To solve this problem, let's analyze each translation step-by-step.

1. Initial Translation:

   • Starting point: $(x, y)$.
   • The point is translated 3 units left and 5 units up.
   • Moving 3 units left means we subtract 3 from the $x$-coordinate: $x - 3$.
   • Moving 5 units up means we add 5 to the $y$-coordinate: $y + 5$.
   • New coordinates after the first translation: $(x - 3, y + 5)$.

2. Second Translation:

   • Starting from the new point $(x - 3, y + 5)$, we translate 5 units right and 2 units up.
   • Moving 5 units right means we add 5 to the $x$-coordinate: $(x - 3) + 5 = x + 2$.
   • Moving 2 units up means we add 2 to the $y$-coordinate: $(y + 5) + 2 = y + 7$.
   • Final coordinates after both translations: $(x + 2, y + 7)$.

Answer:

The coordinates of the point after the translations are: $(x + 2, y + 7)$

Would you like more details on this solution or have further questions?

Here are some related questions for deeper understanding:

1. How would the coordinates change if the initial translation were 5 units left instead?
2. What would happen if the translations were applied in a different order?
3. How would you represent these translations using vector addition?
4. Can you describe this translation process using a matrix?
5. What are other common transformations besides translations?

Tip: Translating a point in a coordinate plane involves adding or subtracting from the x- and y-coordinates based on direction.