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The problem asks whether the vector **b** is in the span of the vectors **a₁** and **a₂**. We need to determine if there exist scalars \( x_1 \) and \( x_2 \) such that:

\[
b = x_1 a_1 + x_2 a_2
\]

This translates to solving the following vector equation:

\[
\begin{pmatrix} -42 \\ -16 \\ -4 \end{pmatrix} = x_1 \begin{pmatrix} -9 \\ -8 \\ 4 \end{pmatrix} + x_2 \begin{pmatrix} -1 \\ -8 \\ 8 \end{pmatrix}
\]

To solve this, we can set up the corresponding system of linear equations:

1. \( -42 = -9x_1 - x_2 \)
2. \( -16 = -8x_1 - 8x_2 \)
3. \( -4 = 4x_1 + 8x_2 \)

Let me solve this system to check if a solution exists.The solution to the system of equations is \( x_1 = 5 \) and \( x_2 = -3 \). This means that the vector **b** is in the span of the vectors **a₁** and **a₂**, and we can express:

\[
b = 5a_1 - 3a_2
\]

Thus, the correct answer is:
- **A. Yes, b is in the span.**

The coefficients are:
- \( x_1 = 5 \)
- \( x_2 = -3 \)

Would you like a further explanation on how to solve the system of equations?

Here are some related questions to expand your understanding:
1. How do you determine if vectors are linearly independent?
2. What does it mean for a set of vectors to span a vector space?
3. Can vectors spanning a space form a basis for that space?
4. How can matrix methods (e.g., row reduction) be used to check if vectors are in a span?
5. What is the geometric interpretation of a vector being in the span of two others?

**Tip:** When solving systems of equations, ensure you align the corresponding components of each vector to form the correct system of linear equations.

Is the vector b = [-42, -16, -4] in the span of the vectors a1 = [-9, -8, 4] and a2 = [-1, -8, 8]? If so, find the coefficients.

Learn how to determine if the vector b = [-42, -16, -4] is in the span of vectors a1 = [-9, -8, 4] and a2 = [-1, -8, 8]. Solve the system of linear equations to find the coefficients. Suitable for students in Grades 10-12.

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