Doppler effect Formula manipulation

[Trec93]

Homework Statement

I have this Doppler effect formula, but I don't know how it was derived, I can't repeat the process myself to solve for speed of the source, I would really appreciate if someone could mathematicly solve this in steps, thank you very much.

Homework Equations

$$f=f_{s}\frac{v}{v-v_{s}} \Rightarrow v_{s}=\frac{v(f-f_{s})}{f}$$

The Attempt at a Solution

I checked and this is the same thing as the equation above, but mine is messy and ugly, I don't know how to prepare "neaty" formulas like the one above, this often confuses me and forces me to do checks whether my formula is right or not.
$$v_{s}=\frac{-f_{s}v}{f}+v$$

 

[gneill]

Show, step by step, what you've tried and where you've become stuck.

 

[Trec93]

Show, step by step, what you've tried and where you've become stuck.

Ok here's how I did it:

$$f=f_{s}\frac{v}{v-v_{s}}$$
I multiplied this by: $$(v-v_{s}) \Rightarrow f(v-v_{s})=f_{s}v$$
Then I divided by F and subtracted v
$$-v_{s}=\frac{f_{s}v}{f}-v$$

Finally I multiplied by the negative sign, that's my result:
$$v_{s}=\frac{-f_{s}v}{f}+v$$

I know they both are equal because I checked, but I don't have the skill to make my formula "neat" I often don't understand how people derive their formulas, I hope you know what I mean, I can't transform my formula into one above.
$$\frac{v(f-f_{s})}{f} = \frac{-f_{s}v}{f}+v$$

 

[gneill]

Combine the terms on the RHS with a common denominator.

 

[Trec93]

Combine the terms on the RHS with a common denominator.

Like this?
$$v_{s}=\frac{-f_{s}v}{f}+\frac{vf}{f}$$
Wait I see where this is going..
$$v_{s}=\frac{-f_{s}v+vf}{f}$$
$$v_{s}=\frac{v(f-f_{s})}{f}$$
Is this right?

 

[gneill]

Yup.

 

[Trec93]

Yup.

Wow thank you.