In mathematics, one of the parameters describing the position of a line on the Cartesian coordinate plane is slope this straight line. This parameter characterizes the slope of the straight line to the abscissa axis. To understand how to find the slope, first remember general form equations of a straight line in the XY coordinate system.
In general, any line can be represented by the expression ax+by=c, where a, b and c are arbitrary real numbers, but a 2 + b 2 ≠ 0.
Using simple transformations, such an equation can be brought to the form y=kx+d, in which k and d are real numbers. The number k is the slope, and the equation of a line of this type is called an equation with a slope. It turns out that to find the slope, you simply need to reduce the original equation to the form indicated above. For a more complete understanding, consider a specific example:
Solution: Let's transform the original equation.
Answer: The required slope of this line is 2.
If, during the transformation of the equation, we received an expression like x = const and as a result we cannot represent y as a function of x, then we are dealing with a straight line parallel to the X axis. The angular coefficient of such a straight line is equal to infinity.
For lines expressed by an equation like y = const, the slope is zero. This is typical for straight lines parallel to the abscissa axis. For example:
Solution: Let's bring the original equation to its general form
24x + 12y - 12y + 28 = 4
It is impossible to express y from the resulting expression, therefore the angular coefficient of this line is equal to infinity, and the line itself will be parallel to the Y axis.
For a better understanding, let's look at the picture:
In the figure we see a graph of a function like y = kx. To simplify, let’s take the coefficient c = 0. In the triangle OAB, the ratio of side BA to AO will be equal to the angular coefficient k. At the same time, the ratio VA/AO is the tangent acute angleα in right triangle OAV. It turns out that the angular coefficient of the straight line is equal to the tangent of the angle that this straight line makes with the abscissa axis of the coordinate grid.
Solving the problem of how to find the angular coefficient of a straight line, we find the tangent of the angle between it and the X axis of the coordinate grid. Boundary cases, when the line in question is parallel to the coordinate axes, confirm the above. Indeed, for a straight line described by the equation y=const, the angle between it and the abscissa axis is zero. The tangent of the zero angle is also zero and the slope is also zero.
For straight lines perpendicular to the x-axis and described by the equation x=const, the angle between them and the X-axis is 90 degrees. Tangent right angle is equal to infinity, and the angular coefficient of similar straight lines is also equal to infinity, which confirms what was written above.
A common task often encountered in practice is also to find the slope of a tangent to the graph of a function at a certain point. A tangent is a straight line, therefore the concept of slope is also applicable to it.
To figure out how to find the slope of a tangent, we will need to recall the concept of derivative. The derivative of any function at a certain point is a constant numerically equal to the tangent of the angle that is formed between the tangent at the specified point to the graph of this function and the abscissa axis. It turns out that to determine the angular coefficient of the tangent at the point x 0, we need to calculate the value of the derivative of the original function at this point k = f"(x 0). Let's look at the example:
Solution: Find the derivative of the original function in general form
y"(0.1) = 24. 0.1 + 2. 0.1. e 0.1 + 2. e 0.1
Answer: The required slope at point x = 0.1 is 4.831
In this lesson we will learn about such a concept as coefficient. We will also look at several problems, using examples of which we can easily find the coefficients of various expressions.
This is the product: the number 2 is multiplied by the letter.
In such a work we agreed to name the number coefficient.
A coefficient is a numerical factor in a product where there is a letter.
For example:
Therefore the coefficient is 4.
Therefore the coefficient is 1.
Therefore the coefficient is -1.
Therefore the coefficient is 5.
In mathematics, we agreed to write the coefficient at the beginning, therefore:
There may be several letters, but this does not affect the coefficient. For example:
Coefficient -17.
Factor 46.
If the product has several numerical factors, then this expression can be simplified:
The coefficient in this expression is 100.
A numerical factor in a product that contains at least one letter is called a coefficient.
If there are several numbers, you need to multiply them, simplify the expression, and thus obtain a coefficient.
There is only one coefficient in one product.
If there is a sum, for example, this:
Then each term has coefficients: and .
If there is no number, then you can put one. This is the coefficient.
, coefficient 1.
Find the coefficient: a) ; b) 
.
a) , coefficient -50.
b) coefficient.
So, coefficient is a number that stands in a product with one or more variables. It can be integer or fractional, positive or negative.
When planting potatoes, the yield is 10 times greater than the number of potatoes planted. What will the harvest be if you planted 65 kg?
Solution
What if 90 kg of potatoes are planted?
What if we don’t know how much has been planted? How then to decide in this case?
If you planted kg, then the harvest will be kg.
So, 10 is a coefficient here (let’s call it yield), and is a variable. can take any value, and the formula will calculate the amount of harvest.
If the yield is different, for example 9, then the formula looks like this: .
The coefficient in the formula has changed.
If we consider different yields, the formula will remain the same in appearance, only the coefficient will change.
This means that we can write down the general form of all such formulas.
Where is the coefficient; - variable.
This is the yield, it can be equal to, for example, 10 or 9, as before, or another number.
So, how to answer the question “what is the coefficient in the entry?”?
If nothing is known about this record, then they are just letters, variables. Coefficient one.
If it is known that this is part of the formula for calculating the potato yield, then this is the coefficient.
In other words, the coefficient can often be denoted by a letter.
In mathematics, physics, and other sciences there are many formulas where one of the letters is a coefficient.
Example
The density of matter in physics is denoted by the letter.
The higher the density, the more the same volume of a substance weighs.
If you know the volume of a substance and its density, then you can easily find the mass using the formula:
Any person who is familiar with this formula, when asked “what is the coefficient here?” will answer "".
A coefficient is a number in a product where there are one or more variables.
There is an agreement to write the coefficient before the variables.
If there is no number in the product, then you can put a factor of 1, which will be the coefficient.
If we have a formula in front of us, then one of the letters may well be a coefficient.
Bibliography
Homework
Hi all!
Having entered the sports betting community, I did not find any articles on betting theory, although I bet myself and know that theoretical material in betting no less than in poker. Therefore, I want to post here some posts about the mathematical and analytical foundations of sports betting. I hope it is useful to someone.
I would like to start where every player starts: with the bookmaker’s line. The first question that arose in my mind when I first picked up a printed line: How does a bookmaker determine all this mass of odds?
Bookmakers operate solely for the purpose of making a profit. And, contrary to popular belief, the bookmaker’s profit does not depend on the number of lost bets, but on the correctly set odds. What does "correct" mean? This means that in case of any, even the most unexpected, outcome of the event, the bookmaker must remain profitable.
Let's look at how the coefficients are formed. First, analysts determine the teams' chances. This is done in many ways, which can be divided into two groups: analytical and heuristic. Analytical are mainly statistics and mathematics (probability theory), heuristic are expert assessments. By combining the results obtained in one way or another, the probabilities of the outcome of the event are derived. Let’s assume that as a result of the activities of analysts and experts, the following probabilities of outcomes were obtained:
These are "pure odds", but these odds will never line up because the bookmaker will not make a profit in this case. The line odds for these events will look something like this:
That is, out of every one hundred thousand rubles bet by all players, 75,000 were bet on victory 1, 15,000 on a draw and 10,000 on victory 2. Most players most often bet on obvious favorites, making up most of the express bets based on such outcomes . What will the bookmaker get for each hundreds of thousands of dollars invested by players in the event of different outcomes?
It can be seen that if the favorite wins, which happens most often, the bookmaker will suffer losses. This is completely unacceptable for business, and the bookmaker is obliged to exclude even the theoretical possibility of such a situation arising.
To do this, he must artificially lower the odds on the favorite. The bookmaker does not know in advance exactly how the bets will be distributed, but he knows for sure that the players will “load” on the favorite, therefore, for insurance, he overestimates the probability of the favorite’s victory.
In reality neither real chances, nor the distribution of funds by players can be accurately calculated; there is always some error. Therefore, bookmakers try to initially lower the odds for the favorite in order to guarantee their profit, i.e. determine the teams' chances and add 10-20% to the calculated probability of victory for the favorite. And as bets are received, depending on their actual current distribution, the odds are varied so that the profit is greatest.
Conclusion: the main principle that guides the bookmaker is the distribution of finances between two or more groups of players in such a way as to pay winnings from the funds of the losers, leaving a certain percentage for themselves. Very often, the coefficients obtained in this way have nothing to do with the probabilities of certain events. Therefore, you need to have your own system for evaluating sporting events.
Thank you for your attention!
The equation of a reaction in chemistry is called the notation chemical process using chemical formulas and mathematical symbols.
This entry is a schema chemical reaction. When the "=" sign appears, it is called an "equation". Let's try to solve it.
There is one atom in calcium, since the coefficient is not worth it. The index is also not written here, which means one. On the right side of the equation, Ca is also one. We don't need to work on calcium.
Let's look at the next element - oxygen. Index 2 indicates that there are 2 oxygen ions. There are no indices on the right side, that is, one particle of oxygen, and on the left there are 2 particles. What are we doing? No additional indexes or fixes in chemical formula You cannot enter it because it is written correctly.
The coefficients are what is written before the smallest part. They have the right to change. For convenience, we do not rewrite the formula itself. On the right side, we multiply one by 2 to get 2 oxygen ions there.
After we set the coefficient, we got 2 calcium atoms. There is only one on the left side. This means that now we must put 2 in front of calcium.
Now let's check the result. If the number of atoms of an element is equal on both sides, then we can put the “equal” sign.
Another clear example: two hydrogens on the left, and after the arrow we also have two hydrogens.
We multiplied the entire formula by 2, and now the amount of hydrogen has changed. We multiply the index by the coefficient, and we get 4. And on the left side there are two hydrogen atoms left. And to get 4, we have to multiply hydrogen by two.
This is the case when the element in one and the other formula is on the same side, up to the arrow.
One sulfur ion on the left, and one ion on the right. Two oxygen particles, plus two more oxygen particles. This means that there are 4 oxygens on the left side. On the right there are 3 oxygens. That is, on one side there is an even number of atoms, and on the other, an odd number. If we multiply the odd number by two times, we get an even number. First we bring it to an even value. To do this, multiply the entire formula after the arrow by two. After multiplication, we get six oxygen ions, and also 2 sulfur atoms. On the left we have one microparticle of sulfur. Now let's equalize it. We put the equations on the left before gray 2.
Called.
This example is more complex because there are more elements of matter.
This is called a neutralization reaction. What needs to be equalized here first:
The conclusion suggests itself is that you need to multiply the entire formula by two.
Now let's see how much sulfur there is. One on the left and right sides. Let's pay attention to oxygen. On the left side we have 6 oxygen atoms. On the other hand - 5. Less on the right, more on the left. An odd number must be brought to an even number. To do this, we multiply the formula of water by 2, that is, from one oxygen atom we make 2.
Now there are already 6 oxygen atoms on the right side. There are also 6 atoms on the left side. Let's check the hydrogen. Two hydrogen atoms and 2 more hydrogen atoms. So there will be four hydrogen atoms on the left side. And on the other side there are also four hydrogen atoms. All elements are equal. We put the equal sign.
Next example.
Here the example is interesting because parentheses appear. They say that if a factor is behind the brackets, then each element in the brackets is multiplied by it. You need to start with nitrogen, since there is less of it than oxygen and hydrogen. On the left there is one nitrogen, and on the right, taking into account the brackets, there are two.
There are two hydrogen atoms on the right, but four are needed. We get out of this by simply multiplying water by two, resulting in four hydrogens. Great, hydrogen equalized. There is oxygen left. Before the reaction there are 8 atoms, after - also 8.
Great, all the elements are equal, we can set “equal”.
Last example.
Next up is barium. It is equalized, you don’t need to touch it. Before the reaction there are two chlorines, after it there is only one. What needs to be done? Place 2 in front of the chlorine after the reaction.
Now, due to the coefficient that was just set, after the reaction we got two sodiums, and before the reaction we also got two. Great, everything else is equalized.
You can also equalize reactions using the electronic balance method. This method has a number of rules by which it can be implemented. The next step is to arrange the oxidation states of all elements in each substance in order to understand where oxidation occurred and where reduction occurred.
The term "numeric coefficient" often appears in mathematical descriptions, for example, when working with literal expressions and expressions with variables. The article below reveals the concept of this term, including the example of solving problems of finding a numerical coefficient.
Yandex.RTB R-A-339285-1
Textbook N.Ya. Vilenkina ( educational material for 6th grade students) sets the following definition of the numerical coefficient of the expression:
Definition 1
If a letter expression is the product of one or more letters and one number, then this number is called numerical coefficient of expression.
Numerical coefficient often referred to simply as a coefficient.
This definition makes it possible to indicate examples of numerical coefficients of expressions.
Example 1
Consider the product of the number 5 and the letter a, which will have next view: 5 a. The number 5 is the numeric coefficient of the expression as defined above.
Another example:
Example 2
In a given work x y 1, 3 x x z decimal 1, 3 is the only numerical factor, which will serve as the numerical coefficient of the expression.
Let's also look at the following expression:
Example 3
7 x + y. Number 7 in in this case does not serve as a numeric coefficient of an expression because the given expression is not a product. But at the same time, the number 7 is the numerical coefficient of the first term in the given expression.
Example 4
Let the product be given 2 a 6 b 9 c.
We see that the expression notation contains three numbers, and in order to find the numerical coefficient of the original expression, it should be rewritten as an expression with a single numerical factor. Actually, this is the process of finding a numerical coefficient.
Note that products of identical letters can be represented as powers with a natural exponent, therefore the definition of a numerical coefficient is also true for expressions with powers.
Eg:
Example 5
Expression 3 x 3 y z 2– essentially an optimized version of the expression 3 · x · x · x · y · z · z, where the coefficient of the expression is the number 3.
Let's talk separately about the numerical coefficients 1 and - 1. They are very rarely written down explicitly, and this is their peculiarity. When a product consists of several letters (without an explicit numerical factor), and is preceded by a plus sign or no sign at all, we can say that the numerical coefficient of such an expression is the number 1. When a minus sign is indicated before the product of letters, it can be argued that in this case the numerical coefficient is the number - 1.
Example 6
For example, in the product - 5 x + 1, the number - 5 will serve as a numerical coefficient.
By analogy, in the expression 8 1 + 1 x x number 8 – coefficient of expression; and in the expression π + 1 4 · sin x + π 6 · cos - π 3 + 2 · x the numerical coefficient is π + 1 4.
We said above that if an expression is a product with a single numerical factor, then this factor will be the numerical coefficient of the expression. In the case when the expression is written in a different form, a series of identical transformations must be performed, which will bring the given expression to the form of a product with a single numerical factor.
Example 7
Expression given − 3 x (− 6). It is necessary to determine its numerical coefficient.
Solution
Let's do it identity transformation, namely, we will group the factors that are numbers and multiply them. Then we get: − 3 x (− 6) = ((− 3) (− 6)) x = 18 x .
In the resulting expression we see an explicit numerical coefficient equal to 18.
Answer: 18
Example 8
The given expression is a - 1 2 · 2 · a - 6 - 2 · a 2 - 3 · a - 3 . It is necessary to determine its numerical coefficient.
Solution
In order to determine the numerical coefficient, we transform the given integer expression into a polynomial. Let's open the brackets and add similar terms, we get:
a - 1 2 2 a - 6 - 2 a 2 - 3 a - 3 = = 2 a 2 - 6 a - a + 3 - 2 a 2 + 6 a - 3 = - a
The numerical coefficient of the resulting expression will be the number - 1.
Answer: - 1 .
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