Equations. To put it another way, the solution of all equations begins with these transformations. When solving 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 the 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. 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 difficult topic in 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 on the appearance.) The trick is that linear equations are not only equations of the form 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 entire solution of linear equations consists of identical transformations of the 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. But in simple terms, this is not true. Rave. But nevertheless, this nonsense is a very good reason for the correct solution of the equation.)
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.
First you need to understand what it is.
There is a simple definition linear equation, which is given in a regular school: “an equation in which the variable occurs only in the first power.” But it is not entirely correct: the equation is not linear, it does not even reduce to that, it reduces to quadratic.
A more precise definition is: linear equation is an equation that, using equivalent transformations can be reduced to the form , where title="a,b in bbR, ~a0">. На деле мы будем приводить это уравнение к виду путём переноса в правую часть и деления обеих частей уравнения на . Осталось разъяснить, какие уравнения и как мы можем привести к такому виду, и, самое главное, что дальше делать с ними, чтобы решить его.!}
In fact, in order to understand whether an equation is linear or not, it must first be simplified, that is, brought to a form where its classification will be unambiguous. Remember, you can do whatever you want with an equation as long as it doesn’t change its roots - that’s what it is. equivalent conversion. The simplest equivalent transformations include:
- opening parentheses
- bringing similar
- multiplying and/or dividing both sides of an equation by a nonzero number
- adding and/or subtracting from both sides of the same number or expression*
*A particular interpretation of the last transformation is the “transfer” of terms from one part to another with a change of sign.
Example 1:
(let's open the brackets)
(add to both parts and subtract/transfer with changing the sign of the number to the left, and the variables to the right)
(let's give similar ones)
(divide both sides of the equation by 3)
So we get an equation that has the same roots as the original one. Let us remind the reader that "solve the equation"- means finding all its roots and proving that there are no others, and "root of the equation"- this is a number that, when substituted for the unknown, will turn the equation into a true equality. Well, in the last equation, finding a number that turns the equation into a true equality is very simple - this is the number. No other number will make an identity from this equation. Answer:
Example 2:
(multiply both sides of the equation by 
, after making sure that we are not multiplying by : title="x3/2"> и title="x3">. То есть если такие корни получатся, то мы их обязаны будем выкинуть.)!}
(let's open the brackets)
(let's move the terms)
(let's give similar ones)
(we divide both parts by )
This is roughly how all linear equations are solved. For younger readers, most likely, this explanation seemed complicated, so we offer a version "linear equations for grade 5"
A linear equation with one variable has the general form
ax + b = 0.
Here x is a variable, a and b are coefficients. In another way, a is called the “coefficient of the unknown,” b is the “free term.”
Coefficients are some kind of numbers, and solving an equation means finding the value of x at which the expression ax + b = 0 is true. For example, we have the linear equation 3x – 6 = 0. Solving it means finding what x must be equal to in order for 3x – 6 to be equal to 0. Carrying out the transformations, we get:
3x = 6
x = 2
Thus the expression 3x – 6 = 0 is true at x = 2:
3 * 2 – 6 = 0
2 is root of this equation. When you solve an equation, you find its roots.
The coefficients a and b can be any numbers, but there are such values when the root of a linear equation with one variable is more than one.
If a = 0, then ax + b = 0 turns into b = 0. Here x is “destroyed”. The expression b = 0 itself can be true only if the knowledge of b is 0. That is, the equation 0*x + 3 = 0 is false, because 3 = 0 is a false statement. However, 0*x + 0 = 0 is the correct expression. From this we conclude that if a = 0 and b ≠ 0 a linear equation with one variable has no roots at all, but if a = 0 and b = 0, then the equation has an infinite number of roots.
If b = 0, and a ≠ 0, then the equation will take the form ax = 0. It is clear that if a ≠ 0, but the result of multiplication is 0, then x = 0. That is, the root of this equation is 0.
If neither a nor b are equal to zero, then the equation ax + b = 0 is transformed to the form
x = –b/a.
The value of x in this case will depend on the values of a and b. Moreover, it will be the only one. That is, it is impossible to obtain two or more different values of x with the same coefficients. For example,
–8.5x – 17 = 0
x = 17 / –8.5
x = –2
No other number other than –2 can be obtained by dividing 17 by –8.5.
There are equations that at first glance do not resemble the general form of a linear equation with one variable, but are easily converted to it. For example,
–4.8 + 1.3x = 1.5x + 12
If you move everything to the left side, then 0 will remain on the right side:
–4.8 + 1.3x – 1.5x – 12 = 0
Now the equation is reduced to standard form and can be solved:
x = 16.8 / 0.2
x = 84
An equation with one unknown, which, after opening the brackets and bringing similar terms, takes the form
ax + b = 0, where a and b are arbitrary numbers, is called linear equation with one unknown. Today we’ll figure out how to solve these linear equations.
For example, all equations:
2x + 3= 7 – 0.5x; 0.3x = 0; x/2 + 3 = 1/2 (x – 2) - linear.
The value of the unknown that turns the equation into a true equality is called decision or root of the equation .
For example, if in the equation 3x + 7 = 13 instead of the unknown x we substitute the number 2, we obtain the correct equality 3 2 +7 = 13. This means that the value x = 2 is the solution or root of the equation.
And the value x = 3 does not turn the equation 3x + 7 = 13 into a true equality, since 3 2 +7 ≠ 13. This means that the value x = 3 is not a solution or a root of the equation.
Solving any linear equations reduces to solving equations of the form
ax + b = 0.
Let's move the free term from the left side of the equation to the right, changing the sign in front of b to the opposite, we get
If a ≠ 0, then x = ‒ b/a .
Example 1. Solve the equation 3x + 2 =11.
Let's move 2 from the left side of the equation to the right, changing the sign in front of 2 to the opposite, we get
3x = 11 – 2.
Let's do the subtraction, then
3x = 9.
To find x, you need to divide the product by a known factor, that is
x = 9:3.
This means that the value x = 3 is the solution or root of the equation.
Answer: x = 3.
If a = 0 and b = 0, then we get the equation 0x = 0. This equation has infinitely many solutions, since when we multiply any number by 0 we get 0, but b is also equal to 0. The solution to this equation is any number.
Example 2. Solve the equation 5(x – 3) + 2 = 3 (x – 4) + 2x ‒ 1.
Let's expand the brackets:
5x – 15 + 2 = 3x – 12 + 2x ‒ 1.
5x – 3x ‒ 2x = – 12 ‒ 1 + 15 ‒ 2.
Here are some similar terms:
0x = 0.
Answer: x - any number.
If a = 0 and b ≠ 0, then we get the equation 0x = - b. This equation has no solutions, since when we multiply any number by 0 we get 0, but b ≠ 0.
Example 3. Solve the equation x + 8 = x + 5.
Let's group terms containing unknowns on the left side, and free terms on the right side:
x – x = 5 – 8.
Here are some similar terms:
0х = ‒ 3.
Answer: no solutions.
On Figure 1 shows a diagram for solving a linear equation
Let's draw up a general scheme for solving equations with one variable. Let's consider the solution to Example 4.
Example 4. Suppose we need to solve the equation
1) Multiply all terms of the equation by the least common multiple of the denominators, equal to 12.
2) After reduction we get
4 (x – 4) + 3 2 (x + 1) ‒ 12 = 6 5 (x – 3) + 24x – 2 (11x + 43)
3) To separate terms containing unknown and free terms, open the brackets:
4x – 16 + 6x + 6 – 12 = 30x – 90 + 24x – 22x – 86.
4) Let us group in one part the terms containing unknowns, and in the other - free terms:
4x + 6x – 30x – 24x + 22x = ‒ 90 – 86 + 16 – 6 + 12.
5) Let us present similar terms:
- 22x = - 154.
6) Divide by – 22, We get
x = 7.
As you can see, the root of the equation is seven.
Generally such equations can be solved using the following scheme:
a) bring the equation to its integer form;
b) open the brackets;
c) group the terms containing the unknown in one part of the equation, and the free terms in the other;
d) bring similar members;
e) solve an equation of the form aх = b, which was obtained after bringing similar terms.
However, this scheme is not necessary for every equation. When solving many simpler equations, you have to start not from the first, but from the second ( Example. 2), third ( Example. 13) and even from the fifth stage, as in example 5.
Example 5. Solve the equation 2x = 1/4.
Find the unknown x = 1/4: 2,
x = 1/8 .
Let's look at solving some linear equations found in the main state exam.
Example 6. Solve the equation 2 (x + 3) = 5 – 6x.
2x + 6 = 5 – 6x
2x + 6x = 5 – 6
Answer: - 0.125
Example 7. Solve the equation – 6 (5 – 3x) = 8x – 7.
– 30 + 18x = 8x – 7
18x – 8x = – 7 +30
Answer: 2.3
Example 8. Solve the equation
3(3x – 4) = 4 7x + 24
9x – 12 = 28x + 24
9x – 28x = 24 + 12
Example 9. Find f(6) if f (x + 2) = 3 7's
Solution
Since we need to find f(6), and we know f (x + 2),
then x + 2 = 6.
We solve the linear equation x + 2 = 6,
we get x = 6 – 2, x = 4.
If x = 4 then
f(6) = 3 7-4 = 3 3 = 27
Answer: 27.
If you still have questions or want to understand solving equations more thoroughly, sign up for my lessons in the SCHEDULE. I will be glad to help you!
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