Determine the complex number $z$ satisfying the equation $2z-3\bar{z}=-2-30i$.  Note that $\bar{z}$ denotes the conjugate of $z$.
Let $z=a+bi$, where $a$ and $b$ are real numbers representing the real and imaginary parts of $z$, respectively.  Then $\bar{z}=a-bi$, so that $-3\bar{z}=-3a+3ib$.  We now find that \[2z-3\bar{z} = (2a-3a) + (2b +3b)i. \]So if $2z-3\bar{z}=-2-30i$ then we must have $2a-3a=-2$ and $2b+3b=-30$.  This immediately gives us $a=2$ and $b=-6$.  Therefore the complex number we are seeking is $z=\boxed{2-6i}$.