Given an equation system when using Lagrange multipliers to find maxima and minima, how does one solve it will all these variables that I cannot isolate because I don't know if they are 0 or not, so I can't divide?
E.g.
$$y = \lambda 2x \\ x = \lambda 18 y \\ x^2 + 9y^2 = 18$$ The function is $f(x,y) = xy$.
If you don't know whether a variable is $0$ or not, simply separate the cases: see what happens if it is $0$ and what happens if it is not.
In your case, if $\lambda = 0$, then $x=0\cdot 18y = 0$ and $y=0\cdot 2x = 0$, so $x=y=0$, which means $0 = x^2+9y^2 = 18$ which is impossible. Thus, you can conclude that $\lambda \neq 0$.