https://file.aiquickdraw.com/mathbot/file/1730156642110rci569j1.jpg

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.

Translate a point (x, y) 3 units left and 5 units up. Then translate the image 5 units right and 2 units up. What are the coordinates of the point after the translations?

Learn how to translate a point (x, y) first 3 units left and 5 units up, followed by 5 units right and 2 units up. This detailed solution shows the step-by-step process for applying translations in coordinate geometry. Suitable for students in Grades 6-8.

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