To find the coordinates of point \( Q \) after the translation defined by the rule \( (X, Y) \rightarrow (X + 7, Y - 5) \), we need to apply the inverse of this translation to find the original coordinates of \( Q \).
Given that \( Q^1 \) is the image of \( Q(0, 6) \) under this translation, we can express this as:
\[ Q^1 = (X + 7, Y - 5) = (0 + 7, 6 - 5) = (7, 1) \]
Now, to find the original coordinates of \( Q \) from \( Q^1(7, 1) \), we apply the inverse translation:
\[ (X, Y) \rightarrow (X - 7, Y + 5) \]
Applying this to \( Q^1(7, 1) \):
\[ Q = (7 - 7, 1 + 5) = (0, 6) \]
So, the coordinates of \( Q \) are \( (0, 6) \).