Variables are the thing that make algebra different from other kinds of mathematics like, say, arithmetic.
In arithmetic, we deal with numbers, and each number has a value, which is itself. And if we want to be more complicated, we can make expressions, like:
What is the value of this expression? (This isn't a trick question.) Answer: 5.
So what is a variable? One simple definition is:
In elementary algegra, the as-yet undetermined values are almost always numbers.
Because we haven't yet decided what the value of a variable is, we can't write down the value, so we have to write the variable some other way, and the usual way to do this is to give the variable a name, and the most common name for a variable is the letter "x", which is usually written in italics, like so: \(x\)
Here is an example of an expression containing the variable x:
So what is the value of \(x+3\) ?
The basic answer is that it is 3 more than whatever \(x\) is. But since we haven't yet decided what the value of \(x\) is yet, we can't know what the value of \(x+3\) is.
What is the point of taking some number that we haven't decided what it is, and then adding 3 to it, to get another number that we don't know what it is, but which must be 3 more than the number we started with?
There are actually a few different reasons for wanting to do this. The most common reasons are among the following:
Give a mathematician a hard problem, and sometimes they'll sit down and think up a harder problem, even before they have properly solved the first problem.
The way to do this with variables is to have more than one variable.
Each variable has to have a name, and different variables have to have different names. Following a common mathematical rule of thumb, which is:
a second variable will generally be called "y", also written in italics, like so: \(y\).
Here's an example of an expression using two variables \(x\) and \(y\):
What this expression means, is take the unknown value of \(x\) and add it to the unknown value of \(y\), to get a third number, whose value is of course not known, but which is equal to the sum of the unknown values of \(x\) and \(y\).
I will highlight a subtle point here, which is that although \(x\) and \(y\) are different variables, they might actually have the same value. Or then again, they might not. I will give an example of this in my article on Variables in English.