I was doing a problem in my textbook when this set of equations came up:
x = 2y - 9
$x^{2}$ = 3y
Obviously, we can substitute for the x in the second equation and get:
(y-3)(4y-27) = 0
which has the solutions 3 or 27/4
this leads us to the pairs (-3, 3) and ($\frac{9}{2}$, $\frac{27}{4}$). My textbook sort of "assumed" that these were the answers without going back to input these values once again into the original equations. I haven't really noticed this until now but don't the solutions 3 and 27/4 (and it's corresponding x values) only represent a superset of the possible solutions to the original system of equations? Isn't this the case since the implication doesn't go back the other way as in
just because we have y that satisfies (y - 3)(4y - 27) doesn't mean that we have the values of y that satisfy
x = 2y - 9
$x^{2}$ = 3y
(Obviously in this case it's true, but I'm speaking of the general situation with polynomials of higher degree and such) right?
My textbooks seem to assume that all solutions to the combination between the original system of equations represent the solutions. Isn't this technically false since even with a simple example of:
$x^{2}$ = 1
x = 1
$x^{2}$ + x - 2 = 0
which has the solutions x = 1 and x = -2 and I have a feeling that one of these are false...
I might really just be confusing myself even more or overcomplicating things. I hope all of that was clear and sorry for the dumb question haha. I know that this question is perhaps a bit too trivial for the average user of this site, but I really didn't know where else to go. Anyways, thanks in advance!
Your concern is valid.
Typically, you find solutions by converting the given equations into some other (simpler) form, and then you repeat that process until you end up with equations whose solutions are obvious. As you go through this transformation process, you have to think about whether each new simplified set of equations has the same solutions as the previous set (no more and no less).
For example, $x^2 = 9 \Leftrightarrow |x|=3$, so the equations $x^2 = 9$ and $|x|=3$ have the same solutions. But, if you're sloppy, you might think that $x^2 = 9 \Leftrightarrow x=3$, which is not true, although it is true that $x=3 \Rightarrow x^2 =9.$ You have to think about the directions of the implications.
And in any "equation" problem, you have to check what the question is asking. It might be ...
In case #1, you have to make sure that every transformation you perform is a $\Leftrightarrow$ one.
In case #2, you can just pluck a solution from thin air, if you want to, check that it actually is a solution, and you're done. No need for $\Leftrightarrow$ reasoning.
Case #3 could be regarded as bad wording. The word "the" implies that there is only a single solution, and it's not clear whether you're allowed to assume this or you must prove it.