Two years ago, I bought the book Mathematics for the Nonmathematican, by Morris Kline.
There I learned a new way of solving equations, which is related to the principle that states that any alteration on both sides won't alter the output, Ex:.
$$x+8=4\tag{1}$$
$$x+8-8=4-8\tag{2}$$
$$x=-4\tag{3}$$
In my school, they didn't teach me this way, they taught me that the numbers walk from one side to another and when they do that, they change their sign and in one side we should put all the terms that have x, in the other, we put all terms that don't have $x$'s. Ex:.
$$x+4=2x+7\tag{1}$$
$$x-2x=7-4\tag{2}$$
$$-x=7-4\tag{3}$$
$$-x=3\tag{4}$$
If the $x$ is negative, then you swap the $-$ for $+$ and vice-versa for both terms:
$$x=-3\tag{5}$$
Some days ago I watched a video that stated the importance of teaching long division, the author argued that only long division could teach the concept of convergence and then I got worried with this method, is this way of teaching how to solve equations dangerous somehow? When I learned the first method two years ago, I felt that I could handle my equations better but that could be only wishful thinking.
EDIT: For the ones curious about the video in which the guy says that only long division teaches convergence, you can see it here. Specially this comment.
Best Answer
The worst thing to teach is that mathematics is a series of recipes to be blindly followed. Mathematics should be about ideas. The reason "walking from one side to the other" works is precisely that it preserves the solutions to the equation. Teaching the mechanics of solving equations without even mentioning that reason should be a criminal offense.