https://file.aiquickdraw.com/mathbot/file/1725637351850d6fqm71n.jpg

The system of equations given in the image is:

\[
\begin{aligned}
-x + 3y - 2z &= -5 \\
x - y - z &= -1 \\
3x + 5y + 4z &= 7 \\
\end{aligned}
\]

We need to determine the best description of the system's solution: whether it has no solution, a unique solution, or infinitely many solutions.

Let me solve this system step-by-step using methods such as substitution or elimination to find the solution.The system of equations has a unique solution:

\[
(x, y, z) = \left( \frac{9}{19}, -\frac{6}{19}, \frac{34}{19} \right)
\]

Thus, the correct description is that the system has a **unique solution**.

Would you like further clarification or details on this? Here are five related questions that might help expand your understanding:

1. How do you solve systems of equations using substitution?
2. What is the elimination method, and how does it differ from substitution?
3. How can we tell if a system has no solution or infinitely many solutions?
4. How does the determinant of a coefficient matrix relate to the number of solutions in a system of equations?
5. What are some methods for solving systems of nonlinear equations?

**Tip:** When dealing with a system of linear equations, checking the determinant of the coefficient matrix is a quick way to assess if a unique solution exists (det ≠ 0).

Consider the system of equations: -x + 3y - 2z = -5, x - y - z = -1, 3x + 5y + 4z = 7. Choose the best description of its solution.

Learn how to solve the 3x3 system of linear equations -x + 3y - 2z = -5, x - y - z = -1, and 3x + 5y + 4z = 7. This detailed solution includes the use of matrix methods and provides the unique solution (x, y, z) = (9/19, -6/19, 34/19). Suitable for high school students.

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