Show that any non-negative integer can be expressed as sums of 2 and 5 $n \ne 1$ and  $n \ne 3$, where $n = \{0,1,2,3,...\}$.  You cannot use $-2$ or $-5$.  For example, $14$ can be expressed as $5 + 5 + 2 + 2$.
I feel like this could be proved inductively using two variables, but my class has no covered that.  Is there some other method that could be used to solve this more easily?  I don't necessary need to see this problem solved, I'd just like a shove in the right direction.
 A: It's easy to write an even number as a sum of $2$s.
So if you have to write an odd number as a sum of $2$s and $5$s, write it as an even number plus $5$.
A: Hint:  if you can do $4$ and $5$, you can gain one more by trading two $2$'s for a $5$, or by ...
As a reference, see the coin problem
A: You can do it by induction. There are a couple of ways to organize it.
For instance, for $n\ge 4$ let $P(n)$ be the statement that if $4\le k\le n$, then $k$ can be expressed as a sum of $2$’s and $5$’s, i.e., in the form $2a+5b$, where $a$ and $b$ are non-negative integers. Start by proving $P(4)$ and $P(5)$ ‘by hand’. Then the induction step is pretty straightforward, since if $n\ge 5$, $n+1\ge 6$, and your induction hypothesis $P(n)$ tells you that you can express $(n+1)-2$ in the desired form.
Alternatively, you can do two inductions. First show that $P(n)$ implies $P(n+2)$, prove $P(4)$ and conclude that every integer of the form $4+2k$ with $k\ge 0$ can be so expressed; that covers every even integer greater than $2$. Then verify $P(5)$ and conclude that every integer of them form $5+2k$ with $k\ge 0$ can be so expressed; that covers every odd integer greater than $3$.
(I suspect that both of these are variants of your ‘two variables’ idea; if so, you were on the right track, or at any rate a right track.)
