Write the solution set of the given homogeneous system in parametric vector form.
4 x 1 plus 4 x 2 plus 8 x 3
equals
0
where the solution set is xequalsleft bracket Start 3 By 1 Matrix 1st Row 1st Column x 1 2nd Row 1st Column x 2 3rd Row 1st Column x 3 EndMatrix right bracket. write as x = x_3 (fill in)

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.

Write the solution set of the given homogeneous system in parametric vector form. 4x1 + 4x2 + 8x3 = 0 where the solution set is x = [x1, x2, x3]. Write as x = x_3 (fill in).

Learn how to express the solution set of a homogeneous linear system in parametric vector form. This problem focuses on understanding key concepts in linear algebra, such as homogeneous systems, parametric vector form, and free parameters. Suitable for advanced high school or college-level students.

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