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The problem states that we have a solution set to an inhomogeneous linear system in terms of variables \( t \) and \( s \) as follows:

\[
\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = \begin{bmatrix} -2 \\ 8 \\ 3 \end{bmatrix} + t \begin{bmatrix} 1 \\ 4 \\ -10 \end{bmatrix} + s \begin{bmatrix} 4 \\ 2 \\ 2 \end{bmatrix}, \quad t, s \in \mathbb{R}.
\]

The task is to find a value for \( c \) so that the vector 

\[
\begin{bmatrix} 4 \\ 9 \\ c \end{bmatrix}
\]

is a solution to the corresponding **homogeneous system**.

### Solution

For this vector to be a solution to the homogeneous system, it must be expressible as a **linear combination** of the basis vectors from the homogeneous part of the solution (the vectors with \( t \) and \( s \)).

Let's set up the vector in terms of \( t \) and \( s \):

\[
\begin{bmatrix} 4 \\ 9 \\ c \end{bmatrix} = t \begin{bmatrix} 1 \\ 4 \\ -10 \end{bmatrix} + s \begin{bmatrix} 4 \\ 2 \\ 2 \end{bmatrix}.
\]

This gives us a system of equations:

1. \( t + 4s = 4 \)
2. \( 4t + 2s = 9 \)
3. \( -10t + 2s = c \)

We can solve the first two equations for \( t \) and \( s \) and then substitute into the third equation to determine \( c \).

#### Step 1: Solve for \( t \) and \( s \)

From the first equation:
\[
t = 4 - 4s.
\]

Substitute \( t = 4 - 4s \) into the second equation:
\[
4(4 - 4s) + 2s = 9,
\]
\[
16 - 16s + 2s = 9,
\]
\[
16 - 14s = 9,
\]
\[
-14s = -7,
\]
\[
s = \frac{1}{2}.
\]

Now, substitute \( s = \frac{1}{2} \) back into \( t = 4 - 4s \):
\[
t = 4 - 4 \cdot \frac{1}{2},
\]
\[
t = 4 - 2 = 2.
\]

#### Step 2: Find \( c \)

Substitute \( t = 2 \) and \( s = \frac{1}{2} \) into the third equation:
\[
c = -10 \cdot 2 + 2 \cdot \frac{1}{2},
\]
\[
c = -20 + 1 = -19.
\]

### Conclusion
The value of \( c \) that makes \( \begin{bmatrix} 4 \\ 9 \\ c \end{bmatrix} \) a solution to the homogeneous system is:

\[
c = -19.
\]

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

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### Related Questions

1. What is the difference between a homogeneous and inhomogeneous system?
2. How do you determine if a vector is in the span of given vectors?
3. What is the role of parameters in expressing the solution set of a system of linear equations?
4. How do linear combinations relate to solutions of linear systems?
5. What is the geometric interpretation of solutions in homogeneous systems?

### Tip
In a homogeneous system, the solution space always includes the zero vector and forms a subspace of the solution space.

Bestem tallet c så [4, 9, c] bliver en løsning til det tilhørende homogene system.

Determine the value of c for which the vector [4, 9, c] is a solution to a homogeneous system. This problem involves key concepts in linear algebra, such as linear combinations and parametric solutions. Suitable for undergraduate students studying linear algebra.

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