"I will remark, in some horror and exasperation with the modern educational system, that I do not recall any math-book of my youth ever once explaining that the reason why you are always allowed to add 1 to both sides of an equation is that it is a kind of step which always produces true equations from true equations."

I can now say that my K-12 education was, at least in this one way, better than yours. I must have been 14 at the time, and the realization that you can do that hit me like a ton of bricks, followed closely by another ton of bricks -- choosing what to add is not governed by the laws of math -- you really can add anything, but not everything is equally useful.

E.g., "solve for x, x+1=5"

You can choose to add -1 to the equation, getting "x+1+-1=5+-1", simplify both sides and get "x=4" and yell "yay", but you can also choose to add, say, 37, and get (after simplification) "x+38=42" which is still true, just not useful. My immediate question after that was "how do you know what to choose" and, long story short, 15 years later I published a math paper... :)