https://file.aiquickdraw.com/mathbot/file/1730383283229ars5uavk.jpg

The system of equations shown in the image is:

\[
\begin{cases}
-x + 3y - z = -9, \\
2x + 3z = 8, \\
5x + 2y = 1.
\end{cases}
\]

The solution provided in the image is \((x, y, z) = (1, -2, 2)\).

### Solution Verification

Let's substitute \(x = 1\), \(y = -2\), and \(z = 2\) into each equation to verify if this solution is correct:

1. **First equation:** 
   \[
   -x + 3y - z = -9
   \]
   Substitute \(x = 1\), \(y = -2\), and \(z = 2\):
   \[
   -1 + 3(-2) - 2 = -1 - 6 - 2 = -9.
   \]
   This holds true.

2. **Second equation:**
   \[
   2x + 3z = 8
   \]
   Substitute \(x = 1\) and \(z = 2\):
   \[
   2(1) + 3(2) = 2 + 6 = 8.
   \]
   This is also correct.

3. **Third equation:**
   \[
   5x + 2y = 1
   \]
   Substitute \(x = 1\) and \(y = -2\):
   \[
   5(1) + 2(-2) = 5 - 4 = 1.
   \]
   This is correct as well.

Since all three equations hold with \((x, y, z) = (1, -2, 2)\), the provided solution is indeed correct.

Would you like further details on solving this system using a particular method, or do you have any questions?

### Related Questions

1. How would you solve this system using matrix methods (Gaussian elimination or Cramer's rule)?
2. Can we solve this system by substitution, and how would that work?
3. What are the possible methods to solve systems of linear equations with three variables?
4. How can we interpret this solution geometrically?
5. How can we verify solutions of larger systems of equations?

**Tip:** When solving systems of equations, always verify by plugging the values back into the original equations to ensure consistency.

Solve the system of linear equations:
\(-x + 3y - z = -9\),
\(2x + 3z = 8\),
\(5x + 2y = 1\).
Solution given: \((x, y, z) = (1, -2, 2)\)

Learn to solve a system of linear equations with three variables: -x + 3y - z = -9, 2x + 3z = 8, 5x + 2y = 1. This solution involves substitution and elimination methods and verifies that (x, y, z) = (1, -2, 2) satisfies all equations. Suitable for students in Grades 10-12.

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