The linear equation has the form. Solving simple linear equations

Learning to solve equations is one of the main tasks that algebra poses for 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 that need to be performed with equations from the first group, it will be difficult to understand the others.

To continue the conversation, you 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 = b,

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

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

In the case where an unknown quantity is in the denominator of a fraction, you need to determine its values at which the expression will not make sense. In other words, you need to know the domain of definition of the equation.

The principle by which all linear equations are solved comes down to dividing the value on the right side of the equation by the coefficient in front of the variable. That is, “x” will be equal to b/a.

Special cases of linear equations and their solutions

During reasoning, moments may arise 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 such an equation will always be x = 0.

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

0 * x = 0.

The answer to such an equation will be 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 it.

General view of a linear equation with two variables

From its name it becomes clear that there are already two unknown quantities in it. Linear equations in two variables look like this:

a * x + b * y = c.

Since there are two unknowns in the record, 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. A pair of quantities for 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 down first. Sometimes they say 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 select 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 steps to simplify the equation. They are called identity transformations. Moreover, the following properties are always true for equations:

each term can be moved 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, as long as it is not equal to zero.

Examples of tasks with linear equations

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

In the equation that comes first on this list, simply divide 20 by 4. The result will be 5. This is the answer: x = 5.

The third equation requires that an identity transformation be performed. It will consist of opening the brackets and bringing similar terms. After the first step, the equation will take the form: 8x - 8 + 2x = 8 - 4x. Then you need to move all the unknowns to the left side of the equation, and the rest to the right. The equation will look like this: 8x + 2x + 4x = 8 + 8. After adding similar terms: 14x = 16. Now it looks the same as the first one, and its solution is easy to find. The answer will be x=8/7. But in mathematics you are 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 at what values the equations are defined. To do this, you need to exclude numbers at which the denominators go to zero. In the first example it is “-4”, in the second it is “-3”. That is, these values need to be excluded from the answer. After this, you need to multiply both sides of the equality by the expressions in the denominator.

Opening the brackets and bringing similar terms, in the first of these equations we get: 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 latter has no solutions.

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

Suppose that the first unknown x = 1, then the equation will take the form -7 * 1 + 2y = 5. Moving to right side equality, the factor is “-7” and changing its sign to plus, it turns out that 2y = 12. This means y = 6. Answer: one of the solutions to 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 ≥b;
a * x ≤в.

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

Rules for identity transformations of inequalities

Just like linear equations, inequalities can be modified according to certain laws. They boil down to the following:

any alphabetic or numerical expression can be added to the left and right sides of the inequality, and the sign of the inequality remains the same;
You can also multiply or divide by the same thing positive number, this again does not change the sign;
when multiplying or dividing by the same thing a negative number equality will remain true provided the inequality sign is reversed.

General view of double inequalities

The following inequalities can be presented in problems:

V< а * х < с;
c ≤ a * x< с;
V< 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 ordinary 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 them is its image on the coordinate axis. There is no need to use this method for simple inequalities. But in difficult cases it may simply be necessary.

To depict an inequality, you need to mark on the axis all the points that were obtained during the reasoning. These are invalid values, which are indicated by punctured dots, and values from inequalities obtained after transformations. Here, too, it is important to draw the dots correctly. If the inequality is strict, that is< или >, then these values are punched out. In non-strict inequalities, the points must be shaded.

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

The second feature is related to its recording. There are two options offered here. The first is ultimate inequality. The second is in the form of intervals. It happens with him that difficulties arise. The answer in spaces always looks like a variable with a membership sign and parentheses with numbers. Sometimes there are several spaces, then between the brackets you need to write the symbol “and”. These signs look like this: ∈ and ∩. Spacing brackets also play a role. The round one is placed when the point is excluded from the answer, and the rectangular one includes this value. The infinity sign is always in parentheses.

Examples of solving inequalities

1. Solve the inequality 7 - 5x ≥ 37.

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

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

First you need to subtract 6 everywhere. You get: -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].

To continue the conversation, you 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 = b,

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

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

Special cases of linear equations and their solutions

During reasoning, moments may arise 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 such an equation will always be x = 0.

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

0 * x = 0.

The answer to such an equation will be 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 it.

General view of a linear equation with two variables

From its name it becomes clear that there are already two unknown quantities in it. Linear equations in two variables look like this:

a * x + b * y = c.

How to solve a linear equation with two unknowns?

To do this, you just need to select 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 steps to simplify the equation. They are called identity transformations. Moreover, the following properties are always true for equations:

each term can be moved 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, as long as it is not equal to zero.

Examples of tasks with linear equations

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

In the equation that comes first on this list, simply divide 20 by 4. The result will be 5. This is the answer: x = 5.

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

Suppose that the first unknown x = 1, then the equation will take the form -7 * 1 + 2y = 5. Moving the factor “-7” to the right side of the equality and changing its sign to plus, it turns out that 2y = 12. This means y =6. Answer: one of the solutions to 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 ≥b;
a * x ≤в.

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

Rules for identity transformations of inequalities

Just like linear equations, inequalities can be modified according to certain laws. They boil down to the following:

any alphabetic or numerical expression can be added to the left and right sides of the inequality, and the sign of the inequality remains the same;
you can also multiply or divide by the same positive number, this again does not change the sign;
When multiplying or dividing by the same negative number, the equality will remain true provided that the inequality sign is reversed.

General view of double inequalities

The following inequalities can be presented in problems:

V< а * х < с;
c ≤ a * x< с;
V< а * х ≤ с;
c ≤ a * x ≤ c.

Features of solving double inequalities

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

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

Examples of solving inequalities

1. Solve the inequality 7 - 5x ≥ 37.

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

Equations. To put it another way, the solution of all equations begins with these transformations. When deciding linear equations, it (the solution) is based on identity transformations and ends with the final answer.

The case of a non-zero coefficient for an unknown variable.

ax+b=0, a ≠ 0

We move terms with X to one side, and numbers to the other side. Be sure to remember that when moving terms to the opposite side of the equation, you need to change the sign:

ax:(a)=-b:(a)

Let's shorten A at X and we get:

x=-b:(a)

This is the answer. If you need to check if a number is -b:(a) root of our equation, then we need to substitute in the initial equation instead X this is the number:

a(-b:(a))+b=0 ( those. 0=0)

Because this equality is correct, then -b:(a) and truth is the root of the equation.

Answer: x=-b:(a), a ≠ 0.

First example:

5x+2=7x-6

We move members with to one side X, and on the other side the numbers:

5x-7x=-6-2

-2x:(-2)=-8:(-2)

For an unknown factor, we reduced the coefficient and got the answer:

This is the answer. If you need to check whether the number 4 is really the root of our equation, we substitute this number instead of X in the original equation:

5*4+2=7*4-6 ( those. 22=22)

Because this equality is true, then 4 is the root of the equation.

Second example:

Solve the equation:

5x+14=x-49

By moving the unknowns and numbers in different directions, we got:

Divide the parts of the equation by the coefficient at x(by 4) and we get:

Third example:

Solve the equation:

First, we get rid of the irrationality in the coefficient for the unknown by multiplying all terms by:

This form is considered to be simplified, because the number has the root of the number in the denominator. We need to simplify the answer by multiplying the numerator and denominator by same number, we have this:

The case of no solutions.

Solve the equation:

2x+3=2x+7

In front of everyone x our equation will not become a true equality. That is, our equation has no roots.

Answer: there are no solutions.

A special case is an infinite number of solutions.

Solve the equation:

2x+3=2x+3

Moving the x's and numbers in different directions and adding similar terms, we get the equation:

Here, too, it is not possible to divide both parts by 0, because it is forbidden. However, putting in place X any number, we get the correct equality. That is, every number is a solution to such an equation. Thus, there is an infinite number of solutions.

Answer: an infinite number of solutions.

The case of equality of two complete forms.

ax+b=cx+d

ax-cx=d-b

(a-c)x=d-b

x=(d-b):(a-c)

Answer: x=(d-b):(a-c), If d≠b and a≠c, otherwise there are infinitely many solutions, but if a=c, A d≠b, then there are no solutions.

Linear equations are a fairly harmless and clear topic school mathematics. But, oddly enough, the number of errors out of the blue when solving linear equations is only slightly less than in other topics - quadratic equations, logarithms, trigonometry and others. The causes of most errors are banal identical transformations of equations. First of all, this is confusion in signs when transferring terms from one part of the equation to another, as well as errors when working with fractions and fractional coefficients. Yes Yes! Fractions also appear in linear equations! All around. Below we will definitely analyze such evil equations.)

Well, let’s not pull the cat by the tail and let’s start figuring it out, shall we? Then we read and delve into it.)

What is a linear equation? Examples.

Typically the linear equation looks like this:

ax + b = 0,

Where a and b are any numbers. Any kind: integers, fractions, negative, irrational - there can be any!

For example:

7x + 1 = 0 (here a = 7, b = 1)

x – 3 = 0 (here a = 1, b = -3)

x/2 – 1.1 = 0 (here a = 1/2, b = -1.1)

In general, you understand, I hope.) Everything is simple, like in a fairy tale. For the time being... And if you take a closer look at the general notation ax+b=0, and think a little? After all, a and b are any numbers! And if we have, say, a = 0 and b = 0 (any numbers can be taken!), then what will we get?

0 = 0

But that's not all the fun! What if, say, a = 0, b = -10? Then it turns out to be some kind of nonsense:

0 = 10.

Which is very, very annoying and undermines the trust in mathematics that we have gained through sweat and blood... Especially during tests and exams. But out of these incomprehensible and strange equalities, you also need to find X! Which doesn’t exist at all! And here, even well-prepared students can sometimes fall into what is called a stupor... But don’t worry! In this lesson we will also look at all such surprises. And we will definitely find an X from such equalities.) Moreover, this same X can be found very, very simply. Yes Yes! Surprising but true.)

Okay, that's understandable. But how can you tell by the appearance of the task that it is a linear equation and not some other equation? Unfortunately, it is not always possible to recognize the type of equation just by appearance. The point is that not only equations of the form ax + b = 0 are called linear, but also any other equations that, in one way or another, can be reduced to this form by identical transformations. How do you know if it adds up or not? Until you can hardly solve the example - almost not at all. This is upsetting. But for some types of equations, you can immediately tell with confidence whether it is linear or not with one quick glance.

To do this, let’s look once again at the general structure of any linear equation:

ax + b = 0

Please note: in the linear equation Always only variable x is present in the first degree and some numbers! That's all! Nothing else. At the same time, there are no X’s in the square, in the cube, under the root, under the logarithm and other exotic things. And (most importantly!) there are no fractions with X in the denominators! But fractions with numbers in the denominators or division per number- easily!

For example:

This is a linear equation. The equation contains only X's to the first power and numbers. And there are no X's in higher powers - squared, cubed, and so on. Yes, there are fractions here, but at the same time the denominators of the fractions contain only numbers. Namely - two and three. In other words, there is no division by x.

And here is the equation

It can no longer be called linear, although here, too, there are only numbers and X’s to the first power. Because, among other things, there are also fractions with X's in the denominators. And after simplifications and transformations, such an equation can become anything: linear, quadratic - anything.

How to solve linear equations? Examples.

So how do you solve linear equations? Read on and be surprised.) The entire solution of linear equations is based on just two main things. Let's list them.

1) A set of elementary actions and rules of mathematics.

These are using parentheses, opening parentheses, working with fractions, working with negative numbers, multiplication tables, and so on. This knowledge and skills are necessary not only for solving linear equations, but for all mathematics in general. And, if you have problems with this, remember the lower grades. Otherwise you will have a hard time...

There are only two of them. Yes Yes! Moreover, these very basic identity transformations underlie the solution of not only linear, but generally any mathematical equations! In a word, the solution to any other equation - quadratic, logarithmic, trigonometric, irrational, etc. – as a rule, it begins with these very basic transformations. But the solution of linear equations, in fact, ends with them (transformations). Ready answer.) So don’t be lazy and take a look at the link.) Moreover, linear equations are also analyzed in detail there.

Well, I think it's time to start looking at examples.

To begin with, as a warm-up, let's look at some basic stuff. Without any fractions or other bells and whistles. For example, this equation:

x – 2 = 4 – 5x

This is a classic linear equation. All X's are at most in the first power and there is no division by X anywhere. The solution scheme in such equations is always the same and terribly simple: all terms with X's must be collected on the left, and all terms without X's (i.e. numbers) must be collected on the right. So let's start collecting.

To do this, we launch the first identity transformation. We need to move -5x to the left, and move -2 to the right. With a change of sign, of course.) So we transfer:

x + 5x = 4 + 2

Here you go. Half the battle is done: the X's have been collected into a pile, and so have the numbers. Now we present similar ones on the left, and we count them on the right. We get:

6x = 6

What do we now lack for complete happiness? Yes, so that the pure X remains on the left! And the six gets in the way. How to get rid of it? Now we run the second identity transformation - divide both sides of the equation by 6. And - voila! The answer is ready.)

x = 1

Of course, the example is completely primitive. To general idea catch. Well, let's decide something more significant. For example, let's look at this equation:

Let's look at it in detail.) This is also a linear equation, although it would seem that there are fractions here. But in fractions there is division by two and there is division by three, but there is no division by an expression with an X! So let's decide. Using the same identical transformations, yes.)

What should we do first? With X's - to the left, without X's - to the right? In principle, this is possible. Fly to Sochi via Vladivostok.) Or you can take the shortest route, immediately using a universal and powerful method. If you know the identity transformations, of course.)

First, I ask a key question: what stands out to you most and dislikes most about this equation? 99 out of 100 people will say: fractions! And they will be right.) So let’s get rid of them first. Safe for the equation itself.) Therefore, let's start right away with second identity transformation- from multiplication. What should we multiply the left side by so that the denominator is successfully reduced? That's right, a two. What about the right side? For three! But... Mathematics is a capricious lady. She, you see, requires multiplying both sides only for the same number! Multiplying each part by its own number doesn’t work... What are we going to do? Something... Look for a compromise. In order to satisfy our desires (to get rid of fractions) and not to offend mathematics.) Let’s multiply both parts by six!) That is, by the common denominator of all fractions included in the equation. Then in one fell swoop both the two and the three will be reduced!)

So let's multiply. The entire left side and the entire right side! Therefore, we use parentheses. This is what the procedure itself looks like:

Now we open these same brackets:

Now, representing 6 as 6/1, let's multiply six by each of the fractions on the left and right. This is the usual multiplication of fractions, but so be it, I’ll describe it in detail:

And here - attention! I put the numerator (x-3) in brackets! This is all because when multiplying fractions, the numerator is multiplied entirely, entirely! And the x-3 expression must be worked as one integral structure. But if you write the numerator like this:

6x – 3,

But we have everything right and we need to finalize it. What to do next? Open the parentheses in the numerator on the left? In no case! You and I multiplied both sides by 6 to get rid of fractions, and not to worry about opening parentheses. At this stage we need reduce our fractions. With a feeling of deep satisfaction, we reduce all the denominators and get an equation without any fractions, in a ruler:

3(x-3) + 6x = 30 – 4x

And now the remaining brackets can be opened:

3x – 9 + 6x = 30 – 4x

The equation keeps getting better and better! Now let’s remember again about the first identical transformation. With a straight face we repeat the spell from junior classes: with X - to the left, without X - to the right. And apply this transformation:

3x + 6x + 4x = 30 + 9

We present similar ones on the left and count on the right:

13x = 39

It remains to divide both parts by 13. That is, apply the second transformation again. We divide and get the answer:

x = 3

The job is done. As you can see, in this equation we had to apply the first transformation once (transferring terms) and the second twice: at the beginning of the solution we used multiplication (by 6) in order to get rid of fractions, and at the end of the solution we used division (by 13), to get rid of the coefficient in front of the X. And the solution to any (yes, any!) linear equation consists of a combination of these same transformations in one sequence or another. Where exactly to start - from specific equation depends. In some places it is more profitable to start with transfer, and in others (as in this example) with multiplication (or division).

We work from simple to complex. Let's now consider outright cruelty. With a bunch of fractions and parentheses. And I’ll tell you how not to overstrain yourself.)

For example, here's the equation:

We look at the equation for a minute, are horrified, but still pull ourselves together! The main problem is where to start? You can add fractions on the right side. You can subtract fractions in parentheses. You can multiply both parts by something. Or divide... So what is still possible? Answer: everything is possible! Mathematics does not prohibit any of the listed actions. And no matter what sequence of actions and transformations you choose, the answer will always be the same - the correct one. Unless, of course, at some step you violate the identity of your transformations and, thereby, make mistakes...

And, in order not to make mistakes, in such sophisticated examples as this one, it is always most useful to evaluate it appearance and figure out in your mind: what can be done in the example so that maximum simplify it in one step?

So let's figure it out. On the left are sixes in the denominators. Personally, I don't like them, and they are very easy to remove. Let me multiply both sides of the equation by 6! Then the sixes on the left will be successfully reduced, the fractions in brackets will not go anywhere yet. Well, that's okay. We'll deal with them a little later.) But on the right, we have the denominators 2 and 3 cancelling. It is with this action (multiplying by 6) that we achieve maximum simplifications in one step!

After multiplication, our whole evil equation becomes like this:

If you don’t understand exactly how this equation came about, then you haven’t understood the analysis of the previous example well. And I tried, by the way...

So, let's reveal:

Now the most logical step would be to isolate the fractions on the left, and send 5x to the right side. At the same time, we will present similar ones on the right side. We get:

Much better already. Now the left side has prepared itself for multiplication. What should we multiply the left side by so that both the five and the four are reduced at once? On 20! But we also have disadvantages on both sides of the equation. Therefore, it will be most convenient to multiply both sides of the equation not by 20, but by -20. Then in one fell swoop both the minuses and the fractions will disappear.

So we multiply:

Anyone who still doesn’t understand this step means that the problem is not in the equations. The problems are in the basics! Let's remember again Golden Rule opening brackets:

If a number is multiplied by some expression in brackets, then this number must be sequentially multiplied by each term of this very expression. Moreover, if the number is positive, then the signs of the expressions are preserved after expansion. If negative, change to the opposite:

a(b+c) = ab+ac

-a(b+c) = -ab-ac

Our cons disappeared after multiplying both sides by -20. And now we multiply the brackets with fractions on the left by quite positive number 20. Therefore, when these brackets are opened, all the signs that were inside them are preserved. But where the brackets in the numerators of fractions come from, I already explained in detail in the previous example.

Now you can reduce fractions:

4(3-5x)-5(3x-2) = 20

Open the remaining brackets. Again, we reveal it correctly. The first brackets are multiplied by the positive number 4 and, therefore, all signs are preserved when they are opened. But the second brackets are multiplied by negative the number is -5 and, therefore, all signs are reversed:

12 - 20x - 15x + 10 = 20

There are mere trifles left. With X's to the left, without X's to the right:

-20x – 15x = 20 – 10 – 12

-35x = -2

That's almost all. On the left you need a pure X, but the number -35 is in the way. So we divide both sides by (-35). Let me remind you that the second identity transformation allows us to multiply and divide both sides by whatever number. Including negative ones.) As long as it’s not zero! Feel free to divide and get the answer:

X = 2/35

This time the X turned out to be fractional. It's OK. Such an example.)

As we can see, the principle of solving linear equations (even the most complicated ones) is quite simple: we take the original equation and, using identical transformations, successively simplify it until we get the answer. With the basics, of course! The main problems here are precisely the failure to follow the basics (for example, there is a minus in front of the brackets, and they forgot to change the signs when expanding), as well as in banal arithmetic. So don't neglect the basics! They are the foundation of all other mathematics!

Some fun things to do when solving linear equations. Or special occasions.

Everything would be fine. However... Among the linear equations there are also such funny pearls that in the process of solving them can drive you into a strong stupor. Even an excellent student.)

For example, here’s an innocuous-looking equation:

7x + 3 = 4x + 5 + 3x - 2

Yawning widely and slightly bored, we collect all the X’s on the left and all the numbers on the right:

7x-4x-3x = 5-2-3

We present similar ones, count and get:

0 = 0

That's it! I gave a sample trick! This equality in itself does not raise any objections: zero really is equal to zero. But X is missing! Without a trace! And we must write down in the answer, why equal to x . Otherwise, the decision does not count, yes.) What to do?

Don't panic! In such non-standard cases, the most general concepts and principles of mathematics. What is an equation? How to solve equations? What does it mean to solve an equation?

Solving an equation means finding All values of the variable x, which, when substituted into original equation will give us the correct equality (identity)!

But we have true equality it's already happened! 0=0, or rather, nowhere!) We can only guess at what X's we get this equality. What kind of X's can be substituted in original equation, if upon substitution all of them will they still be reduced to zero? Haven't you figured it out yet?

Surely! X's can be substituted any!!! Absolutely any. Submit whatever you want. At least 1, at least -23, at least 2.7 - whatever! They will still be reduced and as a result, the pure truth will remain. Try it, substitute it and see for yourself.)

Here's your answer:

x – any number.

In scientific notation this equality is written as follows:

This entry reads like this: “X is any real number.”

Or in another form, at intervals:

Design it the way you like best. This is a correct and completely complete answer!

Now I'm going to change just one number in our original equation. Now let’s solve this equation:

7x + 2 = 4x + 5 + 3x – 2

Again we transfer the terms, count and get:

7x – 4x – 3x = 5 – 2 – 2

0 = 1

And what do you think of this joke? There was an ordinary linear equation, but it became an incomprehensible equality

0 = 1…

Scientifically speaking, we got false equality. But in Russian this is not true. Bullshit. Nonsense.) Because zero is in no way equal to one!

And now let’s figure out again what kind of X’s, when substituted into the original equation, will give us true equality? Which? But none! No matter what X you substitute, everything will still be shortened and everything will remain crap.)

Here is the answer: no solutions.

IN mathematical notation such a response is formatted like this:

It reads: “X belongs to the empty set.”

Such answers in mathematics also occur quite often: not always do any equations have roots in principle. Some equations may not have roots at all. At all.

Here are two surprises. I hope that now the sudden disappearance of X's from the equation will not leave you perplexed forever. This is quite familiar.)

And then I hear a logical question: will they be in the OGE or the Unified State Exam? On the Unified State Examination in itself as a task - no. Too simple. But in the OGE or in word problems - easily! So now let’s train and decide:

Answers (in disarray): -2; -1; any number; 2; no solutions; 7/13.

Everything worked out? Great! You have a good chance in the exam.

Does something not add up? Hm... Sadness, of course. This means there are still gaps somewhere. Either in the basics or in identical transformations. Or it’s just a matter of simple inattention. Read the lesson again. Because this is not a topic that can be so easily dispensed with in mathematics...

Good luck! She will definitely smile at you, believe me!)

A linear equation is an algebraic equation whose total degree of polynomials is equal to one. Solving linear equations - part school curriculum, and not the most difficult. However, some still have difficulty completing this topic. We hope after reading this material, all difficulties for you will be a thing of the past. So, let's figure it out. how to solve linear equations.

General form

The linear equation is represented as:

ax + b = 0, where a and b are any numbers.

Although a and b can be any number, their values affect the number of solutions to the equation. There are several special cases of solution:

If a=b=0, the equation has an infinite number of solutions;
If a=0, b≠0, the equation has no solution;
If a≠0, b=0, the equation has a solution: x = 0.

In the event that both numbers have non-zero values, the equation must be solved to derive final expression for a variable.

How to decide?

Solving a linear equation means finding what the variable is equal to. How to do this? Yes, it’s very simple - using simple algebraic operations and following the rules of transfer. If the equation appears in front of you in general form, you are in luck; all you need to do is:

Move b to the right side of the equation, not forgetting to change the sign (transfer rule!), thus, from an expression of the form ax + b = 0, an expression of the form should be obtained: ax = -b.
Apply the rule: to find one of the factors (x - in our case), you need to divide the product (-b in our case) by another factor (a - in our case). Thus, you should get an expression of the form: x = -b/a.

That's it - a solution has been found!

Now let's look at a specific example:

2x + 4 = 0 - move b equal to in this case 4, to the right
2x = -4 - divide b by a (don’t forget about the minus sign)
x = -4/2 = -2

That's all! Our solution: x = -2.

As you can see, the solution to a linear equation with one variable is quite simple to find, but everything is so simple if we are lucky enough to come across the equation in its general form. In most cases, before solving the equation in the two steps described above, you also need to reduce the existing expression to general appearance. However, this is also not an extremely difficult task. Let's look at some special cases using examples.

Solving special cases

First, let's look at the cases that we described at the beginning of the article and explain what it means to have an infinite number of solutions and no solution.

If a=b=0, the equation will look like: 0x + 0 = 0. Performing the first step, we get: 0x = 0. What does this nonsense mean, you exclaim! After all, no matter what number you multiply by zero, you always get zero! Right! That's why they say that the equation has an infinite number of solutions - no matter what number you take, the equality will be true, 0x = 0 or 0 = 0.
If a=0, b≠0, the equation will look like: 0x + 3 = 0. Perform the first step, we get 0x = -3. Nonsense again! It is obvious that this equality will never be true! That's why they say that the equation has no solutions.
If a≠0, b=0, the equation will look like: 3x + 0 = 0. Performing the first step, we get: 3x = 0. What is the solution? It's easy, x = 0.

Lost in translation

The described special cases are not all that linear equations can surprise us with. Sometimes the equation is difficult to identify at first glance. Let's look at an example:

12x - 14 = 2x + 6

Is this a linear equation? What about the zero on the right side? We will not rush to conclusions, we will act - we will move all the components of our equation to the left side. We get:

12x - 2x - 14 - 6 = 0

Now subtract like from like, we get:

10x - 20 = 0

Learned? The most linear equation ever! The solution to which is: x = 20/10 = 2.

What if we have this example:

12((x + 2)/3) + x) = 12 (1 - 3x/4)

Yes, this is also a linear equation, only more transformations need to be carried out. First, let's open the brackets:

(12(x+2)/3) + 12x = 12 - 36x/4
4(x+2) + 12x = 12 - 36x/4
4x + 8 + 12x = 12 - 9x - now we carry out the transfer:
25x - 4 = 0 - it remains to find a solution using the already known scheme:
25x = 4,
x = 4/25 = 0.16

As you can see, everything can be solved, the main thing is not to worry, but to act. Remember, if your equation contains only variables of the first degree and numbers, you have a linear equation, which, no matter how it looks initially, can be reduced to a general form and solved. We hope everything works out for you! Good luck!