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The given problem provides two vectors:

\[
\mathbf{a} = (2x - 1, y + 2, 3)
\]
\[
\mathbf{b} = (x + 1, y - 2, 2)
\]

You are asked to find \(x + y\) given that these two vectors \(\mathbf{a}\) and \(\mathbf{b}\) are parallel.

### Condition for Parallelism:
For two vectors to be parallel, their components must be proportional. This means there exists a scalar \(k\) such that:

\[
\frac{2x - 1}{x + 1} = \frac{y + 2}{y - 2} = \frac{3}{2}
\]

### Step-by-step solution:

1. **Set up the first proportion:**
   \[
   \frac{2x - 1}{x + 1} = \frac{3}{2}
   \]
   Cross-multiply to get:
   \[
   2(2x - 1) = 3(x + 1)
   \]
   Simplifying this:
   \[
   4x - 2 = 3x + 3
   \]
   Solve for \(x\):
   \[
   4x - 3x = 3 + 2
   \]
   \[
   x = 5
   \]

2. **Set up the second proportion:**
   \[
   \frac{y + 2}{y - 2} = \frac{3}{2}
   \]
   Cross-multiply to get:
   \[
   2(y + 2) = 3(y - 2)
   \]
   Simplifying this:
   \[
   2y + 4 = 3y - 6
   \]
   Solve for \(y\):
   \[
   2y - 3y = -6 - 4
   \]
   \[
   -y = -10
   \]
   \[
   y = 10
   \]

3. **Calculate \(x + y\):**
   \[
   x + y = 5 + 10 = 15
   \]

### Final Answer:
\[
x + y = 15
\]

Would you like more details or have any questions about this solution?

Here are five related questions to further explore this concept:
1. How would the solution change if the vectors were perpendicular instead of parallel?
2. Can you find the magnitude of vector \(\mathbf{a}\) and \(\mathbf{b}\) for \(x = 5\) and \(y = 10\)?
3. What is the scalar factor \(k\) that relates vectors \(\mathbf{a}\) and \(\mathbf{b}\) when they are parallel?
4. What happens if one of the vector components was set to zero?
5. Can two vectors in 3D space with one zero component still be parallel?

**Tip:** When checking if two vectors are parallel, always check if all the ratios between corresponding components are equal.

Si los vectores a y b son paralelos: a = (2x - 1, y + 2, 3), b = (x + 1, y - 2, 2). Halla el valor de x + y.

Learn how to find the value of x + y when two vectors are parallel. This step-by-step solution covers vector algebra and the proportionality rule. Suitable for students in Grades 9-12.

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