As one can see,
In[1]:= Options[SolveAlways]
Out[1]= {InverseFunctions -> Automatic, MakeRules -> False,
Method -> 3, Mode -> Generic, Sort -> True,
WorkingPrecision -> \[Infinity]}
Both InverseFunctions -> Automatic and WorkingPrecision -> Infinity are well-documented; MakeRules -> True has been explained here, Mode -> Modular is explained here, and Sort -> True has been explained here. But what are the meanings of Method -> 3 and Mode -> Generic?
According to Message[SolveAlways::method, "`1`"],
Value of option Method -> `1` should be 1, 2, or 3.
And according to ?Mode,
Mode is an option to Solve and related functions that specifies in what sense the equations are to be solved. The possible settings for Mode are Generic, Modular, and Rational.
So, what is the difference among the three Method options and the three Mode options?
Modealso out-dated? The help page does not mark it as an out-dated option. Besides, inOptions[SolveAlways], the default method isMethod -> 3, while inOptions[Eliminate], the default method isMethod -> 1; is there any difference? $\endgroup$Methodones different? Probably. What do they do that's different? I don't remember. ReMode, I believethe idea was that one could use theGeneric` to getSolve to give generic solutions adReduce` give a full breakdown using theRationalsetting (maybe). One could useMode->Modularand give an extra equation, andSolvewould try to find a modulus for which the system was noy over-determined. My guess is this can instead be handled with an existential quantifier, in a way that might be more reliable. $\endgroup$