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The image you've uploaded seems to involve the process of finding the eigenvector corresponding to a given eigenvalue, likely in the context of solving a system of linear equations derived from a matrix.

### Let's break down the solution in your notes:

1. **Given system of equations:**
   \[
   -2x_1 + 6x_2 = 0
   \]
   \[
   x_1 - 3x_2 = 0
   \]
   These equations are derived from the eigenvalue problem, where we are solving for the eigenvector corresponding to the eigenvalue \(\lambda = 6\).

2. **Step-by-step procedure:**

   - The first equation is solved for \(x_1\):
     \[
     -2x_1 + 6x_2 = 0 \quad \Rightarrow \quad x_1 = \frac{6}{2} x_2 = 3x_2
     \]
     So, \(x_1 = 3x_2\).

   - Let \(x_2 = k\), a scalar parameter. This gives:
     \[
     x_1 = 3k, \quad x_2 = k
     \]
   
   - Therefore, the eigenvector \(x\) corresponding to \(\lambda = 6\) is:
     \[
     x = \begin{pmatrix} 3 \\ 1 \end{pmatrix} k
     \]

   - Taking \(k = 1\), the final eigenvector is:
     \[
     x = \begin{pmatrix} 3 \\ 1 \end{pmatrix}
     \]

### Conclusion:
The eigenvector corresponding to the eigenvalue \(\lambda = 6\) is:
\[
x = \begin{pmatrix} 3 \\ 1 \end{pmatrix}
\]

Would you like more details on this solution, or do you have any further questions?

Here are 5 questions that expand on this topic:

1. How do you find the eigenvalue from a matrix?
2. What is the geometric interpretation of an eigenvector?
3. How can we generalize the process of finding eigenvectors for different eigenvalues?
4. Why do we often choose the scalar \(k = 1\) when finding eigenvectors?
5. How can this process be applied to larger matrices?

**Tip:** When solving systems like this, double-check your matrix manipulation and ensure that you properly simplify equations to avoid mistakes!

Explore the step-by-step process of finding the eigenvector corresponding to the eigenvalue λ = 6, including solving a system of linear equations and understanding the concept of eigenvectors in linear algebra.

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