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You've provided a system of three linear equations. The system is as follows:

\[
\begin{aligned}
1. &\quad x - 4y - 3z = 0 \\
2. &\quad -2x + 9y + 4z = -2 \\
3. &\quad 3x - 13y - 7z = 2
\end{aligned}
\]

Let me solve this system using either substitution or elimination methods.

I'll calculate the values of \(x\), \(y\), and \(z\).The solution to the system of equations depends on the variable \( z \). The values of \( x \) and \( y \) are expressed in terms of \( z \) as follows:

\[
x = 11z - 8
\]
\[
y = 2z - 2
\]

This means that the system has infinitely many solutions, with \( z \) being a free parameter. You can assign any value to \( z \), and then calculate corresponding values for \( x \) and \( y \).

Do you want further details or have any questions? Here are some related questions you might consider:

1. What does it mean for a system to have infinitely many solutions?
2. Can we solve this system using matrix methods like Gaussian elimination?
3. How do we interpret the dependency on \( z \)?
4. What happens if we set \( z = 0 \)? What are the values of \( x \) and \( y \)?
5. Can this system represent a geometric concept, such as planes?

**Tip:** When a system has infinitely many solutions, one of the variables usually acts as a free parameter, allowing for multiple combinations of solutions.

Solve the system of equations: x - 4y - 3z = 0, -2x + 9y + 4z = -2, 3x - 13y - 7z = 2

Explore how to solve the system of equations x - 4y - 3z = 0, -2x + 9y + 4z = -2, and 3x - 13y - 7z = 2. Learn how to find the values of x and y in terms of z, using elimination and substitution methods. This solution involves linear algebra concepts and is suitable for high school students.

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