What's the algebraic property where you can flip the fractions in an equation? Earlier in algebra, we spent over 20 minutes trying to figure out
$$ \frac{1}{R_1} + \frac{1}{R_2} = \frac{1}{R_e} \,\,\,\, \text{solve for }R_2 $$
when the teacher said "What you start out with is the same as what you learned in pre-algebra
$$
\frac{1}{R_1} + \frac{1}{R_2} = \frac{1}{R_e}
$$
subtract $\frac{1}{R_1}$ from both sides:
$$\frac{1}{R_2} = \frac{1}{R_e} - \frac{1}{R_1}$$
and then the math gods said 'you may flip as long as all are flipped'"
$$R_2 = R_e - R_1$$
What is the name of this algebraic property?
(Sorry, I couldn't find any good tags for use here.)

You can't just flip all willy-nilly.
If what you said was true, then the resistance of electrical circuits in series and parallel connections would be the same! That just isn't true.
$$\frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \dots + \frac{1}{R_n}\mathbf{\neq} \frac{1}{R_1 + R_2 + R_3 + \dots + R_n}$$
Fastest way to see this is by setting $R_1 = R_ 2 = \dots = R_n = R$ (say)
Does it feel like $\frac{n}{R} $ and $\frac{1}{nR}$ are the same? 
Now, you see. You must either change your math gods or listen to them more carefully.
The flipping property is called the Invertendo:
$$\frac{a}{b} = \frac{c}{d} \iff \frac{b}{a} = \frac{d}{c} $$
There are many ways you can realize it and I leave it as an exercise to you to try.
The simplest way is by raising both sides of the equation to $-1$ :
$$\frac{a}{b} = \frac{c}{d} 
\implies \left(\frac{a}{b}\right)^{-1} = \left(\frac{c}{d}\right)^{-1} 
\implies \frac{a^{-1}}{b^{-1}} = \frac{c^{-1}}{d^{-1}} 
\implies \frac{1}{a} \cdot \frac{b}{1} = \frac{1}{c} \cdot \frac{d}{1} \implies \frac{b}{a} = \frac{d}{c}$$

The following is the best approach to your question:

$$
\frac{1}{R_1} + \frac{1}{R_2} = \frac{1}{R_e}\\
\implies \frac{1}{R_2} = \frac{1}{R_e} - \frac{1}{R_1} 
= \overbrace{\frac{R_1 - R_e}{R_1 R_e}}^{\text{Taking LCM}} \\
\implies \underbrace{R_2 = \frac{R_1 R_e}{R_1 - R_e}}_{\text{Applying Invertendo}}$$
