https://file.aiquickdraw.com/mathbot/file/1726997301767dlcrnck5.jpg

The system of linear equations presented in the image is:

\[
\begin{aligned}
5x_1 - 6x_2 + 4x_3 + 4x_4 &= 4 \\
-x_1 + x_2 + 3x_3 + 3x_4 &= 2 \\
4x_1 - 5x_2 + 7x_3 + 7x_4 &= 6 \\
-3x_1 + 3x_2 + 9x_3 + 9x_4 &= 6
\end{aligned}
\]

The structure also suggests finding the solution in parametric form, where the solution can be written as:

\[
\begin{pmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \end{pmatrix} = \mathbf{p} + s \mathbf{v_1} + t \mathbf{v_2}
\]
where \(\mathbf{p}\) is a particular solution and \(\mathbf{v_1}, \mathbf{v_2}\) are the vectors representing the free variables.

### Solving this system
1. Write the augmented matrix of the system.
2. Perform row reduction to find the row-reduced echelon form (RREF).
3. Express the solution in terms of free variables if necessary.

I will now proceed with solving it.The row-reduced echelon form (RREF) of the augmented matrix is:

\[
\begin{pmatrix}
1 & 0 & -22 & -22 & -16 \\
0 & 1 & -19 & -19 & -14 \\
0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0
\end{pmatrix}
\]

From this, we can interpret the solution:

1. \( x_1 - 22x_3 - 22x_4 = -16 \), so \( x_1 = 22x_3 + 22x_4 - 16 \).
2. \( x_2 - 19x_3 - 19x_4 = -14 \), so \( x_2 = 19x_3 + 19x_4 - 14 \).
3. \( x_3 \) and \( x_4 \) are free variables.

### Parametric form:
Let \( x_3 = s \) and \( x_4 = t \). The solution can be written as:

\[
\begin{pmatrix}
x_1 \\
x_2 \\
x_3 \\
x_4
\end{pmatrix}
=
\begin{pmatrix}
-16 \\
-14 \\
0 \\
0
\end{pmatrix}
+ s
\begin{pmatrix}
22 \\
19 \\
1 \\
0
\end{pmatrix}
+ t
\begin{pmatrix}
22 \\
19 \\
0 \\
1
\end{pmatrix}
\]

This represents the general solution to the system.

---

Would you like a detailed breakdown of how this form was derived?

Here are five related questions to further explore:
1. How does the row reduction method work for solving linear systems?
2. What is the significance of free variables in a system of linear equations?
3. How can we interpret the solution geometrically in terms of planes?
4. How does parametric form help in understanding the solution set of a system?
5. What is the rank of this matrix, and how does it relate to the number of solutions?

**Tip:** When solving systems of equations, always check the number of free variables to understand if the system has infinite solutions.

Solve the system of linear equations and express the solution in parametric form.

Explore the solution of a complex system of four linear equations with four variables using advanced algebraic techniques including row reduction to RREF and expression in parametric forms. This problem is suitable for college-level students studying linear algebra.

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