>>13800323The method of isolating the variable we want on the LHS, and recursively substituting unknown variables of the RHS by an expression of other variables (that we obtain from another equation in which the variable to be replaced occured) doesn't work generally. It works when you end up with a RHS containing only variables with known numerical value, but doesn't if you reintroduce unknown variable. Like if you were in a case of infinite mutual recursion.
For example, you have a system with 3 variables x, y, z :
(1) 0 = ax + by + cz
(2) 0 = dx + ey + fz
(3) 0 = gx + hy + iz
say you want z,
you take (1) and isolate z to obtain z = F(x,y)
then you take (2) and isolate x to obtain x = G(y,z)
you substitute x = G(y,z) inside z = F(x,y) so that you can eliminate x. It works, but you also reintroduced z on the LHS so that's not good.
the equation you now have is z = H(y,z)
If we stupidly make only recursive substitution like this, it will not work. The solution is to check if we reintroduce in the RHS, the same variable previously isolated on the LHS, and, if we can, make them "fusion" (via factorisation for ex.) and re-isolate it on the LHS.
So from z = H(y,z), you end up with z = I(y) and you're good, you have made progress.
This is the kind of problems I am interested in. To me it feels like Graph Theory, and I'm interested in any input.
