If I have a system of period functions of $x$ and $y$, in this case trigonometric,
$\begin{cases} \sin{(2x + y)} = 0 \\ \sin{(2y + x)} = 0 \end{cases}$
is it okay for me to to use the same variable $k$ to describe the period at which those two conditions are met, or should I use two separate values?
$\begin{cases} 2x + y = k \pi \\ 2y + x = k \pi \end{cases}, k \in \mathbb{Z}$
I would think that using the same variable is fine, since the important aspect is to describe the period at which the original conditions hold true, not specifically how many times you have to travel $\pi$ distance around the circle in each case.