Why does algebra work?

[Gigabitten]

It would seem to be the case that if you have an equation in n variables, performing the same operation to each side of that equation provides an identical equation, in that if you graphed each equation in n dimensions you'd get the same graph.

Why is this, though? The first place I'd think to look would be the fundamental theorem of algebra, but I don't really see how that ties in here if it does at all.

 

[Maschke]

If two things are equal you can do the same thing to both of them and the results will still be equal.

 

[Gigabitten]

If two things are equal you can do the same thing to both of them and the results will still be equal.

That makes perfect sense, but how do I know that this is true? I...

Wait, actually I just figured it out. It's because, in an equality, both sides of the equation are (by definition) equal. So, applying any operation to each of them will give the same result (or results, if that operation isn't a function). Neat. Thanks!

 

[Maschke]

Wait, actually I just figured it out. It's because, in an equality, both sides of the equation are (by definition) equal. So, applying any operation to each of them will give the same result (or results, if that operation isn't a function). Neat. Thanks!

Yes that's exactly it!