When solving a system of equations analytically, one transforms the equations in successive steps until he finds out the expression of the unknown variables. Optionally, some additional constraints may pop up during the solving, and what I would like to know is whether there is a systematic way to verify whether these constraints are required at the end.
In my case, I have solved exponential functions having the shape $y=e^{ax+b}+c$. More precisely, given 3 points $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$, I obtain a system of 3 equations: $$ y_1=e^{ax_1+b}+c\\ y_2=e^{ax_2+b}+c\\ y_3=e^{ax_3+b}+c $$ and I have searched the expressions of $a$, $b$, and $c$ as functions of $x_i$ and $y_i$. Interested people may look at the solving here (it is a draft), but my question is about a specific aspect of solving.
In the process, I obtain the expression $a=\frac{ln(y_1-c)-b}{x_1}$, which requires me to assume the constraint $x_1 \neq 0$, otherwise it is undefined. Later, I assume another constraint to help me solve the system, which is $x_3 - x_2 = x_2 - x_1$. The system is solved, no problem with that, but when I use my solutions on concrete examples, although breaking the second constraint leads indeed to garbage results, breaking the first one has no impact (it still works with $x_1=0$).
Consequently, in this case I would like to know whether the constraint $x_1 \neq 0$ is actually required. If not, I assume I can revise the procedure to make it disappear, but at least I would like to know a systematic way to check whether such a constraint is required. Is there such a systematic validation procedure? I know you can systematically check your final expressions by replacing them in the initial equations and validate the equality, but how to check the constraints? Is it enough to consider that the constraints you don't reuse in the expression validation procedure are useless or do we need something else?