A translation of the plane takes $5 + 5i$ to $6 + 8i.$  Find the complex number that the translation takes $-6$ to.
This translation takes $z$ to $z + w,$ where $w$ is a fixed complex number.  Thus,
\[6 + 8i = (5 + 5i) + w.\]Hence, $w = 1 + 3i.$  Then the translation takes $-6$ to $-6 + (1 + 3i) = \boxed{-5 + 3i}.$