[zozbot123]

> which is to stress the rule you can do the same thing to both sides of an equation (worrying about things like squaring both sides or multiplying by zero later).

It's a pity that they're being so ambiguous here, because explaining why and when "you can do the same thing" to both sides of an equation is not actually hard! You can apply an injective function that's always defined over the appropriate domain to both sides of an arbitrary equation, and this will preserve the equation entirely because (a = b) is equivalent to (f(a) = f(b)) when f has this property. You can apply a non-injective function with no restriction on its domain, and this may introduce extraneous solutions but will not "miss" any, because (a = b) implies (f(a) = f(b)) if f is always defined. You can apply an injective function, perhaps defined over a more limited domain than the original equality, and this will not introduce extraneous solutions but may "miss" some, because (f(a) = f(b)) implies (a = b) if f is injective, but the converse is not true given any restriction on f's domain. Of course, if these functions are defined in terms of x, then you get to worry about whether the function is injective or well-defined given some value of x. For instance, multiplication by x is not injective if (x = 0) but it is otherwise.