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.)
 A: You can flip if you flip correctly. Flipping both sides of
$$\frac{1}{R_2} = \frac{1}{R_e} - \frac{1}{R_1}$$
gives you
$$ R_2 = \frac{1}{\frac{1}{R_e} - \frac{1}{R_1}}$$
Well, that's not quite right: more pedantically, flipping both sides gives
$$ \frac{1}{\frac{1}{R_2}} = \frac{1}{\frac{1}{R_e} - \frac{1}{R_1}}$$
but we know that the left hand side of this is the same thing as $R_2$. (at least in the current setting, where $R_2$ is known to be nonzero)
A: Yes, you can “flip”, so long as the two sides are single fractions: from
$$
\frac{1}{R_2}=\frac{R_1-R_e}{R_1R_e}
$$
you can rightly deduce
$$
R_2=\frac{R_1R_e}{R_1-R_e}
$$
Note that, in general,
$$
\frac{R_1R_e}{R_1-R_e}\ne R_e-R_1
$$
Indeed the equality would imply
$$
R_1R_e=-R_e^2+2R_1R_e-R_1^2
$$
or
$$
R_1^2-R_1R_e+R_e^2=0
$$
Since your numbers are by hypothesis non zero, this would imply
$$
\left(\frac{R_1}{R_e}\right)^2-\frac{R_1}{R_e}+1=0
$$
or, setting $t=R_1/R_e$, $t^2-t+1=0$. This equality is not true for every real $t$. So your conclusion is really wrong.
A: 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:

$$
\frac{1}{R_1} + \frac{1}{R_2} = \frac{1}{R_e}\\
\implies \frac{1}{R_2} = \frac{1}{R_e} - \frac{1}{R_1} 
= \overbrace{\frac{R_1 - R_e}{R_1 R_e}}^{\text{Taking LCM}} \\
\implies \underbrace{R_2 = \frac{R_1 R_e}{R_1 - R_e}}_{\text{Applying Invertendo}}$$
A: This is not an algebraic property, because it is not true. 
For example, let $R_2=R_1=2$ and $R_e =1$. Then, $$\frac{1}{2} = \frac{1}{1} - \frac{1}{2}$$
but
$$2 \neq 1 - 2$$
