How to rearrange a series of equations to express in terms of one variable

Question

I'm working on an equation where, given the following equations:

$nL=TE$,

$\large{R=\frac{VT_f}{2}}$, and

$qf=nT_fT^{-1}$

I should get

$\large{R=\frac{VLqf}{2E}}$

I don't trust my algebra to get $R$ in these terms and was hoping someone could show me the explicit steps involved. Thank you.

Answer 1

Isolating for $T_{f}$ in $qf=nT_{f}T^{-1}$ gives $T_{f}=\frac{qfT}{n}$.

Now, $nl=TE$, so $T=\frac{nl}{E}$.

Therefore, $T_{f}=\frac{qf}{n}\cdot\frac{nl}{E}=\frac{qfl}{E}$.

It follows that $R=\frac{VT_{f}}{2} =\frac{Vqfl}{2E}$, as desired.

Answer 2

In the expression $R = \frac{VT_{f}}{2}$, you want to replace $T_{f}$ with something involving $L$, $q$, $f$, and $E$. You can rewrite your last equation to make $T_{f}$ the subject:

$T_{f} = \frac{qf}{nT^{-1}} = \frac{qfT}{n}$

Substitute this into your expression for $R$ to get:

$R = \frac{VqfT}{2n}$. Now you need to get rid of $n$ and $T$, and replace them with $L$ and $E$. I think you have an equation that has these four variables in.