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To solve \( v + u + 3w \) with the given vectors, we can follow these steps:

### Given Vectors
- \( u = \langle 7, -4 \rangle \)
- \( v = \langle -2, 8 \rangle \)
- \( w = \langle -2, -5 \rangle \)

### Step 1: Calculate \( 3w \)
To find \( 3w \), multiply each component of \( w \) by 3:
\[
3w = 3 \cdot \langle -2, -5 \rangle = \langle 3 \cdot -2, 3 \cdot -5 \rangle = \langle -6, -15 \rangle
\]

### Step 2: Calculate \( v + u \)
Add corresponding components of \( v \) and \( u \):
\[
v + u = \langle -2, 8 \rangle + \langle 7, -4 \rangle = \langle -2 + 7, 8 - 4 \rangle = \langle 5, 4 \rangle
\]

### Step 3: Calculate \( v + u + 3w \)
Now add the result of \( v + u \) with \( 3w \):
\[
v + u + 3w = \langle 5, 4 \rangle + \langle -6, -15 \rangle = \langle 5 - 6, 4 - 15 \rangle = \langle -1, -11 \rangle
\]

### Final Answer
\[
v + u + 3w = \langle -1, -11 \rangle
\]

Would you like further details or have any questions?

Here are some related questions you might find interesting:
1. How do we generalize the formula for scaling and adding multiple vectors?
2. What would happen if we changed the scalar multiplier for \( w \) in this problem?
3. How do vector operations relate to real-world physics applications?
4. What is the geometric interpretation of adding and scaling vectors in two-dimensional space?
5. How would this solution change if the vectors were three-dimensional?

**Tip:** Remember that multiplying a vector by a scalar scales each of its components by that scalar.

Calculate v + u + 3w, where u = ⟨7, -4⟩, v = ⟨-2, 8⟩, and w = ⟨-2, -5⟩.

Calculate the vector expression v + u + 3w, with given vectors u = ⟨7, -4⟩, v = ⟨-2, 8⟩, and w = ⟨-2, -5⟩. This solution includes steps for scalar multiplication and vector addition, perfect for high school students in Grades 10-12.

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