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We are given the solution set for an inhomogeneous linear system, where:

\[
\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = \begin{bmatrix} -6 \\ 8 \\ 7 \end{bmatrix} + t \cdot \begin{bmatrix} 1 \\ 5 \\ -13 \end{bmatrix} + s \cdot \begin{bmatrix} 2 \\ 3 \\ -5 \end{bmatrix}
\]
with \( t, s \in \mathbb{R} \).

We are asked to determine a value \( c \) so that:
\[
\begin{bmatrix} 9 \\ 4 \\ c \end{bmatrix}
\]
becomes a solution to the associated homogeneous system.

### Solution:

To find \( c \), we should assume that the vector \( \begin{bmatrix} 9 \\ 4 \\ c \end{bmatrix} \) can be expressed as a linear combination of the vectors \( \begin{bmatrix} 1 \\ 5 \\ -13 \end{bmatrix} \) and \( \begin{bmatrix} 2 \\ 3 \\ -5 \end{bmatrix} \), since these vectors span the solution space of the homogeneous system. Thus, we need to find \( t \) and \( s \) such that:

\[
\begin{bmatrix} 9 \\ 4 \\ c \end{bmatrix} = t \cdot \begin{bmatrix} 1 \\ 5 \\ -13 \end{bmatrix} + s \cdot \begin{bmatrix} 2 \\ 3 \\ -5 \end{bmatrix}
\]

Expanding this equation gives:

1. \( 9 = t + 2s \)
2. \( 4 = 5t + 3s \)
3. \( c = -13t - 5s \)

We can solve these equations for \( t \), \( s \), and ultimately \( c \).

1. From the first equation, \( t = 9 - 2s \).
2. Substitute \( t = 9 - 2s \) into the second equation:

   \[
   4 = 5(9 - 2s) + 3s
   \]
   \[
   4 = 45 - 10s + 3s
   \]
   \[
   4 = 45 - 7s
   \]
   \[
   -41 = -7s
   \]
   \[
   s = \frac{41}{7}
   \]

3. Substitute \( s = \frac{41}{7} \) back into \( t = 9 - 2s \):

   \[
   t = 9 - 2 \cdot \frac{41}{7}
   \]
   \[
   t = 9 - \frac{82}{7}
   \]
   \[
   t = \frac{63 - 82}{7}
   \]
   \[
   t = -\frac{19}{7}
   \]

4. Finally, substitute \( t = -\frac{19}{7} \) and \( s = \frac{41}{7} \) into the third equation to find \( c \):

   \[
   c = -13 \left(-\frac{19}{7}\right) - 5 \cdot \frac{41}{7}
   \]
   \[
   c = \frac{247}{7} - \frac{205}{7}
   \]
   \[
   c = \frac{42}{7}
   \]
   \[
   c = 6
   \]

### Answer:
\[
c = 6
\]

Would you like more details, or do you have any questions?

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### Related Questions:

1. How would the solution change if the inhomogeneous term were different?
2. What is the geometric interpretation of the solution set for this system?
3. How can we check if a vector belongs to the solution space of a homogeneous system?
4. What are the properties of a homogeneous system's solution set?
5. Can we apply the same method to a higher-dimensional system?

### Tip:
In homogeneous systems, solutions can often be written as linear combinations of vectors that span the null space.

Bestem tallet c så \( \begin{bmatrix} 9 \\ 4 \\ c \end{bmatrix} \) bliver en løsning til det tilhørende homogene system.

Learn to determine the value of c such that the vector \( \begin{bmatrix} 9 \\ 4 \\ c \end{bmatrix} \) is a solution to a homogeneous linear system. This problem involves key concepts in linear algebra, including solution sets, homogeneous systems, and vector spaces.

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