Order of operations is arbitary. You can build a completely consistent and valid system of mathematical notation under which you do addition and subtraction first, then multiplication and division.

Take the NORMALLY notated equation

2 * x + 3 = 3

Under normal notation, this means "the result of multiplying x by two then adding three yields three." Under normal notation, this "x" must be 0.

Now, imagine a new kind of notation, where you do parentheses, then additions and subtractions, then multiplications and divisions. How would you say, under this new kind of notation, "the result of multiplying x by two then adding three yields three"? You'd say it like this:

(2 * x) + 3 = 3.

You have to use parentheses now, because you're supposed to do additions first unless they are superceded by parentheses--and what we're trying to say with this equation is supposed to involve doing a multiplication first. Hence, under our new notational convention, we find we must put parens around the multiplication.

We haven't broken any fundamental laws of mathematics. We've simply adopted a new notation. And it works--it gets you the right solution. For both the old notational convention and the new one, you can see, you must first subtract three from both sides, then divide both sides by two, leaving x = 0 as the answer.

Another example:

Ordinary notation: 2 * (x + 3) = 4

The same expression under the new notation: 2 * x + 3 = 4

Under both forms of notation, the solution is found by first dividing both sides by two then subtracting three from each side (because under both notations, following the rules of the respective notations, you can see that the addition is performed "first," the multiplication "second." Hence when solving, you divide both sides "first," then you will subtract "second.") In both cases, the correct solution is yielded: x = -1

Order of operations is not fundamental in mathematics. It is literally a *mere notational convention* and nothing else. You can adopt a different convention to do exactly the same math. You could use no notation at all and simply write everything out in English. (That's what Newton and his contemporaries did btw.) You could instead write it all out in French. (Descartes did some of that I believe.) Deciding between English and French is simply deciding between two notational conventionss, just as deciding whether to do addition first or multiplication is just a way to decide between two notational conventions, i.e., languages.

Who understood this? Who is like "what?!" Who is like "no way!!" I believe I am willing to stick with this conversation for quite a while til I've explained it adequately.