Proof by contradiction: $r - \frac{1}{r} =5\Longrightarrow r$ is irrational? Prove that any positive real number $r$ satisfying: 
$r - \frac{1}{r} = 5$ must be irrational. 
Using the contradiction that the equation must be rational, we set $r= a/b$, where a,b are positive integers and substitute:
$\begin{align*}
&\frac{a}{b} - \frac{1}{a/b}\\

&\frac{a}{b} - \frac{b}{a}\\

&\frac{a^2}{ab} - \frac{b^2}{ab}\\

&\frac{a^2-b^2}{ab}
\end{align*}$
I am unsure what to do next?

Since you have already seen answers which complete your proof, here is another proof using continued fractions.
If $r$ was rational, it will have a finite continued fraction, say $[a_1, a_2, \dots, a_n]$, but that can be extended to $[5, a_1, a_2, \dots , a_n]$ and $[5,5,a_1, a_2, \dots a_n]$ etc, because
$$ r = 5 + \frac{1}{r} = 5 + \frac{1}{[a_1, a_2, \dots, a_n]} = [5, a_1, a_2, \dots, a_n]$$
In fact, you can easily give the infinite continued fraction of $r$
$$[5,5,5,5,\dots]$$
