What is x if x x 5
Solving / Solving Equations: Explanation and Examples
How can you solve equations? This is exactly what this article provides explanations, examples and exercises. We look at simple linear equations, quadratic equations, and higher-order functions. Systems of equations can be solved with Gaussian methods or substitution methods or addition methods. Overall, the aim is to find the amount of solution with transformations. This article is part of our math section.
How can you solve an equation? Well, it depends on the type of equation. And for this very reason, we logically have to deal with different types of equations here as well. We start by solving linear equations.
Show:Solve or solve equations: Linear equation
Solving linear equations has already driven many students to despair. So let's start very simply. So we start with something that everyone should know from elementary school, an equation. No joke!
It's a very simple equation. Because on the left we get 7 and on the right we get 7. So we get 7 = 7, a true statement. This also applies to solving linear equations. What is new now is that another variable appears in the equation. What is a variable A variable is, so to speak, a "placeholder" for a number. At least in the vast majority of cases it is a number. In mathematics, a letter is usually used for it. This is then e.g. an a, b, x or y. A number will later be used in place of this variable. The goal is to find out what the number of the variable is. And that is exactly what we will deal with in the following sections.
As already explained in the introduction, a linear equation with an unknown is now to be solved. This unknown is usually called "x" in class, other letters (variables) are also possible. The aim is that "x = a number" can be given as a solution at the end. It starts with a very simple task. This is explained below.
Table scrollable to the rightExample 1: | |
x + 2 = 5 | | -2 |
x = 3 |
The first line contains the output equation which is to be solved for the variable x. To do this, so-called Equivalent transformations or sometimes simply transformations are carried out. This means: The appearance of the equation is changed, but still the same value appears on the left side as on the right side of the equation. In order to resolve for "x", the 2 on the left must be "eliminated". To remove a +2, "-2" must be expected. For a better overview, all arithmetic operations are followed by a "|" written. So now "| -2" is written down to clearly see that a 2 should be subtracted. Very, very important: Arithmetic operations must be carried out on both sides. If I calculate "-2" on the left, this must also be done on the right!
Table scrollable to the rightExample 2: | |
x - 5 = 2 | | + 5 |
x = 7 |
Example 3: | |
4 = x + 2 | | -2 |
2 = x |
Example 4: | |
4 - 3 + x = 5 - 2 | |
1 + x = 3 | | -1 |
x = 2 |
So much for the addition and subtraction of equations. Example 4 clearly shows that it often makes sense to first simplify the equation before transformations are carried out.
Multiplication and division:
So far you had to calculate with "-2" to eliminate a "+2" and vice versa. This also applies to multiplication and division in order to solve the corresponding equations. To remove a "· 5" you have to calculate ": 5". Sounds a bit strange at first, but the following task shows you how this works. Here, too, should be resolved for "x".
Table scrollable to the rightExample 5: | |
5 · x = 15 | | :5 |
x = 3 |
So that "x" stands alone, it has to be divided by 5. Because: 5: 5 = 1 and 1 · x = x. If this is too complicated for you, you just have to remember: To get 5 x away, I have to divide by 5. Note: The notation 5 x corresponds mathematically to 5x. If there is no arithmetic symbol between the number and the variable, a multiplication is carried out. So also with the next tasks.
Table scrollable to the rightExample 6: | |
0.5x = 2 | | :0,5 |
x = 4 |
Example 7: | |
5 = 0.2x | | :0,2 |
25 = x |
Let's make the whole thing a little more difficult. For the next tasks it will be necessary that you pay attention to the point before the line and the brackets. Otherwise the results will (usually) be wrong. As always, let's start with an example.
Table scrollable to the rightExample 8 | |
5 · 8 + x = 10 | |
40 + x = 10 | | -40 |
x = -30 |
The same applies here: point calculation before line calculation. Multiplications and division are calculated first, followed by addition and subtraction. And two more tasks:
Table scrollable to the rightExample 9: | |
40 + 20x = 20 | | :20 |
2 + x = 1 | | -2 |
x = -1 |
Example 10: | |
3 + 5 · 2 + 5x = 10 | |
3 + 10 + 5x = 10 | |
13 + 5x = 10 | | -13 |
5x = -3 | | :5 |
x = -0.6 |
Solve or solve equations: Quadratic equations
We had just dealt with solving or solving linear equations. Now let's go to solving quadratic equations. This naturally begs the question: what is a quadratic equation? Well, this is an equation of the form ax^{2} + bx + c = 0 or an equation that can be converted into this form. The variables a, b and c stand for any number, where a must be non-zero. Two examples or tasks follow: 3x^{2} + 5x + 3 = 0 or also x^{2} + 2x + 1 = 0.
In contrast to the "simple" equations that we have got to know so far (example: x + 5 = 0) there is still a quadratic component here. So how do you solve this equation for x? The answer to this question is PQ formula, which we want to explore in this section. Note: In addition to the PQ formula, there are other ways to solve a quadratic equation (midnight formula or ABC formula or polynomial division). In this comprehensive article on solving equations, however, we would like to present only one variant in detail and have decided on the PQ formula.
Solve quadratic equation: solution formula
How can you solve such an equation? To find an equation such as x^{2} + 2x + 1 = 0 to solve for x, we use the PQ formula in the following. First of all, I'll give you the formula and some general information. Don't panic: a few tasks explain this afterwards.
Solve the quadratic equation:
- Put the equation in the form x^{2} + px + q = 0
- Find out "p" and "q"
- Plug this into the PQ formula
- Calculate the solution with it
So much for the plan. Time to clear this up with a few tasks. Follow these examples using the 4-point list from above.
Important NOTE: In order not to confuse students with many fractions, some examples have been rounded.
Example 1:
Explanations: The "3" before the x^{2} disturbs! There must always be a "1", i.e. 1x^{2}. To do this, it is divided by 3. Then read p and q. The numbers of p and q are inserted into the solution formula. Then the expression before and below the root is calculated. Then the root is taken from the value and it is added once and subtracted once. A quadratic equation has a maximum of two real solutions. So in school a quadratic equation has a maximum of two solutions, in studies it always has two solutions (if you allow complex numbers, but we won't deal with them here).
Example 2:
Explanations: The original task is already in the right form. Therefore, p and q can be determined equally. Then insert this into the equation and calculate it. As you can see from the result, there is twice the solution -2, i.e. x_{1} = -2 and x_{2} = -2.
Solve equations with negative root
There are two more small tips from us when solving quadratic equations or quadratic functions with the PQ formula:
- If you calculate the numbers under the root and then there is a negative number under the root, you can cancel. Then the equation has no solution (at least not for schoolchildren, students then have to do imaginary arithmetic).
- Pay attention to the sign! For example, do you have the task x^{2} To solve -5x + 3 = 0, then p = -5. You then have to use this -5 in the PQ formula!
You can find an example for both cases here:
Solve equations: ABC formula or midnight formula
Solving equations - or rather quadratic equations - can also be carried out with the ABC formula or midnight formula. The ABC formula is very similar to the PQ formula and is used to solve quadratic equations. If you calculate correctly, you will get the same result with both formulas. The general formula and solution now follow, and then we turn to an example.
More about this solution method in the article ABC formula.
Solving equations: 3rd power and higher
Polynomial division is a mathematical method used to calculate the zeros of polynomials. It can also be used to solve equations of a higher degree. The method of calculation is similar to the written division that you already got to know in elementary school. For this reason, in the following, we will first briefly discuss the written division and then apply this knowledge to the polynomial division.
In order not to make the article here longer, there is everything else about the polynomial division in the corresponding article: Polynomial division.
Solve systems of linear equations
Let's look at solving equations in a different way. First of all, you should know what is meant by a system of equations with two variables. First of all, a small example: You go shopping and you know that 6 apples and 12 pears of particularly good quality cost 30 euros. And you know that 3 apples and 3 pears cost 9 euros. The question now is: what does an apple or a pear cost? Since the terms apples and pears are too long, we substitute "x" for the price of an apple and "y" for the price of a pear. This results in the following equations (compare these with the information in the text!):
Table scrollable to the right6 | Apples | and | 12 | Pears | costs | 30 euro |
6 | x | + | 12 | y | = | 30 |
3 | Apples | and | 3 | Pears | costs | 9 euros |
3 | x | + | 3 | y | = | 9 |
Of course, that doesn't look that clear yet. For this reason, the following notation has been introduced in mathematics to provide a better overview:
Table scrollable to the right| 6x + 12y | = | 30 | | Equation No. 1 |
| 3x + 3y | = | 9 | | Equation No. 2 |
Such a system of equations indicates: These equations belong to each other. This is also the reason why you have to solve them together. The goal is to get a number for x and y that satisfies both equations. And we'll take care of that now
In order not to make the article longer, there is everything else about linear systems of equations in our article systems of linear equations.
Further articles:
- ABC formula: The ABC formula or the midnight formula can also be used to solve quadratic equations. You can learn how this works in our ABC formula article.
- Polynomial division: Polynomial division is a method of finding zeros in equations with a higher power. You can learn how this works and how you can use it to solve equations in the article Polynomial Division.
- zeropoint: How do you find zeros? A detailed article with various methods, examples and tasks can be found in the article Calculate zeros.
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