https://file.aiquickdraw.com/mathbot/file/1727219418312w5auv540.jpg

Let's break down the problem from the image.

We have two variables:
- \( x \) represents Pedro's current age.
- \( y \) represents Manuel's current age.

**First condition**: "Pedro's age is double Manuel's age."
This translates into the equation:
\[ x = 2y \]
Or equivalently:
\[ x - 2y = 0 \]

**Second condition**: "10 years ago, Pedro's age was triple Manuel's age."
This translates into:
\[ x - 10 = 3(y - 10) \]
Simplifying this:
\[ x - 10 = 3y - 30 \]
\[ x - 3y = -20 \]

So, the system of equations is:
\[
\begin{aligned}
x - 2y &= 0 \\
x - 3y &= -20
\end{aligned}
\]

Now let's compare this with the options provided:

**Option C**:
\[
\begin{aligned}
x - 2y &= 0 \\
x - 3y &= -20
\end{aligned}
\]

Thus, the correct answer is **C**.

---

Would you like more details on how these equations were derived or any additional clarifications? Here are some related questions to consider:

1. What would the solution be if Pedro's age was triple Manuel's age instead of double?
2. How would the equations change if the "10 years ago" condition involved 5 years instead?
3. How can we solve this system of equations to find the exact ages of Pedro and Manuel?
4. How do these types of age-related problems fit into the broader category of linear equations?
5. Could we approach this problem graphically instead of algebraically?

**Tip**: Always check if you can simplify real-world conditions into equations that match known formats like linear systems. This helps reduce complex word problems into manageable mathematical tasks.

Actualmente, la edad de Pedro es el doble de la edad de Manuel. Hace 10 años, la edad de Pedro era el triple de la edad de Manuel. ¿Qué sistema de ecuaciones permite determinar las edades actuales de Pedro (x) y Manuel (y)?

Learn how to determine the ages of Pedro and Manuel using a system of linear equations. This problem explores the relationship between their ages and provides step-by-step solutions to solve the system. Suitable for students in Grades 9-10.

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