seen from the example, the system has no solution, because the graphs are parallel and do not intersect along their entire length.

The systems from Examples 2 and 3 are similar, but when constructed, it becomes obvious that their solutions are different. It should be remembered that it is not always possible to say whether the system has a solution or not, it is always necessary to build a graph.

Matrix and its varieties

Matrices are used to briefly write down a system of linear equations. A matrix is ​​a special type of table filled with numbers. n*m has n - rows and m - columns.

A matrix is ​​square when the number of columns and rows is equal. A matrix-vector is a single-column matrix with an infinitely possible number of rows. A matrix with units along one of the diagonals and other zero elements is called identity.

An inverse matrix is ​​such a matrix, when multiplied by which the original one turns into a unit one, such a matrix exists only for the original square one.

Rules for transforming a system of equations into a matrix

With regard to systems of equations, the coefficients and free members of the equations are written as numbers of the matrix, one equation is one row of the matrix.

A matrix row is called non-zero if at least one element of the row is not equal to zero. Therefore, if in any of the equations the number of variables differs, then it is necessary to enter zero in place of the missing unknown.

The columns of the matrix must strictly correspond to the variables. This means that the coefficients of the variable x can only be written in one column, for example, the first, the coefficient of the unknown y - only in the second.

When multiplying a matrix, all matrix elements are sequentially multiplied by a number.

Options for finding the inverse matrix

The formula for finding the inverse matrix is ​​quite simple: K -1 = 1 / |K|, where K -1 is the inverse matrix and |K| - matrix determinant. |K| must not be equal to zero, then the system has a solution.

The determinant is easily calculated for a two-by-two matrix, it is only necessary to multiply the elements diagonally by each other. For the "three by three" option, there is a formula |K|=a 1 b 2 c 3 + a 1 b 3 c 2 + a 3 b 1 c 2 + a 2 b 3 c 1 + a 2 b 1 c 3 + a 3 b 2 c 1 . You can use the formula, or you can remember that you need to take one element from each row and each column so that the column and row numbers of the elements do not repeat in the product.

Solution of examples of systems of linear equations by the matrix method

The matrix method of finding a solution makes it possible to reduce cumbersome entries when solving systems with a large number of variables and equations.

In the example, a nm are the coefficients of the equations, the matrix is ​​a vector x n are the variables, and b n are the free terms.

Solution of systems by the Gauss method

In higher mathematics, the Gauss method is studied together with the Cramer method, and the process of finding a solution to systems is called the Gauss-Cramer method of solving. These methods are used to find the variables of systems with a large number of linear equations.

The Gaussian method is very similar to substitution and algebraic addition solutions, but is more systematic. In the school course, the Gaussian solution is used for systems of 3 and 4 equations. The purpose of the method is to bring the system to the form of an inverted trapezoid. By algebraic transformations and substitutions, the value of one variable is found in one of the equations of the system. The second equation is an expression with 2 unknowns, and 3 and 4 - with 3 and 4 variables, respectively.

After bringing the system to the described form, the further solution is reduced to the sequential substitution of known variables into the equations of the system.

In school textbooks for grade 7, an example of a Gaussian solution is described as follows:

As can be seen from the example, at step (3) two equations were obtained 3x 3 -2x 4 =11 and 3x 3 +2x 4 =7. The solution of any of the equations will allow you to find out one of the variables x n.

Theorem 5, which is mentioned in the text, states that if one of the equations of the system is replaced by an equivalent one, then the resulting system will also be equivalent to the original one.

The Gauss method is difficult for middle school students to understand, but is one of the most interesting ways to develop the ingenuity of children studying in the advanced study program in math and physics classes.

For ease of recording calculations, it is customary to do the following:

Equation coefficients and free terms are written in the form of a matrix, where each row of the matrix corresponds to one of the equations of the system. separates the left side of the equation from the right side. Roman numerals denote the numbers of equations in the system.

First, they write down the matrix with which to work, then all the actions carried out with one of the rows. The resulting matrix is ​​written after the "arrow" sign and continue to perform the necessary algebraic operations until the result is achieved.

As a result, a matrix should be obtained in which one of the diagonals is 1, and all other coefficients are equal to zero, that is, the matrix is ​​reduced to a single form. We must not forget to make calculations with the numbers of both sides of the equation.

This notation is less cumbersome and allows you not to be distracted by listing numerous unknowns.

The free application of any method of solution will require care and a certain amount of experience. Not all methods are applied. Some ways of finding solutions are more preferable in a particular area of ​​human activity, while others exist for the purpose of learning.

Learning to solve equations is one of the main tasks that algebra poses to students. Starting with the simplest, when it consists of one unknown, and moving on to more and more complex ones. If you have not mastered the actions to be performed with the equations from the first group, it will be difficult to deal with others.

To continue the conversation, we need to agree on notation.

General form of a linear equation with one unknown and the principle of its solution

Any equation that can be written like this:

a * x = in,

called linear. This is the general formula. But often in assignments, linear equations are written in an implicit form. Then it is required to perform identical transformations in order to obtain a generally accepted notation. These actions include:

• opening brackets;
• moving all terms with a variable value to the left side of the equality, and the rest to the right;
• reduction of like terms.

In the case when an unknown value is in the denominator of a fraction, it is necessary to determine its values ​​for which the expression will not make sense. In other words, it is supposed to know the domain of the equation.

The principle by which all linear equations are solved is to divide the value on the right side of the equation by the coefficient in front of the variable. That is, "x" will be equal to / a.

Particular cases of a linear equation and their solutions

During reasoning, there may be moments when linear equations take on one of the special forms. Each of them has a specific solution.

In the first situation:

a * x = 0, and a ≠ 0.

The solution to this equation will always be x = 0.

In the second case, "a" takes the value equal to zero:

0 * x = 0.

The answer to this equation is any number. That is, it has an infinite number of roots.

The third situation looks like this:

0*x=in, where in ≠ 0.

This equation doesn't make sense. Because there are no roots that satisfy him.

General form of a linear equation with two variables

From its name it becomes clear that there are already two unknown quantities in it. Linear Equations with Two Variables look like this:

a * x + b * y = c.

Since there are two unknowns in the entry, the answer will look like a pair of numbers. That is, it is not enough to specify only one value. This will be an incomplete answer. The pair of quantities at which the equation becomes an identity is a solution to the equation. Moreover, in the answer, the variable that comes first in the alphabet is always written first. It is sometimes said that these numbers satisfy him. Moreover, there can be an infinite number of such pairs.

How to solve a linear equation with two unknowns?

To do this, you just need to pick up any pair of numbers that turns out to be correct. For simplicity, you can take one of the unknowns equal to some prime number, and then find the second.

When solving, you often have to perform actions to simplify the equation. They are called identical transformations. Moreover, the following properties are always true for equations:

• each term can be transferred to the opposite part of the equality by replacing its sign with the opposite one;
• the left and right sides of any equation are allowed to be divided by the same number, if it is not equal to zero.

Examples of tasks with linear equations

First task. Solve linear equations: 4x \u003d 20, 8 (x - 1) + 2x \u003d 2 (4 - 2x); (5x + 15) / (x + 4) = 4; (5x + 15) / (x + 3) = 4.

In the equation that comes first in this list, it is enough to simply divide 20 by 4. The result will be 5. This is the answer: x \u003d 5.

The third equation requires that the identity transformation be performed. It will consist in opening brackets and bringing like terms. After the first action, the equation will take the form: 8x - 8 + 2x \u003d 8 - 4x. Then you need to transfer all the unknowns to the left side of the equality, and the rest to the right. The equation will look like this: 8x + 2x + 4x \u003d 8 + 8. After bringing like terms: 14x \u003d 16. Now it looks the same as the first one, and its solution is easy. The answer is x=8/7. But in mathematics it is supposed to isolate the whole part from an improper fraction. Then the result will be transformed, and "x" will be equal to one whole and one seventh.

In the remaining examples, the variables are in the denominator. This means that you first need to find out for what values ​​the equations are defined. To do this, you need to exclude numbers at which the denominators turn to zero. In the first of the examples it is "-4", in the second it is "-3". That is, these values ​​should be excluded from the answer. After that, you need to multiply both sides of the equality by the expressions in the denominator.

Opening the brackets and bringing like terms, in the first of these equations it turns out: 5x + 15 = 4x + 16, and in the second 5x + 15 = 4x + 12. After transformations, the solution to the first equation will be x = -1. The second turns out to be equal to "-3", which means that the last one has no solutions.

Second task. Solve the equation: -7x + 2y = 5.

Suppose that the first unknown x \u003d 1, then the equation will take the form -7 * 1 + 2y \u003d 5. Transferring the multiplier "-7" to the right side of the equality and changing its sign to plus, it turns out that 2y \u003d 12. So, y =6. Answer: one of the solutions of the equation x = 1, y = 6.

General form of inequality with one variable

All possible situations for inequalities are presented here:

• a * x > b;
• a*x< в;
• a*x ≥v;
• a * x ≤c.

In general, it looks like the simplest linear equation, only the equal sign is replaced by an inequality.

Rules for identical transformations of inequality

Just like linear equations, inequalities can be modified according to certain laws. They come down to this:

1. any literal or numeric expression can be added to the left and right parts of the inequality, and the inequality sign will remain the same;
2. it is also possible to multiply or divide by the same positive number, from this again the sign does not change;
3. when multiplying or dividing by the same negative number, the equality will remain true, provided that the inequality sign is reversed.

General form of double inequalities

In tasks, the following variants of inequalities can be presented:

• in< а * х < с;
• c ≤ a * x< с;
• in< а * х ≤ с;
• c ≤ a * x ≤ c.

It is called double because it is limited by inequality signs on both sides. It is solved using the same rules as the usual inequalities. And finding the answer comes down to a series of identical transformations. Until the simplest is obtained.

Features of solving double inequalities

The first of these is its image on the coordinate axis. There is no need to use this method for simple inequalities. But in difficult cases, it may be simply necessary.

To depict the inequality, it is necessary to mark on the axis all the points that were obtained during the reasoning. These are both invalid values, which are denoted by dots, and values ​​from inequalities obtained after transformations. Here, too, it is important to draw the points correctly. If the inequality is strict, then< или >, then these values ​​are punctured. In non-strict inequalities, the points must be painted over.

Then it is necessary to indicate the meaning of inequalities. This can be done with hatching or arcs. Their intersection will indicate the answer.

The second feature is related to its recording. Two options are offered here. The first is ultimate inequality. The second is in the form of gaps. This is where he gets into trouble. The answer in gaps always looks like a variable with an ownership sign and parentheses with numbers. Sometimes there are several gaps, then you need to write the “and” symbol between the brackets. These signs look like this: ∈ and ∩. The spacing brackets also play a role. Round is placed when the point is excluded from the answer, and rectangular includes this value. The infinity sign is always in parentheses.

Examples of solving inequalities

1. Solve the inequality 7 - 5x ≥ 37.

After simple transformations, it turns out: -5x ≥ 30. Dividing by “-5”, you can get the following expression: x ≤ -6. This is already an answer, but it can be written in another way: x ∈ (-∞; -6].

2. Solve the double inequality -4< 2x + 6 ≤ 8.

First you need to subtract 6 everywhere. It turns out: -10< 2x ≤ 2. Теперь нужно разделить на 2. Неравенство примет вид: -5 < x ≤ 1. Изобразив ответ на числовой оси, сразу можно понять, что результатом будет промежуток от -5 до 1. Причем первая точка исключена, а вторая включена. То есть ответ у неравенства такой: х ∈ (-5; 1].

And so on, it is logical to get acquainted with equations of other types. Next in line are linear equations, the purposeful study of which begins in algebra lessons in grade 7.

It is clear that first you need to explain what a linear equation is, give a definition of a linear equation, its coefficients, show its general form. Then you can figure out how many solutions a linear equation has depending on the values ​​of the coefficients, and how the roots are found. This will allow you to move on to solving examples, and thereby consolidate the studied theory. In this article we will do this: we will dwell in detail on all theoretical and practical points regarding linear equations and their solution.

Let's say right away that here we will consider only linear equations with one variable, and in a separate article we will study the principles of solving linear equations in two variables.

Page navigation.

What is a linear equation?

The definition of a linear equation is given by the form of its notation. Moreover, in different textbooks of mathematics and algebra, the formulations of the definitions of linear equations have some differences that do not affect the essence of the issue.

For example, in an algebra textbook for grade 7 by Yu. N. Makarycheva and others, a linear equation is defined as follows:

Definition.

Type equation ax=b, where x is a variable, a and b are some numbers, is called linear equation with one variable.

Let us give examples of linear equations corresponding to the voiced definition. For example, 5 x=10 is a linear equation with one variable x , here the coefficient a is 5 , and the number b is 10 . Another example: −2.3 y=0 is also a linear equation, but with the variable y , where a=−2.3 and b=0 . And in the linear equations x=−2 and −x=3.33 a are not explicitly present and are equal to 1 and −1, respectively, while in the first equation b=−2 and in the second - b=3.33 .

And a year earlier, in the textbook of mathematics by N. Ya. Vilenkin, linear equations with one unknown, in addition to equations of the form a x = b, were also considered equations that can be reduced to this form by transferring terms from one part of the equation to another with the opposite sign, as well as by reducing like terms. According to this definition, equations of the form 5 x=2 x+6 , etc. are also linear.

In turn, the following definition is given in the algebra textbook for 7 classes by A. G. Mordkovich:

Definition.

Linear equation with one variable x is an equation of the form a x+b=0 , where a and b are some numbers, called the coefficients of the linear equation.

For example, linear equations of this kind are 2 x−12=0, here the coefficient a is equal to 2, and b is equal to −12, and 0.2 y+4.6=0 with coefficients a=0.2 and b =4.6. But at the same time, there are examples of linear equations that have the form not a x+b=0 , but a x=b , for example, 3 x=12 .

Let's, so that we do not have any discrepancies in the future, under a linear equation with one variable x and coefficients a and b we will understand an equation of the form a x+b=0 . This type of linear equation seems to be the most justified, since linear equations are algebraic equations first degree. And all the other equations indicated above, as well as equations that are reduced to the form a x+b=0 with the help of equivalent transformations, will be called equations reducing to linear equations. With this approach, the equation 2 x+6=0 is a linear equation, and 2 x=−6 , 4+25 y=6+24 y , 4 (x+5)=12, etc. are linear equations.

How to solve linear equations?

Now it's time to figure out how the linear equations a x+b=0 are solved. In other words, it's time to find out if the linear equation has roots, and if so, how many and how to find them.

The presence of roots of a linear equation depends on the values ​​of the coefficients a and b. In this case, the linear equation a x+b=0 has

• the only root at a≠0 ,
• has no roots for a=0 and b≠0 ,
• has infinitely many roots for a=0 and b=0 , in which case any number is a root of a linear equation.

Let us explain how these results were obtained.

We know that in order to solve equations, it is possible to pass from the original equation to equivalent equations, that is, to equations with the same roots or, like the original one, without roots. To do this, you can use the following equivalent transformations:

• transfer of a term from one part of the equation to another with the opposite sign,
• and also multiplying or dividing both sides of the equation by the same non-zero number.

So, in a linear equation with one variable of the form a x+b=0, we can move the term b from the left side to the right side with the opposite sign. In this case, the equation will take the form a x=−b.

And then the division of both parts of the equation by the number a suggests itself. But there is one thing: the number a can be equal to zero, in which case such a division is impossible. To deal with this problem, we will first assume that the number a is different from zero, and consider the case of zero a separately a bit later.

So, when a is not equal to zero, then we can divide both parts of the equation a x=−b by a , after that it is converted to the form x=(−b): a , this result can be written using a solid line as .

Thus, for a≠0, the linear equation a·x+b=0 is equivalent to the equation , from which its root is visible.

It is easy to show that this root is unique, that is, the linear equation has no other roots. This allows you to do the opposite method.

Let's denote the root as x 1 . Suppose that there is another root of the linear equation, which we denote x 2, and x 2 ≠ x 1, which, due to definitions of equal numbers through the difference is equivalent to the condition x 1 − x 2 ≠0 . Since x 1 and x 2 are the roots of the linear equation a x+b=0, then the numerical equalities a x 1 +b=0 and a x 2 +b=0 take place. We can subtract the corresponding parts of these equalities, which the properties of numerical equalities allow us to do, we have a x 1 +b−(a x 2 +b)=0−0 , whence a (x 1 −x 2)+( b−b)=0 and then a (x 1 − x 2)=0 . And this equality is impossible, since both a≠0 and x 1 − x 2 ≠0. So we have come to a contradiction, which proves the uniqueness of the root of the linear equation a·x+b=0 for a≠0 .

So we have solved the linear equation a x+b=0 with a≠0 . The first result given at the beginning of this subsection is justified. There are two more that meet the condition a=0 .

For a=0 the linear equation a·x+b=0 becomes 0·x+b=0 . From this equation and the property of multiplying numbers by zero, it follows that no matter what number we take as x, when we substitute it into the equation 0 x+b=0, we get the numerical equality b=0. This equality is true when b=0 , and in other cases when b≠0 this equality is false.

Therefore, with a=0 and b=0, any number is the root of the linear equation a x+b=0, since under these conditions, substituting any number instead of x gives the correct numerical equality 0=0. And for a=0 and b≠0, the linear equation a x+b=0 has no roots, since under these conditions, substituting any number instead of x leads to an incorrect numerical equality b=0.

The above justifications make it possible to form a sequence of actions that allows solving any linear equation. So, algorithm for solving a linear equation is:

• First, by writing a linear equation, we find the values ​​of the coefficients a and b.
• If a=0 and b=0 , then this equation has infinitely many roots, namely, any number is a root of this linear equation.
• If a is different from zero, then
• the coefficient b is transferred to the right side with the opposite sign, while the linear equation is transformed to the form a x=−b ,
• after which both parts of the resulting equation are divided by a non-zero number a, which gives the desired root of the original linear equation.

The written algorithm is an exhaustive answer to the question of how to solve linear equations.

In conclusion of this paragraph, it is worth saying that a similar algorithm is used to solve equations of the form a x=b. Its difference lies in the fact that when a≠0, both parts of the equation are immediately divided by this number, here b is already in the desired part of the equation and it does not need to be transferred.

To solve equations of the form a x=b, the following algorithm is used:

• If a=0 and b=0 , then the equation has infinitely many roots, which are any numbers.
• If a=0 and b≠0 , then the original equation has no roots.
• If a is non-zero, then both sides of the equation are divided by a non-zero number a, from which the only root of the equation equal to b / a is found.

Examples of solving linear equations

Let's move on to practice. Let us analyze how the algorithm for solving linear equations is applied. Let us present solutions of typical examples corresponding to different values ​​of the coefficients of linear equations.

Example.

Solve the linear equation 0 x−0=0 .

Decision.

In this linear equation, a=0 and b=−0 , which is the same as b=0 . Therefore, this equation has infinitely many roots, any number is the root of this equation.

Answer:

x is any number.

Example.

Does the linear equation 0 x+2.7=0 have solutions?

Decision.

In this case, the coefficient a is equal to zero, and the coefficient b of this linear equation is equal to 2.7, that is, it is different from zero. Therefore, the linear equation has no roots.