Trial-and-Error with thermodynamics tables

[ShizukaSm]

I know there`s a method in thermodynamics when you need two variables, have one state equation and know one information of the final state, where you can progressively approximate and discover the two wanted variables. I know about that because my professor did it in class, and while I also admit that it's a stretch come here with incomplete information, I couldn't find any information about the method, I can't replicate and I feel it's really important (I failed to copy it, as you might imagine)

Now, to provide an example of what I am talking about, here's the situation where it was originally applied: (it was a cilinder with vapor, a piston and a linear coil, but that's irrelevant)
I knew that:
$P_2-P_1=\frac{km}{A^2}\left(v_2-v_1\right)$
$P = \text{Pressure}; k = \text{Hook's constant};A = \text{Piston area};m = \text{Vapor mass};v = \text{Specific volume}$
I also had the temperature at state 2 ($T2$)

The point is, we're supposed to do the calculation using paper and real tables, so the method was more or less like that:
Guess a value(P or v), look in the table with temperature (e.g. P,T), get specific volume (in the table) and input it again in the formula, it will spit out another P, then I look it up again (P,T) and so on, it's supposed to converge to a value.

Unfortunately, whenever I try it doesn't really converge, it diverges to absurd and impossible values and I believe I am doing it in a completely wrong way. Is this a widely know method? I apologize to come here with so scarce information, but you guys are my last hope on this matter.

 

[mfb]

Iteration? That is a very common method to solve equations if there is no analytic way (or the analytic method is too complicated).
If the process does not converge, the initial choice could have been bad.