Equation is an algebraic expression that contains an equality. It was created to help people find solutions to problems where a number is not known. Knowing that the sum of two consecutive numbers equals 11, for example, it is possible to find these two numbers using equations.
Before learning to solve equations, one must understand the meaning of the definition given above.
algebraic expressions
algebraic expressions are a set of basic mathematical operations applied to both known and unknown numbers. To represent these unknown numbers, letters are used. It's more common to use the letters x and y, but that doesn't mean they're the only ones. In some cases, letters from the Greek alphabet and even different symbols are used.
Note the examples of algebraic expressions below:
1) 12x2 + 16y + 4ab
2) x + y
3) 4 + 7th
All these expressions have letters representing numbers and numbers being added and multiplied.
Equality
All algebraic expression who have one equality in its composition it will be called an equation. Take a look at some examples:
1) x + 2 = 7
2) 12x2 + 16y + 4ab = 7
3) 1:x = 3
THE equality is what allows you to find the results of a equation. It is the equality that relates a mathematical operation applied to some numbers with its result. Therefore, equality is key when looking for the results of an equation.
For example: Given the equation x – 14 = 8, what is the value of x?
Now we know that x is a number that, subtracted by 14, has 8 as a result. Note that it is possible to think of a result “in your head” or think of a strategy to solve this equation. The strategy can be obtained as follows: If x is a number that, subtracted from 14, results in 8, then, to find x, just add 14 to 8. In this way, we can write the following line of reasoning:
x – 14 = 8
x = 8 + 14
x = 22
Adding 14 and 8 together, we have 22 as a result.
degree of an equation
O degree of an equation it is related to the amount of unknowns it has. We say that an equation is of degree 1 when the largest exponent of its unknowns is 1. An equation has degree 2 when the largest exponent of its unknowns is 2, and so on. The grade can also be given by the product of incognitos many different. For example, the equation xy + 2 = y is an equation of degree 2 because it has a product between two unknowns of exponent 1.
O degree of an equation determines how many solutions the equation has. Thus, an equation of degree 1 has only 1 result (a possible value for the unknown); an equation of degree 2 has two results, and so on.
Solution of equations
One of the resolution strategies of a equation makes use of the above thought. Note that, looking at the two equations (x – 14 = 8 and x = 8 + 14), it is possible to imagine that the number 14 switched sides of the equality with a side effect: it changed its sign from negative to positive. This is one of the rules for solving equations which are listed below:
Rule 1 - on the right side of equality, only numbers that do not have an unknown remain; on the left side, only numbers they have;
Rule 2 – To change numbers sideways, having or not unknown, it is necessary to change their sign;
Rule 3 – After steps 1 and 2, perform the calculations that are possible. Remember that numbers that have an unknown can be added together if the unknown is the same. To do this, just add the number that accompanies them.
Rule 4 – At the end, the unknown must be isolated. For this, the number that accompanies it must be passed to the right side of the equation dividing its components.
Rule 5 – If it is necessary to switch sides a number that is in the denominator of a fraction, it must switch to the other side by multiplying.
Examples
1) What is the value of x in the equation 4x + 4 = 2x – 8?
Solution: Following the first and second rules, we will obtain the following line of reasoning:
4x + 4 = 2x - 8
4x – 2x = – 8 – 4
Now, perform the third rule to get:
2x = – 12
Finally, perform rule 4:
2x = – 12
x = –12
2
x = – 6
Therefore, the value of x is – 6.
2) Knowing that the sum of two consecutive numbers equals 11, what are these two numbers?
Solution: Note that the numbers are unknown, but they are consecutive. Being consecutive means that the second is a unit greater than the first. For example, 1 and 2 are consecutive because 2 is a unit greater than 1. If consecutive numbers are unknown, we will represent them by a letter (in this case x) and add 1 to the first to get the second. Also, knowing that the sum between the two has 11 as a result, we can write:
x + (x + 1) = 11
x + x + 1 = 11
By rules 1 and 2, get:
x + x = 11 - 1
By rule 3, note the result:
2x = 10
Using rule 4, get:
2x = 10
x = 10
2
x = 5
Since x represented the first number, then the consecutive numbers that add up to 11 are 5 and 6.
By Luiz Paulo Moreira
Graduated in Mathematics
Source: Brazil School - https://brasilescola.uol.com.br/o-que-e/matematica/o-que-e-equacao.htm
