«Equality" - a topic that the students are still in elementary school.Attendant as she "inequalities".These two concepts are closely linked.Moreover, with them linked terms such as equations identity.So what is equality?
concept of equality
It is defined as statements in the record that there is a sign "=".Equality is divided into right and wrong.If the entry is in place = & lt ;, & gt ;, when it comes to inequality.Incidentally, the first equality sign indicates that both parts are identical in expression result or recording.
addition to the concept of equality in schools are also studying the theme of "numerical equality."Under this statement to understand two numerical expressions that stand on either side of the = sign.For example, 2 * 5 + 7 = 17.Both parts of the record are equal.
In numeric expressions of this type can be used braces influencing procedures.So, there are four rules that should be taken into account when calculating the results of numerical expressions.
- If the record is not bracket, then the actions are performed with the highest level: III → II → I.If there are several steps one category, then they are left to right.
- If the entry has brackets, then the action is performed in parentheses, and then considering the steps.Perhaps in the brackets will be some action.
- If the expression is presented as a fraction, then you must first calculate the numerator, then the denominator, then the numerator divided by the denominator.
- If records are nested parentheses, then the first expression is evaluated in the inner brackets.
So, now it is clear that such equality.In the future, will be considered the concept of the equation, the identity and methods of their calculation.
Properties numerical equations
What is equality? study of this concept requires a knowledge of the properties of numerical identities.The following text formulas allow to better understand this topic.Of course, these properties are more suitable for the study of mathematics in high school.
1. Numerical equality would not be violated if in both its parts add the same number to an existing expression.
A = ↔ A + B + 5 = 5
2. Do not be disturbed equation if both sides multiplied or divided by the same number or expression, which are different from zero.
P = O ↔ P ∙ O ∙ 5 = 5
P = O ↔ R 5 = O: 5
3. Adding to both sides of the identity of the same function, which makes sense whenall the possible values of a variable, we obtain a new equation, which is equivalent to the original.
F (X) = Ψ (X) ↔ F (X) + R (X) = Ψ (X) + R (X)
4. Any term or expression can bemove on the other side of the equal sign, you will need to change the sign.
5 = X + Y - 20 ↔ X = Y - 20 - 5 ↔ X = Y - 25
5. multiply or divide both sides by the same function that is different from zerohaving the meaning for each value of X of the DHS, we obtain a new equation equivalent to the original.
F ( X) = Ψ ( X) ↔ F ( X) ∙ R ( X) = Ψ ( X) ∙ R ( X)
F (X) = Ψ (X) ↔ F (X): G (X) = Ψ (X): G (X)
These rules expresslyan indication of the principle of equality that exists under certain conditions.
concept proportion
In mathematics there is such a thing as equality relationships.In this case, it implies a certain proportion.If the section A to B, then the result is the ratio of the number of A to B. The proportions referred to the equality of two relations:
Sometimes ratio is written as follows: A: B = C: D. Hence the main property of a proportion: A * D = D * C , where A and D - the proportion of the extreme terms, and B and C - medium.
Identities
identities called equality, which will be true for all possible values of those variables are included in the job.Identities can be presented as a letter or numerical equality.
identically equal is an expression containing both sides of the unknown variable, which can equate the two parts of one whole.
If you spend replacing one another expression, which will be equal to, if it comes to the identity transformation.In this case, you can use the formulas of abridged multiplication, the laws of arithmetic and other identities.
To reduce the fraction, you need to carry out identity transformations.For example, a given fraction.To get results, you should use the formulas of abridged multiplication, factorization, simplification and reduction of expression of fractions.
It is worth considering that the expression will be identical when the denominator is not equal to 3.
5 ways to prove the identity
order to prove identity, it is necessary to carry out the transformation of expressions.
I method
necessary to conduct amounting to convert the left side.The result is the right side, and we can say that the identity is proved.
II method
All actions to transform the expression occur in the right-hand side.The result of the manipulation is the left-hand side.If both sides are identical, then the identity proved.
III method
«transformation" taking place in both parts of the expression.If as a result we get two identical parts, identity is proved.
IV method
From the left side is deducted right.As a result of equivalent transformations should get zero.Then we can talk about the identity expression.
V method
From the right side of the left is subtracted.All tantamount transformation reduced to the fact that the answer was zero.Only in this case we can speak about the identity of equality.
Basic properties identities
In mathematics often use properties of equality, to speed up the process of calculation.Through basic algebraic identities of the process of calculating certain expressions it takes minutes instead of long hours.
- x + y = y + x
- X + (Y + C) = (x + y) + C
- X + 0 = X
- X + (-x) = 0
- X ∙ (S + C) = A ∙ V + X ∙ With
- X ∙ (U - C) = x ∙ y - x ∙ With
- (X + Y) ∙ (C + E) = A ∙ C +X ∙ E + V ∙ C + V ∙ E
- X + (Y + S) = X + Y + C
- X + (Y - C) = X + Y - With
- X - (Y + C)= x - y - With
- X - (Y - C) = x - y + C
- X ∙ V = V ∙ X
- X ∙ (V ∙ C) = (A ∙ V) ∙ With
- X∙ 1 = X X
- ∙ 1 / x = 1, where x ≠ 0
reduction formula multiplying
At its core formula are abridged multiplication equations.They help to solve many problems in mathematics because of its simplicity and ease of use.
- (A + B) 2 = A2 + 2 ∙ A ∙ B + B2 - the sum of the square of the pairs;
- (A - B) 2 = A2 - 2 ∙ A ∙ B + B2 - squared difference pairs of numbers;
- (C + B) ∙ (C - B) = C2 - B2 - the difference between the squares;
- (A + B) = 3 A3 + A2 3 ∙ ∙ B + 3 ∙ A ∙ B2 + B3 - cubic amount;
- (A - B) = 3 A3 - A2 3 ∙ ∙ B + 3 ∙ A ∙ B2 - B3 - cube difference;
- (P + B) ∙ (P2 - P ∙ B + B2) = P3 + B3 - the sum of the cubes;
- (P - In) ∙ (P2 + p ∙ B + B2) = P3 - B3 - the difference between the cubes.
reduction formula multiplying often used if you want to lead a polynomial to the usual form, simplifying it in all possible ways.The presented formulas are proved, simply open the brackets and cause similar terms.
equations
After studying the question, what is the equality, you can proceed to the next step: what is the equation.Under the equation refers to equality, in which there are unknown quantities.Solution of the equation is called to find all the values of a variable in which the two parts of the whole expression will be equal.Also, there are jobs in which it is impossible to find solutions to the equation.In this case we say that there are no roots.
Usually equality with unknown as a solution to give integers.However, there are cases where the root is a vector function and other objects.
equation is one of the most important concepts in mathematics.Most of the scientific and practical problems do not measure or calculate any amount.Therefore, you must be the ratio which will satisfy all the conditions of the task.In the process of drawing up this relationship appears equation or system of equations.
Usually the decision of equality with unknown reduces to the transformation of a complex equation, and reduce it to a simple shape.It must be remembered that the conversion should be carried out with respect to both parts, otherwise the output will turn the wrong result.
4 ways to solve the equation
By a solution of the given equation understand replace another that is equivalent to the first.Such a substitution is known as the identity transformation.To solve the equation, you must use one of the ways.
1. One expression is replaced by another, which is mandatory to be identical to the first.Example (3 ∙ x + 3) = 2 x 15 + 10 ∙.This expression can be converted to 9 ∙ 18 ∙ x2 + x + 9 = 15 ∙ x + 10.
2. Transfer of equality with unknown members from one side to the other.In this case, you must correctly change the signs.The slightest mistake ruined all the work done.As an example, take the previous "sample".
9 ∙ x2 + 12 ∙ x + 4 = 15 ∙ x + 10
9 ∙ x2 + 12 ∙ x + 4 - 15 ∙ x - 10 = 0
9 ∙ x2 - 3 ∙ x - 6 = 0
Next equation is solved using the discriminant.
3. Multiply both sides of an equal number or expression that is not equal to 0. However, it is worth recalling that if the new equation is not equivalent to the equality before the reforms, then the number of roots could change significantly.
4. Squaring both sides of the equation.This method is just wonderful, especially when there is equality of irrational expression, that is, the square root of the expression below.There is one caveat: if you build an equation in even degree, then may appear foreign roots, which distort the essence of the job.And if it is wrong to remove the root, then the meaning of the question in the problem is unclear.Example: │7 ∙ h│ = 35 → 1) 7 ∙ x = 35 and 2) - 7 ∙ x = 35 → equation is solved correctly.
So, in this article is about such terms as the equations and identities.All of them come from the concept of "equality".Through various kinds of expressions equivalent to the solution of some problems largely alleviated.
