βEqualityβ is a topic that students go through in elementary school. She is also accompanied by "Inequalities." These two concepts are closely related. In addition, such terms as equations, identities are associated with them. So what is equality?
Concept of equality
This term refers to statements in the record of which there is a β=β sign. Equalities are divided into true and false. If the entry instead of = is <,>, then we are talking about inequalities. By the way, the first sign of equality indicates that both parts of the expression are identical in their result or record.
In addition to the concept of equality, the school also studies the topic of βNumerical Equalityβ. This statement means two numerical expressions that are on both sides of the = sign. For example, 2 * 5 + 7 = 17. Both parts of the record are equal to each other.
In numeric expressions of this type, brackets can be used that affect the order of actions. So, there are 4 rules that should be considered when calculating the results of numerical expressions.
- If there are no brackets in the record, then the actions are performed from the highest level: III β II β I. If there are several actions of the same category, then they are performed from left to right.
- If there are brackets in the record, then the action is performed in brackets, and then taking into account the steps. There may be several actions in parentheses.
- If the expression is represented as a fraction, then you need to calculate the numerator first, then the denominator, then the numerator is divided by the denominator.
- If there are nested parentheses in the record, then the expression in parentheses is evaluated first.
So now itβs clear what equality is. In the future, concepts of equations, identities, and methods for calculating them will be considered.
Properties of numerical equalities
What is equality? The study of this concept requires knowledge of the properties of numerical identities. The following text formulas allow you to better study this topic. Of course, these properties are more suitable for studying mathematics in high school.
1. Numerical equality will not be violated if in both its parts the same number is added to the existing expression.
A = B β A + 5 = B + 5
2. The equation will not be violated if both its parts are multiplied or divided by the same number or expression that are non-zero.
P = O β P β 5 = O β 5
P = O β P: 5 = O: 5
3. Adding the same function to both sides of the identity, which makes sense for any admissible values ββof the variable, we obtain a new equality equivalent to the original one.
F (X) = Ξ¨ (X) β F (X) + R (X) = Ξ¨ (X) + R (X)
4. Any term or expression can be moved to the other side of the equal sign, and you need to change the signs to the opposite.
X + 5 = Y - 20 β X = Y - 20 - 5 β X = Y - 25
5. Multiplying or dividing both sides of the equation by the same function, nonzero and meaningful for each value of X from the ODZ, we will get a new equation equivalent to the original one.
F ( X) = Ξ¨ ( X) β F ( X) β R ( X) = Ξ¨ ( X) β R ( X)
F (X) = Ξ¨ (X) β F (X): G (X) = Ξ¨ (X): G (X)
The above rules clearly indicate the principle of equality, which exists under certain conditions.
Concept of proportion
In mathematics, there is such a thing as equality of relations. In this case, the definition of proportion is implied. If we divide A by B, the result will be the ratio of A to B. Proportion is the equality of two relations:
Sometimes a proportion is written as follows: A: B = C: D. The main property of the proportion follows from here: A * D = D * C , where A and D are the extreme members of the proportion, and B and C are average.
Identities
Identity is the equality that will be true for all admissible values ββof those variables that are included in the task. Identities can be represented as alphabetic or numerical equalities.
Identically equal are called expressions containing an unknown variable in both sides of the equality, which is able to equate two parts of one whole.
If we replace one expression with another, which will be equal to it, then we are talking about an identical transformation. In this case, you can use the formulas of abbreviated multiplication, the laws of arithmetic and other identities.
To reduce the fraction, it is necessary to carry out identical transformations. For example, a fraction is given. To get the result, you should use the formulas of abbreviated multiplication, factorization, simplification of expressions and reduction of fractions.
It should be borne in mind that this expression will be identical when the denominator is not equal to 3.
5 Ways to Prove Identity
To prove the identity is identical, it is necessary to carry out the transformation of expressions.
I way
It is necessary to carry out equivalent transformations on the left side. The result is the right-hand side, and we can say that the identity is proved.
II method
All actions to transform the expression occur on the right side. The result of the manipulations is the left side. If both parts are identical, then the identity is proved.
III method
βTransformationsβ occur in both parts of the expression. If the result is two identical parts, the identity is proved.
IV method
The right side is subtracted from the left. As a result of equivalent transformations, zero should be obtained. Then we can talk about the identity of the expression.
V way
The left side is subtracted from the left. All equivalent transformations are reduced to ensuring that the answer is zero. Only in this case can we talk about the identity of equality.
The main properties of identities
In mathematics, they often use the properties of equalities to speed up the computation process. Thanks to basic algebraic identities, the process of computing some expressions will take a matter of minutes instead of long hours.
- X + Y = Y + X
- X + (Y + C) = (X + Y) + C
- X + 0 = X
- X + (-X) = 0
- X β (Y + C) = X β Y + X β C
- X β (Y - C) = X β Y - X β C
- (X + Y) β (C + E) = X β C + X β E + Y β C + Y β E
- X + (Y + C) = X + Y + C
- X + (Y - C) = X + Y - C
- X - (Y + C) = X - Y - C
- X - (Y - C) = X - Y + C
- X β Y = Y β X
- X β (Y β C) = (X β Y) β C
- X β 1 = X
- X β 1 / X = 1, where X β 0
Abbreviation Formulas
At their core, abbreviated multiplication formulas are equalities. They help to solve many problems in mathematics due to its simplicity and ease of use.
- (A + B) 2 = 2 + 2 β β + 2 - the square of the sum of a pair of numbers;
- (A - B) 2 = A 2 - 2 β A β B + B 2 - the square of the difference of a pair of numbers;
- (C + B) β (C - B) = C 2 - B 2 - the difference of squares;
- (A + B) 3 = A 3 + 3 β A 2 β B + 3 β A β B 2 + B 3 - the cube of the sum;
- (A - B) 3 = A 3 - 3 β A 2 β B + 3 β A β B 2 - B 3 - cube of difference;
- ( + ) β ( 2 - β + 2 ) = 3 + 3 - the sum of cubes;
- (P - B) β (P 2 + P β B + B 2 ) = P 3 - B 3 - the difference of cubes.
Abbreviated multiplication formulas are often used if it is necessary to bring the polynomial to its usual form, simplifying it in all possible ways. The formulas presented are proved simply: it is enough to open the brackets and give similar terms.
Equations
After studying the question of what equality is, we can proceed to the next point: what is the equation. By an equation is meant an equality in which unknown quantities are present. The solution to an equation is the finding of all values ββof a variable for which both parts of the whole expression will be equal. There are also tasks in which finding solutions to the equation is impossible. In this case, they say that there are no roots.
As a rule, equalities with unknowns give integers as a solution. However, there may be cases where the root is a vector, a function, and other objects.
Equation is one of the most important concepts in mathematics. Most scientific and practical tasks do not allow to measure or calculate any value. Therefore, it is necessary to draw up a ratio that will satisfy all the conditions of the task. In the process of compiling such a relation, an equation or system of equations appears.
Usually the solution of equality with the unknown is reduced to transforming a complex equation and reducing it to simple forms. It must be remembered that the transformations must be carried out with respect to both parts, otherwise the result will be the wrong result.
4 ways to solve the equation
By a solution of an equation is meant the replacement of a given equality by another, which is equivalent to the first. Such a substitution is known as the identity transformation. To solve the equation, you must use one of the methods.
1. One expression is replaced by another, which will necessarily be identical to the first. Example: (3 β x + 3) 2 = 15 β x + 10. This expression can be converted to 9 β x 2 + 18 β x + 9 = 15 β x + 10.
2. Transfer of members of equality with the unknown from one side to another. In this case, it is necessary to correctly change the signs. The slightest mistake will ruin all the work done. As an example, take the previous βsampleβ.
9 β x 2 + 12 β x + 4 = 15 β x + 10
9 β x 2 + 12 β x + 4 - 15 β x - 10 = 0
9 β x 2 - 3 β x - 6 = 0
Further, the equation is solved using the discriminant.
3. The multiplication of both parts of the equality by an equal number or expression that does not equal 0. However, it is worth recalling that if the new equation is not equivalent to equality before the transformations, then the number of roots can change significantly.
4. Squaring both sides of the equation. This method is simply wonderful, especially when there are irrational expressions in the equality, that is, the square root and the expression under it. There is one caveat: if you raise the equation to an even degree, then extraneous roots may appear that will distort the essence of the task. And if it is wrong to extract the root, then the meaning of the question in the problem will be unclear. Example: β7 β xβ = 35 β 1) 7 β x = 35 and 2) - 7 β x = 35 β the equation will be solved correctly.
So, this article refers to terms such as equations and identities. They all come from the concept of "equality." Thanks to various equivalent expressions, the solution of some problems is greatly facilitated.