Let us analyze two types of solutions to systems of equations:
1. Solving the system using the substitution method.
2. Solving the system by term-by-term addition (subtraction) of the system equations.
In order to solve the system of equations by substitution method you need to follow a simple algorithm:
1. Express. From any equation we express one variable.
2. Substitute. We substitute the resulting value into another equation instead of the expressed variable.
3. Solve the resulting equation with one variable. We find a solution to the system.
To decide system by term-by-term addition (subtraction) method need to:
1. Select a variable for which we will make identical coefficients.
2. We add or subtract equations, resulting in an equation with one variable.
3. Solve the resulting linear equation. We find a solution to the system.
The solution to the system is the intersection points of the function graphs.
Let us consider in detail the solution of systems using examples.
Example #1:
Let's solve by substitution method
Solving a system of equations using the substitution method2x+5y=1 (1 equation)
x-10y=3 (2nd equation)
1. Express
It can be seen that in the second equation there is a variable x with a coefficient of 1, which means that it is easiest to express the variable x from the second equation.
x=3+10y
2.After we have expressed it, we substitute 3+10y into the first equation instead of the variable x.
2(3+10y)+5y=1
3. Solve the resulting equation with one variable.
2(3+10y)+5y=1 (open the brackets)
6+20y+5y=1
25y=1-6
25y=-5 |: (25)
y=-5:25
y=-0.2
The solution to the equation system is the intersection points of the graphs, therefore we need to find x and y, because the intersection point consists of x and y. Let's find x, in the first point where we expressed it, we substitute y there.
x=3+10y
x=3+10*(-0.2)=1
It is customary to write points in the first place we write the variable x, and in the second place the variable y.
Answer: (1; -0.2)
Example #2:
Let's solve using the term-by-term addition (subtraction) method.
Solving a system of equations using the addition method3x-2y=1 (1 equation)
2x-3y=-10 (2nd equation)
1. We choose a variable, let’s say we choose x. In the first equation, the variable x has a coefficient of 3, in the second - 2. We need to make the coefficients the same, for this we have the right to multiply the equations or divide by any number. We multiply the first equation by 2, and the second by 3 and get a total coefficient of 6.
3x-2y=1 |*2
6x-4y=2
2x-3y=-10 |*3
6x-9y=-30
2. Subtract the second from the first equation to get rid of the variable x. Solve the linear equation.
__6x-4y=2
5y=32 | :5
y=6.4
3. Find x. We substitute the found y into any of the equations, let’s say into the first equation.
3x-2y=1
3x-2*6.4=1
3x-12.8=1
3x=1+12.8
3x=13.8 |:3
x=4.6
The intersection point will be x=4.6; y=6.4
Answer: (4.6; 6.4)
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Equations
How to solve equations?
In this section we will recall (or study, depending on who you choose) the most elementary equations. So what is the equation? In human terms, this is some kind of mathematical expression where there is an equal sign and an unknown. Which is usually denoted by the letter "X". Solve the equation- this is to find such values of x that, when substituted into original expression will give us the correct identity. Let me remind you that identity is an expression that is beyond doubt even for a person who is absolutely not burdened with mathematical knowledge. Like 2=2, 0=0, ab=ab, etc. So how to solve equations? Let's figure it out.
There are all sorts of equations (I’m surprised, right?). But all their infinite variety can be divided into only four types.
4. Everyone else.)
All the rest, of course, most of all, yes...) This includes cubic, exponential, logarithmic, trigonometric and all sorts of others. We will work closely with them in the appropriate sections.
I’ll say right away that sometimes the equations of the first three types they will cheat you so much that you won’t even recognize them... Nothing. We will learn how to unwind them.
And why do we need these four types? And then what linear equations solved in one way square others, fractional rationals - third, A rest They don’t dare at all! Well, it’s not that they can’t decide at all, it’s that I was wrong with mathematics.) It’s just that they have their own special techniques and methods.
But for any (I repeat - for any!) equations provide a reliable and fail-safe basis for solving. Works everywhere and always. This foundation - Sounds scary, but it's very simple. And very (Very!) important.
Actually, the solution to the equation consists of these very transformations. 99% Answer to the question: " How to solve equations?" lies precisely in these transformations. Is the hint clear?)
Identical transformations of equations.
IN any equations To find the unknown, you need to transform and simplify the original example. And so that when changing appearance the essence of the equation has not changed. Such transformations are called identical or equivalent.
Note that these transformations apply specifically to the equations. There are also identity transformations in mathematics expressions. This is another topic.
Now we will repeat all, all, all basic identical transformations of equations.
Basic because they can be applied to any equations - linear, quadratic, fractional, trigonometric, exponential, logarithmic, etc. etc.
First identity transformation: you can add (subtract) to both sides of any equation any(but one and the same!) number or expression (including an expression with an unknown!). This does not change the essence of the equation.
By the way, you constantly used this transformation, you just thought that you were transferring some terms from one part of the equation to another with a change of sign. Type:
The case is familiar, we move the two to the right, and we get:
Actually you taken away from both sides of the equation is two. The result is the same:
x+2 - 2 = 3 - 2
Moving terms left and right with a change of sign is simply a shortened version of the first identical transformation. And why do we need such deep knowledge? – you ask. Nothing in the equations. For God's sake, bear it. Just don’t forget to change the sign. But in inequalities, the habit of transference can lead to a dead end...
Second identity transformation: both sides of the equation can be multiplied (divided) by the same thing non-zero number or expression. Here an understandable limitation already appears: multiplying by zero is stupid, and dividing is completely impossible. This is the transformation you use when you solve something cool like
It's clear X= 2. How did you find it? By selection? Or did it just dawn on you? In order not to select and not wait for insight, you need to understand that you are just divided both sides of the equation by 5. When dividing the left side (5x), the five was reduced, leaving pure X. Which is exactly what we needed. And when dividing the right side of (10) by five, the result is, of course, two.
That's it.
It's funny, but these two (only two!) identical transformations are the basis of the solution all equations of mathematics. Wow! It makes sense to look at examples of what and how, right?)
Examples of identical transformations of equations. Main problems.
Let's start with first identity transformation. Transfer left-right.
An example for the younger ones.)
Let's say we need to solve the following equation:
3-2x=5-3x
Let's remember the spell: "with X's - to the left, without X's - to the right!" This spell is instructions for using the first identity transformation.) What is the expression with an X on the right? 3x? The answer is incorrect! On our right - 3x! Minus three x! Therefore, when moving to the left, the sign will change to plus. It will turn out:
3-2x+3x=5
So, the X’s were collected in a pile. Let's get into the numbers. There is a three on the left. With what sign? The answer “with none” is not accepted!) In front of the three, indeed, nothing is drawn. And this means that before the three there is plus. So the mathematicians agreed. Nothing is written, which means plus. Therefore, in right side the troika will be transferred with a minus. We get:
-2x+3x=5-3
There are mere trifles left. On the left - bring similar ones, on the right - count. The answer comes straight away:
In this example, one identity transformation was enough. The second one was not needed. Well, okay.)
An example for older children.)
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The online equation solving service will help you solve any equation. Using our site, you will not only receive the answer to the equation, but also see detailed solution, that is, a step-by-step display of the process of obtaining the result. Our service will be useful to high school students and their parents. Students will be able to prepare for tests and exams, test their knowledge, and parents will be able to monitor the solution of mathematical equations by their children. Ability to solve equations – mandatory requirement to schoolchildren. The service will help you educate yourself and improve your knowledge in the field of mathematical equations. With its help you can solve any equation: quadratic, cubic, irrational, trigonometric, etc. Benefit online service and is priceless, because in addition to the correct answer, you receive a detailed solution to each equation. Benefits of solving equations online. You can solve any equation online on our website absolutely free. The service is completely automatic, you don’t have to install anything on your computer, you just need to enter the data and the program will give you a solution. Any errors in calculations or typos are excluded. With us, solving any equation online is very easy, so be sure to use our site to solve any kind of equations. You only need to enter the data and the calculation will be completed in a matter of seconds. The program works independently, without human intervention, and you receive an accurate and detailed answer. Solving the equation in general view. In such an equation, the variable coefficients and the desired roots are interconnected. The highest power of a variable determines the order of such an equation. Based on this, various methods and theorems are used for equations to find solutions. Solving equations of this type means finding the required roots in general form. Our service allows you to solve even the most complex algebraic equation online. You can obtain both a general solution to the equation and a particular one for the numerical values of the coefficients you specify. To solve an algebraic equation on the website, it is enough to correctly fill in only two fields: the left and right sides of the given equation. U algebraic equations with variable coefficients there is an infinite number of solutions, and by setting certain conditions, private ones are selected from the set of solutions. Quadratic equation. The quadratic equation has the form ax^2+bx+c=0 for a>0. Solving quadratic equations involves finding the values of x at which the equality ax^2+bx+c=0 holds. To do this, find the discriminant value using the formula D=b^2-4ac. If the discriminant less than zero, then the equation has no real roots (the roots are from the field complex numbers), if equal to zero, then the equation has one real root, and if the discriminant is greater than zero, then the equation has two real roots, which are found by the formula: D= -b+-sqrt/2a. To solve a quadratic equation online, you just need to enter the coefficients of the equation (integers, fractions or decimals). If there are subtraction signs in an equation, you must put a minus sign in front of the corresponding terms of the equation. You can solve a quadratic equation online depending on the parameter, that is, the variables in the coefficients of the equation. Our online service for finding general solutions copes well with this task. Linear equations. To solve linear equations (or systems of equations), four main methods are used in practice. We will describe each method in detail. Substitution method. Solving equations using the substitution method requires expressing one variable in terms of the others. After this, the expression is substituted into other equations of the system. Hence the name of the solution method, that is, instead of a variable, its expression is substituted through the remaining variables. In practice, the method requires complex calculations, although it is easy to understand, so solving such an equation online will help save time and make calculations easier. You just need to indicate the number of unknowns in the equation and fill in the data from the linear equations, then the service will make the calculation. Gauss method. The method is based on the simplest transformations of the system in order to arrive at an equivalent triangular system. From it, the unknowns are determined one by one. In practice, it is required to solve such an equation online with detailed description, thanks to which you will have a good understanding of the Gaussian method for solving systems of linear equations. Write down the system of linear equations in the correct format and take into account the number of unknowns in order to accurately solve the system. Cramer's method. This method solves systems of equations in cases where the system has a unique solution. The main mathematical action here is the calculation of matrix determinants. Solving equations using the Cramer method is carried out online, you receive the result instantly with a complete and detailed description. It is enough just to fill the system with coefficients and select the number of unknown variables. Matrix method. This method consists of collecting the coefficients of the unknowns in matrix A, the unknowns in column X, and the free terms in column B. Thus, the system of linear equations is reduced to a matrix equation of the form AxX = B. This equation has a unique solution only if the determinant of matrix A is different from zero, otherwise the system has no solutions, or an infinite number of solutions. Solving equations using the matrix method involves finding the inverse matrix A.
Solving exponential equations. Examples.
Attention!
There are additional
materials in Special section 555.
For those who are very "not very..."
And for those who “very much…”)
What's happened exponential equation? This is an equation in which the unknowns (x's) and expressions with them are in indicators some degrees. And only there! This is important.
Here you go examples of exponential equations:
3 x 2 x = 8 x+3
Pay attention! In the bases of degrees (below) - only numbers. IN indicators degrees (above) - a wide variety of expressions with an X. If, suddenly, an X appears in the equation somewhere other than an indicator, for example:
this will be an equation mixed type. Such equations do not have clear rules for solving them. We will not consider them for now. Here we will deal with solving exponential equations in its purest form.
In fact, even pure exponential equations are not always solved clearly. But there are certain types of exponential equations that can and should be solved. These are the types we will consider.
Solving simple exponential equations.
First, let's solve something very basic. For example:
Even without any theories, by simple selection it is clear that x = 2. Nothing more, right!? No other value of X works. Now let's look at the solution to this tricky exponential equation:
What have we done? We, in fact, simply threw out the same bases (triples). Completely thrown out. And, the good news is, we hit the nail on the head!
Indeed, if in an exponential equation there are left and right identical numbers in any powers, these numbers can be removed and the exponents can be equalized. Mathematics allows. It remains to solve a much simpler equation. Great, right?)
However, let us remember firmly: You can remove bases only when the base numbers on the left and right are in splendid isolation! Without any neighbors and coefficients. Let's say in the equations:
2 x +2 x+1 = 2 3, or
twos cannot be removed!
Well, we have mastered the most important thing. How to move from evil exponential expressions to simpler equations.
"Those are the times!" - you say. “Who would give such a primitive lesson on tests and exams!?”
I have to agree. Nobody will. But now you know where to aim when solving tricky examples. It is necessary to bring it to the form where the same base number is on the left and on the right. Then everything will be easier. Actually, this is a classic of mathematics. We take the original example and transform it to the desired one us mind. According to the rules of mathematics, of course.
Let's look at examples that require some additional effort to reduce them to the simplest. Let's call them simple exponential equations.
Solving simple exponential equations. Examples.
When solving exponential equations, the main rules are actions with degrees. Without knowledge of these actions nothing will work.
To actions with degrees, one must add personal observation and ingenuity. Do we need the same base numbers? So we look for them in the example in explicit or encrypted form.
Let's see how this is done in practice?
Let us be given an example:
2 2x - 8 x+1 = 0
The first keen look is at grounds. They... They are different! Two and eight. But it’s too early to become discouraged. It's time to remember that
Two and eight are relatives in degree.) It is quite possible to write:
8 x+1 = (2 3) x+1
If we recall the formula from operations with degrees:
(a n) m = a nm ,
this works out great:
8 x+1 = (2 3) x+1 = 2 3(x+1)
The original example began to look like this:
2 2x - 2 3(x+1) = 0
We transfer 2 3 (x+1) to the right (no one has canceled the elementary operations of mathematics!), we get:
2 2x = 2 3(x+1)
That's practically all. Removing the bases:
We solve this monster and get
This is the correct answer.
In this example, knowing the powers of two helped us out. We identified in eight there is an encrypted two. This technique (encoding common bases under different numbers) is a very popular technique in exponential equations! Yes, and in logarithms too. You must be able to recognize powers of other numbers in numbers. This is extremely important for solving exponential equations.
The fact is that raising any number to any power is not a problem. Multiply, even on paper, and that’s it. For example, anyone can raise 3 to the fifth power. 243 will work out if you know the multiplication table.) But in exponential equations, much more often it is not necessary to raise to a power, but vice versa... Find out what number to what degree is hidden behind the number 243, or, say, 343... No calculator will help you here.
You need to know the powers of some numbers by sight, right... Let's practice?
Determine what powers and what numbers the numbers are:
2; 8; 16; 27; 32; 64; 81; 100; 125; 128; 216; 243; 256; 343; 512; 625; 729, 1024.
Answers (in a mess, of course!):
5 4 ; 2 10 ; 7 3 ; 3 5 ; 2 7 ; 10 2 ; 2 6 ; 3 3 ; 2 3 ; 2 1 ; 3 6 ; 2 9 ; 2 8 ; 6 3 ; 5 3 ; 3 4 ; 2 5 ; 4 4 ; 4 2 ; 2 3 ; 9 3 ; 4 5 ; 8 2 ; 4 3 ; 8 3 .
If you look closely you can see strange fact. There are significantly more answers than tasks! Well, it happens... For example, 2 6, 4 3, 8 2 - that's all 64.
Let us assume that you have taken note of the information about familiarity with numbers.) Let me also remind you that to solve exponential equations we use all stock of mathematical knowledge. Including those from junior and middle classes. You didn’t go straight to high school, right?)
For example, when solving exponential equations, putting the common factor out of brackets often helps (hello to 7th grade!). Let's look at an example:
3 2x+4 -11 9 x = 210
And again, the first glance is at the foundations! The bases of the degrees are different... Three and nine. And we want them to be the same. Well, in this case the desire is completely fulfilled!) Because:
9 x = (3 2) x = 3 2x
Using the same rules for dealing with degrees:
3 2x+4 = 3 2x ·3 4
That’s great, you can write it down:
3 2x 3 4 - 11 3 2x = 210
We gave an example for the same reasons. And what next!? You can't throw out threes... Dead end?
Not at all. Remember the most universal and powerful decision rule everyone math tasks:
If you don’t know what you need, do what you can!
Look, everything will work out).
What's in this exponential equation Can do? Yes, on the left side it just begs to be taken out of brackets! The overall multiplier of 3 2x clearly hints at this. Let's try, and then we'll see:
3 2x (3 4 - 11) = 210
3 4 - 11 = 81 - 11 = 70
The example keeps getting better and better!
We remember that to eliminate grounds we need a pure degree, without any coefficients. The number 70 bothers us. So we divide both sides of the equation by 70, we get:
Oops! Everything got better!
This is the final answer.
It happens, however, that taxiing on the same grounds works out, but their elimination does not. This happens in other types of exponential equations. Let's master this type.
Replacing a variable in solving exponential equations. Examples.
Let's solve the equation:
4 x - 3 2 x +2 = 0
First - as usual. Let's move on to one base. To a deuce.
4 x = (2 2) x = 2 2x
We get the equation:
2 2x - 3 2 x +2 = 0
And this is where we hang out. The previous techniques will not work, no matter how you look at it. We'll have to pull out another powerful and universal method from our arsenal. It's called variable replacement.
The essence of the method is surprisingly simple. Instead of one complex icon (in our case - 2 x) we write another, simpler one (for example - t). Such a seemingly meaningless replacement leads to amazing results!) Everything just becomes clear and understandable!
So let
Then 2 2x = 2 x2 = (2 x) 2 = t 2
In our equation we replace all powers with x's by t:
Well, does it dawn on you?) Have you forgotten the quadratic equations yet? Solving through the discriminant, we get:
The main thing here is not to stop, as happens... This is not the answer yet, we need x, not t. Let's return to the X's, i.e. we make a reverse replacement. First for t 1:
Therefore,
One root was found. We are looking for the second one from t 2:
Hm... 2 x on the left, 1 on the right... Problem? Not at all! It is enough to remember (from operations with powers, yes...) that a unit is any number to the zero power. Any. Whatever is needed, we will install it. We need a two. Means:
That's it now. We got 2 roots:
This is the answer.
At solving exponential equations at the end sometimes you end up with some kind of awkward expression. Type:
From seven to two through simple degree it doesn't work. They are not relatives... How can we be? Someone may be confused... But here is a person who read the topic on this site "What is a logarithm?", just smiles sparingly and writes down with a firm hand the absolutely correct answer:
There cannot be such an answer in tasks “B” on the Unified State Examination. There a specific number is required. But in tasks “C” it’s easy.
This lesson provides examples of solving the most common exponential equations. Let's highlight the main points.
1. First of all, we look at grounds degrees. We are wondering if it is possible to make them identical. Let's try to do this by actively using actions with degrees. Don't forget that numbers without x's can also be converted to powers!
2. We try to bring the exponential equation to the form when on the left and on the right there are identical numbers in any powers. We use actions with degrees And factorization. What can be counted in numbers, we count.
3. If the second tip doesn’t work, try using variable replacement. The result may be an equation that can be easily solved. Most often - square. Or fractional, which also reduces to square.
4. To successfully solve exponential equations, you need to know the powers of some numbers by sight.
As usual, at the end of the lesson you are invited to decide a little.) On your own. From simple to complex.
Solve exponential equations:
More difficult:
2 x+3 - 2 x+2 - 2 x = 48
9 x - 8 3 x = 9
2 x - 2 0.5x+1 - 8 = 0
Find the product of roots:
2 3's + 2 x = 9
Did it work?
Well then the most complicated example(decided, however, in the mind...):
7 0.13x + 13 0.7x+1 + 2 0.5x+1 = -3
What's more interesting? Then here's a bad example for you. Quite tempting for increased difficulty. Let me hint that in this example, what saves you is ingenuity and the most universal rule for solving all mathematical problems.)
2 5x-1 3 3x-1 5 2x-1 = 720 x
A simpler example, for relaxation):
9 2 x - 4 3 x = 0
And for dessert. Find the sum of the roots of the equation:
x 3 x - 9x + 7 3 x - 63 = 0
Yes, yes! This is a mixed type equation! Which we did not consider in this lesson. Why consider them, they need to be solved!) This lesson is quite enough to solve the equation. Well, you need ingenuity... And may seventh grade help you (this is a hint!).
Answers (in disarray, separated by semicolons):
1; 2; 3; 4; there are no solutions; 2; -2; -5; 4; 0.
Is everything successful? Great.
Any problems? No question! IN Special section 555 all these exponential equations are solved with detailed explanations. What, why, and why. And, of course, there is additional valuable information on working with all sorts of exponential equations. Not just these ones.)
One last fun question to consider. In this lesson we worked with exponential equations. Why didn’t I say a word about ODZ here? In equations, this is a very important thing, by the way...
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.