In Mathematics, the equation is a equality that involves one or more unknowns. Who determines the "degree" of this equation is the exponent of this unknown, that is, if the exponent is 1, we have the 1st degree equation. If the exponent is 2, the equation is 2nd degree; if the exponent is 3, the equation is 3rd degree.
To exemplify:
4x + 2 = 16 (1st degree equation)
x² + 2x + 4 = 0 (2nd degree equation)
x³ + 2x² + 5x – 2 = 0 (3rd degree equation)
The 1st degree equation is presented as follows:
ax + b = 0
It is important to say that The and B represent any real number and The is nonzero (to 0). the unknown x can be represented by any letter, however, we usually use x or y as the value to be found for the final result of the equation. The first member of the equation is the numbers on the left side of the equality, and the second member, the numbers on the right side of the equality.
See too:Practical method for solving equations
How to Solve a First Degree Equation
To solve an equation of the first degree, we must find the unknown value (which we will call x) and, for this to be possible, just isolate the value of the x on equality, that is, the xmust be alone in one of the members of the equation.
The next step is to analyze which operation is being performed on the same member as it is. x and "play" to the other side of equality by making the operationopposite and isolating x.
First example:
x + 4 = 12
In this case, the number that appears on the same side of x it's the 4 and he's adding up. To isolate the unknown, it goes to the other side of the equality doing the inverse operation (subtraction):
x = 12 – 4
x = 8
Second example:
x – 12 = 20
The number that's on the same side as x is 12 and it's subtracting. In this example, it goes to the other side of equality with the operationinverse, which is the sum:
x = 20 + 12
x = 32
Third example:
4x + 2 = 10
Let's look at the numbers that are on the same side of the unknown, the 4 and the 2. The number 2 is adding and goes to the other side of the equality by subtracting and the number 4, which is multiplying, goes to the other side by dividing.
4x = 10 – 2
x = 10 – 2
4
x = 8
4
x = 2
Fourth example:
-3x = -9
This example involves negative numbers and, before passing the number to the other side, we must always leave the side of the unknown positive, so let's multiply the whole equation by -1.
-3x = -9 .(-1)
3x = 9
Passing the number 3, which is multiplying x, to the other side, we will have:
x = 9
3
x = 3
Fifth example:
2x + 4 = 7
3 5 8
In this case, we must do the MMC of the denominators so that they are equaled and later canceled (always with the intention of isolating the unknown x):
The next step is to match the denominators with the MMC result. The numerators are found by dividing the MMC by the denominator and multiplying by the numerator:
(120 ÷ 3.2x) + (120 ÷ 5.4) = (120 ÷ 8.7)
120 120 120
80x + 96 = 105
120 120 120
After the denominators are equaled, they can be canceled, leaving the equation:
80x + 96 = 105
O 96 is adding and goes to the other side of equality by subtracting:
80x = 105 - 96
80x = 9
Finally, the 80 that is multiplying x goes to the other side of equality by dividing:
x = 9
80
x = 0.1125
Note: Where the unknown x is in parentheses and there is some outside number that is multiplying those parentheses, we should distribute the multiplication of the number for all components that are inside the parentheses (this process is called a property distributive). For example:
5(3x - 9 + 5) = 0
In this case, the 5 must multiply all the components inside the parentheses and then isolate the unknown x:
15x – 45 + 25 = 0
15x – 20 = 0
15x = 20
x = 20
15
x = 4 or x = 1.33333...
3
Also know: Equations that have exponent 2 in the unknown
Fundamental property of equations
The fundamental property of equations is also called scale rule. It is not widely used in Brazil, but it has the advantage of being a single rule. The idea is that everything that is done in the first member of the equation must also be done in the second member in order to isolate the unknown to obtain the final result. See the demo in this example:
3x + 12 = 27
We'll start with the elimination of number 12. Since it's adding, let's subtract the number 12 in the two members of the equation:
3x + 12 - 12 = 27 – 12
3x = 15
Finally, the number 3 that is multiplying the unknown will be divided by 3 in the two members of the equation:
3x = 15
3 3
x = 5
solved exercises
Exercise 1
Solve the following equations:
THE. x + 4 = 15
Resolution:
x = 15 – 4
x = 11
B. 2x – 5 = x + 10
Resolution:
2x – x = 10 + 5
x = 15
Ç. 5x – 3x – 8 = – 29 + 9x
Resolution:
2x – 9x = – 29 + 8
– 7x = – 21 .( –1) Multiply all by -1
7x = 21
x = 21
7
x = 3
Exercise 2
Find the unknown value in the following equation:
5 - (4x + 2) = 8 + 2(x - 1)
5 – 4x – 2 = 8 + 2x – 2
– 4x + 3 = 6 + 2x
– 4x – 2x = 6 – 3
– 6x = 3.( –1)
6x = – 3
x = - 3 ÷ 3 (SIMPLIFIED)
6 3
x = - 1
2
