I should be able to explain if I understand what you're asking, but I'm not sure I do. An example will help: let me try to give one and explain it, and if it isn't an example of the right thing, then please give a better example!
Suppose we know
3a + b = 7
and
a - b = 1
If I understand correctly, you're saying that you know the thing to do is to add up these equations to get 4a = 8, telling you that a = 2, and then substitute back to get b = 1, but you're not sure why that is a legitimate thing to do - is that right?
Technically it works because the + operator is a congruence with respect to = (that's computer science terminology - you almost have to program a computer that doesn't know this to notice that it needs to be said!) but I think it's useful. That is, if you know that a = b then, for any c, a + c = b + c, etc. (And this is so universally true of anything for which we use these symbols that I don't even have to say whether I mean natural numbers, complexes, processes in some weird process calculus or what!) It's a special case of "it's OK to do the same thing to both sides [oh, except divide by something that might be zero or take roots, but, err, don't worry about that at school]". In this case, since we know 3a + b = 7 we can deduce that 3a + b + 1 = 7 + 1 = 8, and since we know a - b = 1 we can deduce that 3a + b + 1 = 3a + b + a - b = 4a, and oh look, we're also using the fact that = is an equivalence relation (that is, three things: for any x and y and z (1) x = x, (2) x = y if and only if y = x (3) if x = y and y = z then x = z) to deduce 4a = 8.
I'll stop there in case I'm barking up the wrong tree entirely...
ETA: gosh, you have to be fast round here. Well, there are more long words in my version ;-)
Last edited by ColinsMum; 02/21/12 07:59 AM.
