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?

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.