How do I write a variable, x, when I mean 'any x' so that it's clear I don't mean a particular number. In high-school, we usually used letters (literals?), such as $y$ to designate particular unknown numbers. In functions, $y$ could designate various numbers, but it seems to me that in these cases $y$ depended upon the value of another letter, $x$. For example, if $x=3$, then $y=6$. It was never the case that $y$ unconditionally designated various numbers. 
I began university math this year and enrolled in the first-year math reasoning class. Proofs, or definitions, seem to use letters to designate various numbers. For example, $ (x=2k+1)\rightarrow (7x-11=2k)$. If I were to substitute $2k+1$ into $7x-11$, I would get $2(7k-2)=2k$, which is false if $k \in \mathbb Z$. However, as far as I know, $2k$ is meant to define any number that one can multiply by two as an even number, and $2(7k-2)$ satisfies those criteria. I think the problem arises from (my own?) equivocal usage of letters. How do can I avoid this problem?
 A: Use quantifiers such as for all $ \ \forall$ (\forall ), exists $ \ \exists$ ( \exists ), exists a unique $\ \exists !$ ( \exists ! ), etc. For example:


*

*$x$ is even if and only if $\exists  \ k \in \mathbb{Z}$ such that $x = 2k$;

*$\forall \ x \in \Bbb R$, we have $(x+1)^2 = x^2 + 2x + 1 $, etc. 
I hope this helps. 
A: Another way that avoids quantifiers is to say "let $x$ be arbitrary" or "for any $x$ we have...". 
A: People do sometimes get wrong results because they use the same letter to write general forms of two different expressions, and then put these expressions together and proceed to evaluate the result as if all occurrences of that letter were known to be equal.
One way to avoid this is to use different letters to represent the "arbitrary number" in different expressions.  For example, you may say $x = 2k + 1$ to express the fact that $x$ is an odd integer (provided that somehow it is clear that $k$ is an integer). You might also say $7x - 11 = 2k$ to express the fact that $7x-11$ is even. But you could just as easily write $7x - 11 = 2m$ to mean the same thing. So when you put the statements about $x$ and $7x-11$ together, you can write them in a way that doesn't say the same arbitrary number occurs on the right-hand side both times:
$$x = 2k+1 \implies 7x-11 = 2m.$$
Sometimes, when one needs to be more formal or needs more precise control over how the values of unknowns are defined, one can use quantifiers. For example,
$$\exists k\in\mathbb N.(x=2k+1)$$
is another way of saying $x$ is odd, but you don't have to say anything elsewhere to make it understood that $k$ is an integer (because it says so, right there in the formula). When it is already understood that the symbol $k$ could only stand for an integer, we could write
$$\exists k.(x=2k+1)$$
to say the same thing. 
In either of these last two formulas, the quantifier $\exists$ "binds" the variable $k$ to this expression. A practical implication of this is that the value of $k$ inside this parentheses is unrelated to any occurrence of the letter $k$ in any other expression we might combine with this, so that (technically) we can correctly write
$$\exists k.(x = 2k+1) \implies \exists k.(7x-11 = 2k).$$
But another property of a bound variable (such as $k$ in either part of this formula) is that you can use any name you want for that variable as long as you do not duplicate one of the other names in the formula. That is, $\exists m.(7x-11 = 2m)$ says exactly the same thing as $\exists k.(7x-11 = 2k).$ So there's no need to use the same name in two separate existential quantifiers, and we might as well write
$$\exists k.(x = 2k+1) \implies \exists m.(7x-11 = 2m)$$
which I think is clearer and less likely to lead to mistakes. But this is just the first formula I suggested, with the addition of a couple of $\exists$ quantifiers to express what you already knew we meant by $k$ and $m$ in that formula.
There is one other option when you want to make a simple statement such as that $x$ odd implies that $7x-11$ is even: use the same arbitrary value, like this:
$$x = 2k+1 \implies 7x-11 = 2(7k-2).$$
This is correct when $x$ and $k$ on the right-hand side have the same values as they do on the left-hand side, and since it's easy to see that $7k-2$ is an integer, the right-hand side adequately expresses the fact that $7x-11$ is even.  It is not always necessary to introduce a symbol for each interesting quantity in your formulas.  Sometimes a new symbol will simplify the formulas so much that it makes the difference between something reasonable to write out and something far too complicated to write out (or to read), but sometimes you can just let an expression represent that quantity.
