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The problem in your image is asking to create a **linear combination** of two vectors \( \vec{v} = \begin{bmatrix} 1 \\ 1 \end{bmatrix} \) and \( \vec{w} = \begin{bmatrix} 3 \\ 1 \end{bmatrix} \), by adjusting the scalars \(a\) and \(b\), so that the result matches the vector \( \begin{bmatrix} 9.4 \\ 6.4 \end{bmatrix} \). The combination of the two vectors can be written as:

\[
a \vec{v} + b \vec{w} = \begin{bmatrix} 9.4 \\ 6.4 \end{bmatrix}
\]

Substituting the components of \( \vec{v} \) and \( \vec{w} \) into the equation, we get:

\[
a \begin{bmatrix} 1 \\ 1 \end{bmatrix} + b \begin{bmatrix} 3 \\ 1 \end{bmatrix} = \begin{bmatrix} 9.4 \\ 6.4 \end{bmatrix}
\]

This leads to the following system of equations:

1. \( a + 3b = 9.4 \)
2. \( a + b = 6.4 \)

We can solve this system of equations to find the values of \(a\) and \(b\). Let me calculate them for you.The solution to the system of equations is:

- \( a = 4.9 \)
- \( b = 1.5 \)

Therefore, the linear combination that matches the vector \( \begin{bmatrix} 9.4 \\ 6.4 \end{bmatrix} \) is:

\[
4.9 \cdot \vec{v} + 1.5 \cdot \vec{w}
\]

Would you like a more detailed explanation or have any further questions?

Here are five related questions to expand your understanding:
1. How can you geometrically interpret the linear combination of two vectors?
2. What happens if you choose different values for \(a\) and \(b\)?
3. How do linear combinations help in determining vector spaces?
4. Can you visualize what it means for a set of vectors to span a space?
5. How would the solution change if the target vector were different?

**Tip:** Linear combinations of vectors form the foundation for many applications, such as solving systems of equations and defining vector spaces.

Consider the two vectors v = [1, 1] and w = [3, 1]. Adjust the scalars a and b to find a linear combination of v and w that gives the vector [9.4, 6.4].

Explore how to find scalars a and b for the linear combination of vectors v = [1,1] and w = [3,1] that results in the vector [9.4, 6.4]. This problem involves solving a system of linear equations and understanding vector operations. Suitable for high school students.

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