What's the algebraic property where you can flip the fractions in an equation?

Question

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.)

Answer 1

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)

Answer 2

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 $$2 \neq 1 - 2$$

Answer 3

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.

Answer 4

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 $-1$ :

$$\frac{a}{b} = \frac{c}{d} \implies \left(\frac{a}{b}\right)^{-1} = \left(\frac{c}{d}\right)^{-1} \implies \frac{a^{-1}}{b^{-1}} = \frac{c^{-1}}{d^{-1}} \implies \frac{1}{a} \cdot \frac{b}{1} = \frac{1}{c} \cdot \frac{d}{1} \implies \frac{b}{a} = \frac{d}{c}$$

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The following is the best approach to your question:

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