Variaveis

Variables

A variable is a placeholder for an unknown quantity. Usually, variables are denoted by English or Greek letters or symbols such as $x$ or $\theta.$

Examples:

• In the equation $10=2x,$ $x$ is the variable.
• In the equation $y + 2 = 6,$ $y$ is the variable.
• In the equation $2P+13\theta=24,$ $P$ and $\theta$ are variables.

Variables are often used in expressions and equations! Equations with several variables are called literal equations.

What is a variable? For example, in $x^2 + 2x$, what is $x$? Or in $x^5 - 10 x-12=0$, what is $x$?

The quick answer is, $x$ is a place-holder. It's a blank slot, in which we can place a number, and then we get either a numerical expression or a numerical sentence (which may be true or false). We may write a number in place of $x$ in the expression $x^2 + 2x$. If we choose $5$, we get $5^2 +2\cdot 5$, which simplifies to $35$. If we choose $2$, we get $2^2 + 2\cdot 2$, which simplifies to $8$. Or we may write a number in place of $x$ in the equation $x^5 -10 x - 12=0$. If we choose $5$, we get $5^5 -50 -10 =0$, which is a false statement. If we choose $2$, we get $2^5 -20-12=0$, which is true.

To go deeper into this, we need to recognize that letters in algebra may be used in two ways: as constant symbols or as variable symbols.

Constant symbols

A letter may stand for a number. Some letters stand for particular number. For example $e$ by convention stands for a number near $2.7$. Sometimes, we use a letter in the context of a particular problem to stand for a specific number.

Example. If Ann's age is one quarter Bill's today and will be one third of Bill's five years from now, then how old is Ann? Let $a$ stand for Ann's age today and let $b$ stand for Bill's age today. Then $4a=b$ and $3(a+5) = b+5$, so $3a+15 = 4a+5$, so $10=a$.

In the example, from the moment we introduced the symbol $a$, it stood for the number $10$. However, we did not know that until we had done a little thinking.

In this case, the symbol $a$ is NOT a variable, it is a name for $10$. The symbol $a$, in this context, is called a constant symbol. The example shows that we may have a constant symbol whose value we do not immediately know. (Let $c$ stand for my father's age on January 1, 2016. Then $c$ is a number, but you have no way of knowing what is is, unless I tell you. But nonetheless, as long as we agree to give the symbol $c$ this meaning, it is a constant symbol.)

Note: some people call any letter in a math problem a variable. But this just leads to confusion, because sometimes (as the example shows) a letter may refer to one fixed and unchanging thing.

Variable symbols

In contrast to constant symbols, we have variable symbols. A variable symbol does not stand for anything at all. It simply holds a place in an expression or an equation, in the same way a blank space or a box does. If we write an expression with a variable symbol in it, then we may think about all the different values that the expression acquires when specific numbers are written in place of the variable symbol. If we write an equation with a variable symbol in it, then when we write a number in place of the variable symbol, we get a sentence that may be true or may be false.

Independent Variables vs Dependent Variables

The term "variable" also refers to something a little different. It might be talking about independent variables and dependent variables.

An independent variable is the one whose value you can change. The dependent variable has its value changed by independent variables. Here's an example. Look at the equation

$$y = x^2$$

This equation uses exponents, so review those if you need to. Here we have $x$ be the independent variable. Make $x$'s value any real number you want. Then $y$'s value will change based on what you chose for $x$. This makes $y$ a dependent variable. More specifically, it depends on $x$.

(Video) Definitions

by mathman1024

Here's a video by mathman1024 where he goes over some basic definitions in algebra.

A constant is any number. A constant doesn't change. $$\text{Ex: }5,\ -12,\ \frac87,\ \pi,\ e$$

A variable is any letter or symbol that represents some unknown value. $$\text{Ex: }x,\ y,\ n,\ \lambda, \text{☺}$$

A term is either a single constant or variable or, the product of constants and variables. I am not entirely sold on his definition in the video. For now just consider a term to be the product of constants and variables. $$\text{Ex: }5x,\ \frac78y^5,\ z,\ -15$$ Note that even though we don't write the multiplication symbol between the number and variable it is implied. So $5\cdot x=5x$.

A coefficient is the constant factor of a term. So if you have the term $5x$ the $5$ would be the coefficient.