Variables

What are variables, constants and coefficients?

Imagine that you encounted the following equation:

$\frac{x}{4} + 2 = 3$

We know that this is an equations because of the equal sign. The equation includes variables, constants and coefficents. We will describe each in turn.

Variables: Abstractly, a variable is any characteristic or quantity that can be counted. This could be the number of jelly-beans in a jar, or the number of instances of some disease in a given metropolitan area. In a statistics context, variables are the observable or known. The example above about diseases is a prime example. In a pure mathematics context variables are the unknown. In the example equation above, this is $x$ . A variable is only meaningful once defined. Before this point, a variable has no meaning, it is simply a letter or a symbol.
Constants: Constants are just that, constant numbers. In the above example, both 2 and 3 are constants. The value of these does not changes as a function of something else. They are fixed, known entities.
Coefficients: Coefficients are constant, multiplicative factors that interact with variables. In the above example, the coefficient to $x$ is $\frac{1}{4}$ .

Supplemental Video

Independent vs. Dependent

Dependent and independent variables are, intuitively, either independent and not influenced by other variables, or dependent upon the value of other variables. For example, if we have the linear equation $y = 3x + 2$ , the value of $y$ is dependent upon the value of $x$ . Likewise, if we define the function $f(x) = 3x + 2$ , the output (value) $f(x)$ is dependent upon (a function of) $x$ .

To solve an equation, it is important to try and isolate the dependent variable(s). This frequently requires that we identify which variable(s) are dependent upon which other variables. If we are performing an experiment, we know that the dependent variables are those that we are measuring. For example, the time required for water to boil on the stove. The independent variable(s) are those that we are manipulating, for example the quantity of water, type of pot, or altitude. By varying one (and thereby introducing two constant variables) or all of the independent variables were can change (presumably) the dependent variable.