What is algebra in math
We'll cover the basics of algebra in this article. It first briefly explains what is meant by algebra and then explains some topics. This article is part of our Mathematics section.
Note: This is an overview page about algebra. For many of the topics briefly presented here, there are links below with further information + examples and videos.
Algebra is a field of mathematics that is devoted to structure, relation, and quantity. In school and in everyday life, algebra is often referred to as calculating with unknowns in equations. We'll take a quick look at some important terms and then go into equations, inequalities, and systems of equations.
A variable is, so to speak, a "placeholder" for a number. In mathematics, a letter is usually used for it. This is then e.g. an a, b, x or y. Instead of this variable, a number will be used later. A value can be assigned to a variable. Here are a few examples of what an assignment can look like.
- a = 3
- b = 10
- x = 3.12
- y = 5.89
In the first example this would mean that the number 3 is used instead of the "a". For the "y" it would be the number 5.89 (last example). This knowledge of the variables is required in the Functions section.
In mathematics (and thus also in algebra) the term term describes a meaningful expression that can contain numbers, variables, symbols (for mathematical connections) and brackets. This means that terms are, so to speak, the grammatically correct words or groups of words in the language of mathematics. For a better understanding of this statement, a number of examples of terms and of what is not called a term now follow.
- 2 + 0,5
- a2 + b2 + c3
- cos (x)
- sin (x)
In mathematics and in the exact natural sciences, the equal sign "=" stands between two expressions that are identical in value. Everyone knows something like this from elementary school, because on the left side of the equal sign is the same value as on the right side. Some examples:
- 3 + 2 = 5 and thus 5 = 5
- 7 + 1 = 8 and therefore 8 = 8
- 2 + 2 = 4 and thus 4 = 4
Algebra: linear equations
As already indicated in the introduction to the topic of algebra, an equation with one unknown (there are also equations with two or more unknowns, but we do not want to torment you here) should now be solved. This unknown is usually called "x" in class. The aim is to have "x = a number" at the end. Here's a very simple example to get you started. This is explained below.Table scrollable to the right
|x + 2 = 5||| -2|
|x = 3|
In the first line there is the starting equation, which is solved for x. To do this, so-called Equivalent transformations be performed. This means: The appearance of the equation is changed, but the same value is on the left as on the right. In order to resolve for "x", the 2 on the left must be "removed". To get a +2 away, "-2" has to be expected. All arithmetic operations are followed by a "|" written. So now "| -2" is written to make it clear that a 2 should be subtracted. Very important: Arithmetic operations must be carried out on both sides. If I calculate "-2" on the left, I have to do the same on the right!Show:
Algebra: solving inequalities
In the case of inequalities, one side of the equation is usually larger or smaller than the other. This is expressed by a "<" (smaller) or ">" (larger), as already discussed in the basics of mathematics. In addition, there is a less than or equal to "≤" and a greater than or equal to "≥". In principle, inequalities are calculated in the same way as normal equations. Only one special rule still has to be observed:
- If you multiply or divide both sides of an inequality by a negative number, "<" and ">" or "≤" and "≥" are exchanged for each other.
It is essential to observe this rule when calculating with inequalities. Otherwise, a few examples should explain this best.Table scrollable to the right
|4x + 10 ≥ 14||| -10|
|4x ≥ 4||| :4|
|x ≥ 1|
Algebra: systems of linear equations
And another topic from the field of algebra, this time linear systems of equations: 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 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 right
|6||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 very 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 one another. 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.
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