Let's say we had a linear equation of the form $ax+b=c$ we then solve it for $x$ getting, let's say, $x=5$.
Just for fun, let's pretend we haven't realized we had solved the problem, so we square both sides of this equality getting $x^2=25$, we then try to solve it by square rooting and we get $|x|=|5| \implies x= \pm5$.
If we try this in the original equation, only one of the $x$'s will be a solution.
Generalizing this process, is there any way to know exactly how many "solutions" are we "adding" to any system of equations by applying irreversible operations?
The general philosophy concerning this question is as follows:
An equation, or system of equations, $$\Phi(x)=0\tag{1}$$ making sense for elements $x$ of an agreed ground set $X$ defines a solution set $$S=\{x\in X\>|\>\Phi(x)=0\}$$ in an implicit way: For each $x\in X$ it is easy to test whether it belongs to $S$ or not, but we do not have an a priori overview over $S$. The solution set $S$ may be empty, it can contain exactly one element, a certain finite number of elements, or infinitely many elements.
Solving the equation or system $(1)$ means producing an explicit description of $S$ in the form of a list, say $S=\{-1, 3,17\}$, or a parametric representation ("production scheme"): $S=\{ x(\iota)\>|\>\iota\in I\}$, whereby $I$ is an index set and $\iota\mapsto x(\iota)\in X$ is an explicitly given function.
In order to solve $(1)$ it is very common to set up a chain of true statements $$\bigl(x\in S\ \Leftrightarrow\bigr)\quad\Phi(x)=0\quad\Rightarrow\ldots\Rightarrow\ldots\Rightarrow\ldots\Rightarrow \quad x\in \hat S\ ,\tag{2}$$ whereby subsequent statements are related by some algebraic manipulation or simplification, and the resulting set $\hat S\subset X$ is an explicitly described set. There is, however, the following caveat: If the individul steps are not all reversible then we have not proven $S=\hat S$, but only $S\subset \hat S$, and we have to check each individual element $x\in\hat S$ whether it actually satisfies the original equation $(1)$.
This check may be omitted only if all steps in the chain $(2)$ are reversible, or if the given equation $(1)$ is an instance of equations for which we have a general theory. If this theory guarantees, e.g., "exactly one solution", and the obtained $\hat S$ is a singleton $\{a\}$ then this $a$ is "the" solution of $(1)$. Similarly, if the general theory guarantees a two dimensional vector space of solutions, and we have found by whatever means two linearly independent solutions $y_1$, $y_2$, then $S=\{y=c_1 y_1+c_2 y_2\>|\>c_1, c_2 \in{\mathbb R}\}$ without further ado.