# Solving Lagrange equation systems?

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 $f(x,y) = xy$.

• It seems you are asking how to solve a problem using Lagrange multipliers, but your question is more than a little jumbled. If the example problem illustrates the difficulty, please state the optimization problem clearly and proceed from there to ask about the difficulty you encounter applying the method of Lagrange multipliers. – hardmath Oct 30 '14 at 10:16

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 $\lambda \neq 0$.