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The problem involves finding scalars \( a \) and \( b \) such that the linear combination of the given vectors \(\mathbf{v} = \begin{bmatrix} 1 \\ 1 \end{bmatrix}\) and \(\mathbf{w} = \begin{bmatrix} 3 \\ 1 \end{bmatrix}\) equals the target vector \(\begin{bmatrix} 9.4 \\ 6.4 \end{bmatrix}\).

### Step-by-Step Solution:

1. **Define the linear combination:**
   The linear combination of vectors \(\mathbf{v}\) and \(\mathbf{w}\) is given by:
   \[
   a\mathbf{v} + b\mathbf{w} = a\begin{bmatrix} 1 \\ 1 \end{bmatrix} + b\begin{bmatrix} 3 \\ 1 \end{bmatrix}.
   \]

2. **Expand the combination:**
   This can be rewritten as:
   \[
   \begin{bmatrix} a + 3b \\ a + b \end{bmatrix}.
   \]

3. **Set up the system of equations:**
   We need this to equal the target vector:
   \[
   \begin{bmatrix} a + 3b \\ a + b \end{bmatrix} = \begin{bmatrix} 9.4 \\ 6.4 \end{bmatrix}.
   \]
   This gives us two equations:
   \[
   a + 3b = 9.4,
   \]
   \[
   a + b = 6.4.
   \]

4. **Solve the system:**
   Subtract the second equation from the first:
   \[
   (a + 3b) - (a + b) = 9.4 - 6.4,
   \]
   \[
   2b = 3,
   \]
   \[
   b = 1.5.
   \]

   Substitute \( b = 1.5 \) into the second equation:
   \[
   a + 1.5 = 6.4,
   \]
   \[
   a = 4.9.
   \]

### Final Answer:
The scalars are \( a = 4.9 \) and \( b = 1.5 \).

Would you like more details or have any questions?

Here are some related questions you might find helpful:
1. What is the geometric interpretation of linear combinations?
2. How do different values of \( a \) and \( b \) affect the resulting vector?
3. Can vectors \(\mathbf{v}\) and \(\mathbf{w}\) span all of \(\mathbb{R}^2\)?
4. What is the significance of solving systems of linear equations in vector spaces?
5. How can you determine if two vectors are linearly independent?

**Tip:** Understanding the geometric representation of vectors helps in visualizing the effect of different scalar values on vector combinations.

Consider the two vectors v = [1, 1] and w = [3, 1]. Adjust the scalars a and b such that the linear combination a*v + b*w equals the target vector [9.4, 6.4]. What are the values of a and b?

Learn how to find the scalars a and b such that the linear combination of vectors v = [1, 1] and w = [3, 1] equals the target vector [9.4, 6.4]. This problem involves solving systems of equations and understanding the concept of vector spanning in R^2.

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