1. This is going to sound repetitive, but the answer is the field axioms, along with more basic axioms, such as substitution. Every single question you asked is answered by the field axioms or some other axioms. Opalg already proved that the only possible solution is -4.

I am assuming that we're working over the field $\mathbb{R}$. The field axioms aren't always relevant (our coefficients might not be from a field), but for now it's a reasonable assumption.

2. Originally Posted by epson1
Btw you used the assumption that the equivalence between for example the equations 3x-3y=2x-2y and x=y is valid. But that was exactly what was to be proven, that the equivalence between those two equations is true.
No This is what I was responding to.

Originally Posted by epson1
Actually it only proves, that x=-4 is a solution, but not necessarily the only solution. How do you know there aren't any other solutions unless you have plugged every other number into the equation?
This is a standard technique to show a solution must be unique. P.S equality is an equvilence relation. Equivalence relation - Wikipedia, the free encyclopedia

3. Every single question you asked is answered by the field axioms or some other axioms. Opalg already proved that the only possible solution is -4.
Opalg proved that the solution is 4, not that it can't be anything else than 4, or have I missed something?

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