# How do I show that one equation is equivalent to another?

I am being asked to show an equation (a) for thermal expansion is equivalent to another equation (b).

Show that the following equation for thermal expansion is equivalent to

I have never come across this type of question before.

What am I being asked to demonstrate in answering the question and how might I start? If there are examples I can be pointed towards, that would be appreciated. Note that I'm not necessarily asking for anybody to do my work for me!

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Try to use algebraic manipulations/identities to turn one into the other. Could you maybe show the actual equations you are considering so that we have something concrete to discuss? –  Ｊ. Ｍ. Feb 8 '12 at 23:25
This is really not the tag to use here. I have absolutely no idea what is the proper tag, but I can exclude (amongst other tags) this one from the set of fitting tags. –  Asaf Karagila Feb 8 '12 at 23:26
I've updated the question with the equations. –  James Feb 8 '12 at 23:57
If in fact your book has defined/stated that $\Delta X=X-X_0$ to begin with, then the proof is not terribly difficult... –  Ｊ. Ｍ. Feb 9 '12 at 0:03
Added the homework tag. James, please do this yourself in the future when posting homework questions. –  Ben Crowell Feb 9 '12 at 0:08

Because $\Delta X = X - X_0$ as specified, it is a simple algebraic manipulation to show equivalence:

$$X = X_0(1 + \alpha \Delta T) = X_0 + X_0\alpha\Delta T$$

Subtract $X_0$ from both sides to get:

$$\Delta X = X - X_0 = X_0\alpha\Delta T$$

This all falls back on basic algebra. The reason you may be asked to prove such an equivalence is just to clarify that the same formula can be expressed in several ways, and when using such a formula you can choose a form which is most conducive to your data (or the problem space).

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Thank you, Kurtis. In demonstrating this, is the question simply asking me to show pretty much what you've shown there? It just looks... simpler than what I understood the question to be asking. –  James Feb 9 '12 at 0:18
Without any further context, this appears to be what you are asked to show. If you have a different understanding, please share it and maybe that will reveal a different interpretation of the question. –  Kurtis Zimmerman Feb 9 '12 at 2:28
Hi Kurtis. That's the question as it appears. I honestly don't know what it's expecting me to show. I imagined perhaps a step-by-step approach, but that was assuming that the answer required would have looked a bit more complex than that. It would be reasonable to assume that what you've offered there is probably what I'm expected to set out. Thanks for your help. –  James Feb 9 '12 at 10:08