The given homogeneous system of equations is:

$4x_1 + 4x_2 + 8x_3 = 0$

We are asked to find the solution set in parametric vector form, where the solution is expressed in terms of $x_3$.

Step-by-Step Solution:

1. Rewrite the equation: $4x_1 + 4x_2 + 8x_3 = 0$

2. Simplify the equation: $x_1 + x_2 + 2x_3 = 0$

3. Solve for one variable in terms of $x_3$: Let's solve for $x_1$: $x_1 = -x_2 - 2x_3$

4. Parametric form: We have $x_1$ in terms of $x_2$ and $x_3$. Now, let's express the general solution for the vector $\mathbf{x} = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}$.

   Let $x_2 = t$ (a free parameter), and $x_3 = s$ (another free parameter). Then: $x_1 = -t - 2s$ Thus, the solution vector becomes: $\mathbf{x} = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = \begin{bmatrix} -t - 2s \\ t \\ s \end{bmatrix}$

5. Express in parametric vector form: We can factor out the parameters $t$ and $s$ to write the solution in parametric vector form: $\mathbf{x} = t \begin{bmatrix} -1 \\ 1 \\ 0 \end{bmatrix} + s \begin{bmatrix} -2 \\ 0 \\ 1 \end{bmatrix}$

Final Parametric Vector Form:

$\mathbf{x} = t \begin{bmatrix} -1 \\ 1 \\ 0 \end{bmatrix} + s \begin{bmatrix} -2 \\ 0 \\ 1 \end{bmatrix}$ Where $t$ and $s$ are free parameters representing the solution set.

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Do you want further details or have any questions?

Here are 5 related questions:

1. How would you solve a system of equations with more than one equation?
2. What are the free variables in a homogeneous system?
3. How can you verify if a solution is correct for a system of equations?
4. What does it mean for a system of equations to have infinitely many solutions?
5. How do you interpret the parametric vector form geometrically?

Tip: The parametric vector form helps visualize the solution space as a combination of basis vectors representing the free parameters.