In Noetherian local rings $(R,\mathfrak{m})$ one can always find a system of parameters $a_1, \dots, a_n \in \mathfrak{m}$, i.e. elements such that $$ \dim(R) = \min \left\{n \in \mathbb{N} : \exists a_1, \dots, a_n \in \mathfrak{m}: \mathfrak{m} = \sqrt{(a_1,\dots,a_n)} \right \}.$$ We had an exercise, where we were supposed to find a system of parameters.
As an example, let $X = \{(x,y) \in \mathbb{C}^2 : xy = 0 \}$. Then, we were supposed to find such a system for $\mathbb{C}[X]_{(0,0)}$.
It turns out that it's not too difficult to guess a solution ($x+y$ does the job). However, is there a more systematic way?