Quadratic equations are studied in 8th grade, so there is nothing complicated here. The ability to solve them is absolutely necessary.
A quadratic equation is an equation of the form ax 2 + bx + c = 0, where the coefficients a, b and c are arbitrary numbers, and a ≠ 0.
Before studying specific methods solutions, note that all quadratic equations can be divided into three classes:
- Have no roots;
- Have exactly one root;
- They have two different roots.
This is an important difference between quadratic equations and linear ones, where the root always exists and is unique. How to determine how many roots an equation has? There is a wonderful thing for this - discriminant.
Discriminant
Let the quadratic equation ax 2 + bx + c = 0 be given. Then the discriminant is simply the number D = b 2 − 4ac.
You need to know this formula by heart. Where it comes from is not important now. Another thing is important: by the sign of the discriminant you can determine how many roots a quadratic equation has. Namely:
- If D< 0, корней нет;
- If D = 0, there is exactly one root;
- If D > 0, there will be two roots.
Please note: the discriminant indicates the number of roots, and not at all their signs, as for some reason many people believe. Take a look at the examples and you will understand everything yourself:
Task. How many roots do quadratic equations have:
- x 2 − 8x + 12 = 0;
- 5x 2 + 3x + 7 = 0;
- x 2 − 6x + 9 = 0.
Let's write out the coefficients for the first equation and find the discriminant:
a = 1, b = −8, c = 12;
D = (−8) 2 − 4 1 12 = 64 − 48 = 16
So the discriminant is positive, so the equation has two different roots. We analyze the second equation in a similar way:
a = 5; b = 3; c = 7;
D = 3 2 − 4 5 7 = 9 − 140 = −131.
The discriminant is negative, there are no roots. The last equation left is:
a = 1; b = −6; c = 9;
D = (−6) 2 − 4 1 9 = 36 − 36 = 0.
The discriminant is zero - the root will be one.
Please note that coefficients have been written down for each equation. Yes, it’s long, yes, it’s tedious, but you won’t mix up the odds and make stupid mistakes. Choose for yourself: speed or quality.
By the way, if you get the hang of it, after a while you won’t need to write down all the coefficients. You will perform such operations in your head. Most people start doing this somewhere after 50-70 solved equations - in general, not that much.
Roots of a quadratic equation
Now let's move on to the solution itself. If the discriminant D > 0, the roots can be found using the formulas:
Basic formula for the roots of a quadratic equation
When D = 0, you can use any of these formulas - you will get the same number, which will be the answer. Finally, if D< 0, корней нет — ничего считать не надо.
- x 2 − 2x − 3 = 0;
- 15 − 2x − x 2 = 0;
- x 2 + 12x + 36 = 0.
First equation:
x 2 − 2x − 3 = 0 ⇒ a = 1; b = −2; c = −3;
D = (−2) 2 − 4 1 (−3) = 16.
D > 0 ⇒ the equation has two roots. Let's find them:
Second equation:
15 − 2x − x 2 = 0 ⇒ a = −1; b = −2; c = 15;
D = (−2) 2 − 4 · (−1) · 15 = 64.
D > 0 ⇒ the equation again has two roots. Let's find them
\[\begin(align) & ((x)_(1))=\frac(2+\sqrt(64))(2\cdot \left(-1 \right))=-5; \\ & ((x)_(2))=\frac(2-\sqrt(64))(2\cdot \left(-1 \right))=3. \\ \end(align)\]
Finally, the third equation:
x 2 + 12x + 36 = 0 ⇒ a = 1; b = 12; c = 36;
D = 12 2 − 4 1 36 = 0.
D = 0 ⇒ the equation has one root. Any formula can be used. For example, the first one:
As you can see from the examples, everything is very simple. If you know the formulas and can count, there will be no problems. Most often, errors occur when substituting negative coefficients into the formula. Here again, the technique described above will help: look at the formula literally, write down each step - and very soon you will get rid of mistakes.
Incomplete quadratic equations
It happens that a quadratic equation is slightly different from what is given in the definition. For example:
- x 2 + 9x = 0;
- x 2 − 16 = 0.
It is easy to notice that these equations are missing one of the terms. Such quadratic equations are even easier to solve than standard ones: they don’t even require calculating the discriminant. So, let's introduce a new concept:
The equation ax 2 + bx + c = 0 is called an incomplete quadratic equation if b = 0 or c = 0, i.e. the coefficient of the variable x or the free element is equal to zero.
Of course, a very difficult case is possible when both of these coefficients are equal to zero: b = c = 0. In this case, the equation takes the form ax 2 = 0. Obviously, such an equation has a single root: x = 0.
Let's consider the remaining cases. Let b = 0, then we obtain an incomplete quadratic equation of the form ax 2 + c = 0. Let us transform it a little:
Since arithmetic Square root exists only from a non-negative number, the last equality makes sense only for (−c /a) ≥ 0. Conclusion:
- If in an incomplete quadratic equation of the form ax 2 + c = 0 the inequality (−c /a) ≥ 0 is satisfied, there will be two roots. The formula is given above;
- If (−c /a)< 0, корней нет.
As you can see, the discriminant was not required - in incomplete quadratic equations There are no complex calculations at all. In fact, it is not even necessary to remember the inequality (−c /a) ≥ 0. It is enough to express the value x 2 and see what is on the other side of the equal sign. If there positive number- there will be two roots. If it is negative, there will be no roots at all.
Now let's look at equations of the form ax 2 + bx = 0, in which the free element is equal to zero. Everything is simple here: there will always be two roots. It is enough to factor the polynomial:
Removal common multiplier out of bracketThe product is zero when at least one of the factors is zero. This is where the roots come from. In conclusion, let’s look at a few of these equations:
Task. Solve quadratic equations:
- x 2 − 7x = 0;
- 5x 2 + 30 = 0;
- 4x 2 − 9 = 0.
x 2 − 7x = 0 ⇒ x · (x − 7) = 0 ⇒ x 1 = 0; x 2 = −(−7)/1 = 7.
5x 2 + 30 = 0 ⇒ 5x 2 = −30 ⇒ x 2 = −6. There are no roots, because a square cannot be equal to a negative number.
4x 2 − 9 = 0 ⇒ 4x 2 = 9 ⇒ x 2 = 9/4 ⇒ x 1 = 3/2 = 1.5; x 2 = −1.5.
How to learn to solve simple and complex equations
Dear parents!
Without basic mathematical training, education is impossible modern man. At school, mathematics serves as a supporting subject for many related disciplines. In post-school life, continuous education becomes a real necessity, which requires basic school-wide training, including mathematics.
IN primary school not only knowledge on the main topics is laid, but also develops logical thinking, imagination and spatial representations, as well as the formation of interest in this subject.
Observing the principle of continuity, we will focus on the most important topic, namely, “The relationship between the components of actions in solving compound equations.”
With this lesson you can easily learn how to solve complex equations. In this lesson you will learn in detail about step by step instructions solving complicated equations.
Many parents are perplexed by the question of how to get their children to learn to solve simple and complex equations. If the equations are simple, that’s half the problem, but there are also complex ones - for example, integral ones. By the way, for information, there are also equations that the best minds of our planet are struggling to solve, and for the solution of which very significant monetary bonuses are given. For example, if you rememberPerelman and an unclaimed cash bonus of several million.
However, let us first return to simple mathematical equations and repeat the types of equations and the names of the components. A little warm-up:
_________________________________________________________________________
WARM-UP
Find the extra number in each column:
2) What word is missing in each column?
3) Connect the words from the first column with the words from the 2nd column.
"Equation" "Equality"
4) How do you explain what “equality” is?
5) What about the “equation”? Is this equality? What's special about it?
sum term
minuend difference
subtractive product
factorequality
dividend
the equation
Conclusion: An equation is an equality with a variable whose value must be found.
_______________________________________________________________________
I invite each group to write equations on a piece of paper with a felt-tip pen: (on the board)
Group 1 - with an unknown term;
group 2 - with an unknown decrement;
Group 3 – with an unknown subtrahend;
group 4 – with an unknown divisor;
Group 5 – with an unknown dividend;
Group 6 – with an unknown multiplier.
1 group x + 8 = 15
Group 2 x – 8 = 7
Group 3 48 – x = 36
4 group 540: x = 9
5 group x: 15 = 9
6 group x * 10 = 360
One of the group must read their equation in mathematical language and comment on their solution, i.e., speak out the operation being performed with the known components of the actions (algorithm).
Conclusion: We can solve simple equations of all types using an algorithm, read and write literal expressions.
I propose to solve a problem in which a new type of equation appears.
X + 2kg 5kg and 3kg
What quantity is the drawing associated with?
Create and write an equation based on this picture:
Choose the appropriate equation for the resulting equation:
x + a = b a: x = b
x: a = b x * a = b
x – a = in a – x = in
Conclusion: We got acquainted with the solution of equations, one of the parts of which contains a numerical expression, the value of which must be found and a simple equation must be obtained.
________________________________________________________________________
Let's consider another version of the equation, the solution of which is reduced to solving a chain of simple equations. Here's one introduction to compound equations.
a + b * c (x – y) : 3 2 * d + (m – n)
Are equations written?
Why?
What are such actions called?
Read them, naming the last action:
No. These are not equations because the equation must have an “=” sign.
Expressions
a + b * c - the sum of the number a and the product of the numbers b and c;
(x – y): 3 - quotient of the difference between the numbers x and y;
2 * d + (m – n) - the sum of double the number d and the difference between the numbers m and n.
I suggest everyone write down a sentence in mathematical language:
The product of the difference between the numbers x and 4 and the number 3 is 15.
Write a sentence in mathematical language: the product of the difference between the numbers x and 4 and the number 3 is equal to 15
(x – 4) * 3 = 15
CONCLUSION: The problematic situation that has arisen motivates the setting of the lesson goal: to learn to solve equations in which the unknown component is an expression. Such equations are compound equations.
__________________________________________________________________________
Or maybe the types of equations we have already studied will help us? (algorithms)
Which of the famous equations is our equation similar to? X * a = b
VERY IMPORTANT QUESTION : What is the expression on the left side - sum, difference, product or quotient?
(x – 4) * 3 = 15 (Product)
Why? (since the last action is multiplication)
Conclusion: Such equations have not yet been considered. But we can solve it if the expression x – 4 put a card (y - igrek), and you get an equation that can be easily solved using a simple algorithm for finding the unknown component.
When solving compound equations, it is necessary at each step to select an action at an automated level, commenting and naming the components of the action.
Find last action
Select unknown component
Apply rule
Simplify part
Found the root of the equation?↓ Yes
Make a check
(y – 5)
*
4
=
28
y – 5
= 28: 4
y – 5 = 7
y = 5 +7
y = 12
(12 - 5) * 4 = 28
28 = 28 (i)
Conclusion: In classes with different backgrounds, this work can be organized differently. In more prepared classes, even for primary consolidation, expressions can be used in which not two, but three or more actions, but their solution requires more steps, each step simplifying the equation until you get a simple equation. And each time you can observe how the unknown component of actions changes.
_____________________________________________________________________________
CONCLUSION:
When we are talking about something very simple and understandable, we often say: “The matter is as clear as two and two are four!”
But before they figured out that two and two equal four, people had to study for many, many thousands of years.
Many rules from school textbooks arithmetic and geometry were known to the ancient Greeks more than two thousand years ago.
Wherever you need to count, measure, compare something, you cannot do without mathematics.
It is difficult to imagine how people would live if they did not know how to count, measure, and compare. Mathematics teaches this.
Today you plunged into school life, have been in the role of students and I invite you, dear parents, to rate your skills on a scale:
My skills
Date and rating
Action components.
Drawing up an equation with an unknown component.
Reading and writing expressions.
Find the root of a simple equation.
Find the root of an equation where one of the parts contains a numerical expression.
Find the root of an equation in which the unknown component of the action is an expression.
An equation is one of the cornerstone concepts of all mathematics. Both school and higher education. It makes sense to figure it out, right? Moreover, this is a very simple concept. See for yourself below. :) So what is the equation?
The fact that this word has the same root as the words “equal”, “equality”, I think, does not raise any objections from anyone.
An equation is two mathematical expressions connected by the "=" (equal) sign.
But... not just any. And those in which (at least one) contains unknown quantity. Or, in another way, variable value. Or simply "variable" for short. Which is usually denoted by the letter "X".
There may be one variable, or there may be several. In school mathematics, equations with one variable. And for now we will also consider equations with one variable. With two or more variables - in special lessons.
What does it mean to solve an equation?
The variable included in the equation can take any mathematically acceptable values. That's why it's variable. :) For some values of the variable, the correct numerical equality is obtained, but for others, it is not.
So here it is:
To solve an equation means to find ALL such values of a variable that, when substituted into original the equation turns out to be a correct equality. Or, more scientifically, true identity. Or prove that such values of the variable do not exist.
What's happened true equality? This is an equality that is beyond doubt even for a person who is absolutely not burdened with deep mathematical knowledge. For example, 5=5, 0=0, -10=-10. And so on. :)
The values of the variable, the substitution of which achieves this same thing true equality, are called very beautifully and scientifically - roots of the equation.
There may be one root, there may be several. Or maybe infinitely many roots- an entire interval or even the entire number line from –∞ before +∞ . Yes, this happens too! It all depends on the specific equation.)
And it also happens that it is forbidden find X's that would give us true equality. In principle it is impossible. For certain reasons. There are no such X's...
In such cases it is usually said that the equation has no roots.
What are equations for?
The question is funny. For life! At school, as a rule, equations are needed to solve word problems . Let me remind you, these are tasks for work, for interest and many others.
And in adult life, without equations, it would be impossible to answer even the most ordinary, but vitally important questions of everyday life: what the weather will be like tomorrow, whether the building will withstand the given load. Or an elevator. Or a plane. Where will the rocket hit... And now there would be no weather forecasters, no engineers, no accountants, no economists, no programmers among us... As unnecessary. Does it inspire?)
Why is this so? But because equations describe almost everything known to man natural phenomena and processes. Change in air pressure and temperature with altitude, law universal gravity, bacterial growth, radioactive decay, chemical reactions, electricity, supply and demand - at the heart of it all are math equations! Simple, complex - all kinds. Whatever the phenomenon or situation, such is the equation.)
So, let's remember:
Equations are a very powerful and versatile tool for solving a wide variety of applied problems.
What are the equations?
There are countless equations in mathematics. Most different types. But the whole variety of equations can be divided into only 4 categories:
1. ,
2. ,
3. (or fractional rational),
4. Others.
Different categories of equations require different approaches to solving them: linear equations are solved in one way, quadratic equations in another, fractional equations in a third, trigonometric, logarithmic, exponential and others are also solved using their own methods.
Of course, there are more other equations, yes...) These are both irrational and trigonometric , and , and , and many other equations. And even differential equations(for students), where the role of the unknown is played not by a number, but function. Or even a family of functions. :)
In the corresponding lessons we will analyze all these types of equations in detail. And here we have basic techniques and rules.
These rules are called - identical (or equivalent) transformations of equations . There are only two of them. And there is no way around them. So let's get acquainted!
How to solve equations? Identical (equivalent) transformations of equations.
Solution any equation consists in a step-by-step transformation of the expressions included in it. But not just any transformations, but such that from step to step the essence of the whole equation has not changed. Despite the fact that after each transformation the equation will change and, ultimately, will become completely different from the original one.
Such transformations in mathematics are called equivalent or identical. There are quite a lot of them, but among the whole variety of identical transformations of equations, one stands out two basic. They will be discussed in this lesson. Yes, yes, only two! But – extremely important! And each of them deserves special attention.
Applying these two identical transformations in one order or another guarantees success in solving 99% of mathematical equations. Tempting, isn't it?
So, go ahead!
First identity transformation:
You can add (or subtract) any (but identical!) number or expression (including those with a variable) to both sides of the equation. This will not change the essence of the equation.
You apply this transformation everywhere, naively thinking that you are transferring some terms from one part of the equation to another, changing signs. :)
For example, this cool equation:
There’s nothing to think about here, we move the three to the right, changing the minus to a plus:
But what is really happening? But in reality you... add three to both sides of the equation!
Here's what's going on:
And the result is the same:
That's all. On the left there remains a pure X (which is what we, in fact, are trying to achieve), and on the right - whatever happens. But the most important thing is that from adding three to both parts the essence of the whole equation has not changed!
The fact is that the usual transfer of terms from one part to another with a change of sign is simply abridged version first identity transformation.
And why do we need to dig so deep? There is no need in equations. Take it easy and don't worry. Just don’t forget to change the signs.) But in inequalities, the habit of transference can be a little discouraging, yes...
This was the first identical transformation. Let's move on to the second.
Second identity transformation:
Both sides of the equation can be multiplied (divided) by the same non-zero number or expression.
We constantly use this identical transformation when we solve something really creepy like:
It is clear to everyone here that x=3. How did you get this answer? Did you pick it up? Did you guess it?
In order not to select and guess (we are mathematicians, not fortune tellers), you need to understand that you are simply divided both sides of the equation for a four. Which is what bothers us.
Like this:
This division stick means that they are divided by four. both parts our equation. Through fractions this procedure looks like this:
On the left, the fours are safely reduced, leaving the x in splendid isolation. And on the right, when dividing 12 by 4, the result is, of course, three. :)
And that's all.)
It sounds incredible, but these two (just two!) simple transformations are at the heart of the solution all math equations! Yes yes exactly everyone, I'm not exaggerating at all! From linear and quadratic at school to differential at university.)
Well, let's look at the identical transformations of equations in action?
Application of identity transformations to solving equations.
Let's start with first identity transformation. Transfer left and right.
Example for beginners:
1 – x = 3 – 2x
It's not a complicated matter. This . We work directly according to the spell: "With X's to the left, without X's to the right."
This mantra is a universal instruction for applying the first identity transformation. So let's look at the equation. What term with X is on the right? What? 2x? Nope!) On our right -2x (minus two x)! Therefore, when moved to the left side, the minus will change to plus:
1 – x +2x = 3
Half the job is done, the X's have been collected on the left. All that remains is to collect all the numbers on the right. There is a one on the left side of the equation. Again the question is - with what sign? The answer “with none” does not work.) There really is nothing written on the left before the 1. And this means that there is a sign in front of her "plus". That’s how it works in mathematics: nothing is written, which means it’s a plus.)
And therefore the one will move to the right with a minus:
-x + 2x = 3 - 1
That's almost all. On the left we present similar ones, and on the right we count them. And we get:
x = 2
It was a very primitive equation.
Now a cooler example, for high school students:
Solve the equation:
The equation . So what? Who cares? Anyway, the first step is to do the basic identity transformation ("With X's to the left...."). To do this, the term with X (that is, - log 3 x) move to the left. With sign change:
And the numerical expression ( log 3 4 ) move to the right. Also with a change of sign, of course:
That's all. On the right is the pure formula. Whoever is friends with , will complete the equation in his head and get:
x=3
What? Do you want sines? Please, here are the sines:
And again all the same! We perform the first identical transformation - we transfer sin x to the left (with a minus), and move -0.25 to the right (with a plus):
We got the simplest trigonometric equation with sine, which (for those in the know) is also not difficult to solve.
See how universal the first equivalent transformation is! It is found everywhere and everywhere and there is no way to get around it... That is why it is so important to be able to do it automatically and without errors.
Actually, you can only make one mistake here - forgetting to change the sign when transferring. Which is what happens all the time. Nobody canceled attentiveness, yes...)
Well, let's continue our games? Let's have fun now second transformation!)
Solve the equation:
7x=28
Cool guy, to be honest.) Okay, these are emotions...
We look and think: what is stopping us in this equation? What, what... Yes, seven is in the way! It would be nice to get rid of her. Yes, so as not to spoil the original equation.)
But how? Move right? Uh... Stop! No.) Seven with an X multiplication connected. Coefficient, you see.) You can’t tear it away from the X and move it to the right. That's the whole expression 7x in full - please (question - why?). But seven separately – no way.
It's time to remember about multiplication/division! We need pure X in the answer, don’t we? And seven is a hindrance. So we divide the left side by seven. We “clear” X from the coefficient. So us necessary. But then right side also need to be divided by seven: this is already mathematics requires. Whatever happens there will work out. But the example is good. I tried.) 28 is perfectly divisible by 7. You get 4.
Answer: x=4
Or this equation:
What is stopping us here? The fraction is 1/6, isn't it? So let's get rid of it. Safe for the equation.) How? Well, you can do the same - divide both parts by this same 1/6. But this is not very convenient in the mind. Some people will get confused...
But we not only divide, we also know how to multiply!) We remember from junior classes, after what action do we have does the fraction disappear? Right! Our fraction disappears when multiplication by a number equal to (or a multiple of) its denominator. So let’s multiply both sides of our equation by 6. On the left you’ll still get a pure X, but multiplying the right side by 6 is not the hardest job.)
That's it.) Multiplication both parts equations for the required number allows you to immediately get rid of fractions, bypassing intermediate calculations, in which, by the way, you can easily make mistakes. Shorter road - fewer mistakes!
Now back to the time machine and - to high school:
Solve the equation:
To get to X and thereby solve this cool trigonometric equation , we first need to get a pure cosine on the left, without any coefficients. But the deuce gets in the way. :) So we divide the entire left side by 2:
But then the right side will also have to be divided by two: MATHEMATICS needs this. Divide:
Got it on the right table value of cosine. And now the equation is solved for the sweet soul.)
That's all the wisdom. As you can see, identical transformations of equations are a useful thing. And at the same time not the most difficult. Transfer and multiplication/division. However, not everyone succeeds the first time and without errors, oh, not everyone... There are two main problems here.
Problem one (for the inexperienced):
Sometimes a student thinks that simplifying equations is done according to one, once and for all established rule. And he just can’t grasp and understand this rule: in some examples they start with multiplication (or division), in others they start with transfer. They transfer it about three times and never multiply it...
For example, this linear equation:
10x + 5 = 5x – 20
Where to start? You can start with the transfer:
10x – 5x = -20 - 5
Or you can first divide both parts by five, and then transfer. Then the numbers will immediately become simpler:
As we see, it is possible to decide this way and that. And this is in primitive example! This raises a question for inexperienced students: "Which is correct?"
In every way correct! Whichever is more convenient for you. :) There is no universal recipe here and cannot be. Mathematics offers you a choice of two types of equation transformations. And the order of these very transformations depends solely on the original equation, as well as on the personal preferences and habits of the decider.
Problem two (for everyone...well...almost):
Errors in calculations. In transformations you constantly have to multiply brackets. Enclose expressions in parentheses and open parentheses. Multiply and divide fractions. Work with degrees... In short, the whole set of elementary mathematics operations is available. With all the consequences...
Both of these problems can be eliminated in only one way - practice. Doubts and mistakes disappear. The examples become simpler, the tasks become easier. And in the end, it is not mathematics that commands you, but you that command mathematics. :)
Linear equations. Solution, examples.
Attention!
There are additional
materials in Special Section 555.
For those who are very "not very..."
And for those who “very much…”)
Linear equations.
Linear equations are not the most complex topic school mathematics. But there are some tricks there that can puzzle even a trained student. Let's figure it out?)
Typically a linear equation is defined as an equation of the form:
ax + b = 0 Where a and b– any numbers.
2x + 7 = 0. Here a=2, b=7
0.1x - 2.3 = 0 Here a=0.1, b=-2.3
12x + 1/2 = 0 Here a=12, b=1/2
Nothing complicated, right? Especially if you don't notice the words: "where a and b are any numbers"... And if you notice and carelessly think about it?) After all, if a=0, b=0(any numbers are possible?), then we get a funny expression:
But that's not all! If, say, a=0, A b=5, This turns out to be something completely out of the ordinary:
Which is annoying and undermines confidence in mathematics, yes...) Especially during exams. But out of these strange expressions you also need to find X! Which doesn't exist at all. And, surprisingly, this X is very easy to find. We will learn to do this. In this lesson.
How to recognize a linear equation by its appearance? It depends what appearance.) The trick is that not only equations of the form are called linear equations ax + b = 0 , but also any equations that can be reduced to this form by transformations and simplifications. And who knows whether it comes down or not?)
A linear equation can be clearly recognized in some cases. Let's say, if we have an equation in which there are only unknowns to the first degree and numbers. And in the equation there is no fractions divided by unknown , it is important! And division by number, or a numerical fraction - that's welcome! For example:
This is a linear equation. There are fractions here, but there are no x's in the square, cube, etc., and no x's in the denominators, i.e. No division by x. And here is the equation
cannot be called linear. Here the X's are all in the first degree, but there are division by expression with x. After simplifications and transformations, you can get a linear equation, a quadratic equation, or anything you like.
It turns out that it is impossible to recognize the linear equation in some complicated example until you almost solve it. This is upsetting. But in assignments, as a rule, they don’t ask about the form of the equation, right? The assignments ask for equations decide. This makes me happy.)
Solving linear equations. Examples.
The whole solution linear equations consists of identical transformations of equations. By the way, these transformations (two of them!) are the basis of the solutions all equations of mathematics. In other words, the solution any the equation begins with these very transformations. In the case of linear equations, it (the solution) is based on these transformations and ends with a full answer. It makes sense to follow the link, right?) Moreover, there are also examples of solving linear equations there.
First, let's look at the simplest example. Without any pitfalls. Suppose we need to solve this equation.
x - 3 = 2 - 4x
This is a linear equation. The X's are all in the first power, there is no division by X's. But, in fact, it doesn’t matter to us what kind of equation it is. We need to solve it. The scheme here is simple. Collect everything with X's on the left side of the equation, everything without X's (numbers) on the right.
To do this you need to transfer - 4x to the left side, with a change of sign, of course, and - 3 - to the right. By the way, this is the first identical transformation of equations. Surprised? This means that you didn’t follow the link, but in vain...) We get:
x + 4x = 2 + 3
Here are similar ones, we consider:
What do we need for complete happiness? Yes, so that there is a pure X on the left! Five is in the way. Getting rid of the five with the help the second identical transformation of equations. Namely, we divide both sides of the equation by 5. We get a ready answer:
An elementary example, of course. This is for warming up.) It’s not very clear why I remembered identical transformations here? OK. Let's take the bull by the horns.) Let's decide something more solid.
For example, here's the equation:
Where do we start? With X's - to the left, without X's - to the right? Could be so. Small steps along a long road. Or you can do it right away, in a universal and powerful way. If, of course, you have identical transformations of equations in your arsenal.
I ask you a key question: What do you dislike most about this equation?
95 out of 100 people will answer: fractions ! The answer is correct. So let's get rid of them. Therefore, we start immediately with second identity transformation. What do you need to multiply the fraction on the left by so that the denominator is completely reduced? That's right, at 3. And on the right? By 4. But mathematics allows us to multiply both sides by the same number. How can we get out? Let's multiply both sides by 12! Those. to a common denominator. Then both the three and the four will be reduced. Don't forget that you need to multiply each part entirely. Here's what the first step looks like:
Expanding the brackets:
Note! Numerator (x+2) I put it in brackets! This is because when multiplying fractions, the entire numerator is multiplied! Now you can reduce fractions:
Expand the remaining brackets:
Not an example, but pure pleasure!) Now let’s remember a spell from elementary school: with an X - to the left, without an X - to the right! And apply this transformation:
Here are some similar ones:
And divide both parts by 25, i.e. apply the second transformation again:
That's all. Answer: X=0,16
Please note: to bring the original confusing equation into a nice form, we used two (just two!) identity transformations– translation left-right with a change of sign and multiplication-division of an equation by the same number. This is a universal method! We will work in this way with any equations! Absolutely anyone. That’s why I tediously repeat about these identical transformations all the time.)
As you can see, the principle of solving linear equations is simple. We take the equation and simplify it using identical transformations until we get the answer. The main problems here are in the calculations, not in the principle of the solution.
But... There are such surprises in the process of solving the most elementary linear equations that they can drive you into a strong stupor...) Fortunately, there can only be two such surprises. Let's call them special cases.
Special cases in solving linear equations.
First surprise.
Suppose you come across a very basic equation, something like:
2x+3=5x+5 - 3x - 2
Slightly bored, we move it with an X to the left, without an X - to the right... With a change of sign, everything is perfect... We get:
2x-5x+3x=5-2-3
We count, and... oops!!! We get:
This equality in itself is not objectionable. Zero really is zero. But X is missing! And we must write down in the answer, what is x equal to? Otherwise, the solution doesn't count, right...) Deadlock?
Calm! In such doubtful cases, the most general rules will save you. How to solve equations? What does it mean to solve an equation? This means, find all the values of x that, when substituted into the original equation, will give us the correct equality.
But we have true equality already happened! 0=0, how much more accurate?! It remains to figure out at what x's this happens. What values of X can be substituted into original equation if these x's will they still be reduced to zero? Come on?)
Yes!!! X's can be substituted any! Which ones do you want? At least 5, at least 0.05, at least -220. They will still shrink. If you don’t believe me, you can check it.) Substitute any values of X into original equation and calculate. All the time you will get the pure truth: 0=0, 2=2, -7.1=-7.1 and so on.
Here's your answer: x - any number.
The answer can be written in different mathematical symbols, the essence does not change. This is a completely correct and complete answer.
Second surprise.
Let's take the same elementary linear equation and change just one number in it. This is what we will decide:
2x+1=5x+5 - 3x - 2
After the same identical transformations, we get something intriguing:
Like this. We solved a linear equation and got a strange equality. In mathematical terms, we got false equality. And speaking in simple language, this is not true. Rave. But nevertheless, this nonsense is a very good reason for the right decision equations.)
Again we think based on general rules. What x's, when substituted into the original equation, will give us true equality? Yes, none! There are no such X's. No matter what you put in, everything will be reduced, only nonsense will remain.)
Here's your answer: there are no solutions.
This is also a completely complete answer. In mathematics, such answers are often found.
Like this. Now, I hope, the disappearance of X's in the process of solving any (not just linear) equation will not confuse you at all. This is already a familiar matter.)
Now that we have dealt with all the pitfalls in linear equations, it makes sense to solve them.
If you like this site...
By the way, I have a couple more interesting sites for you.)
You can practice solving examples and find out your level. Testing with instant verification. Let's learn - with interest!)
You can get acquainted with functions and derivatives.
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