−3x+8y−z=
2
minus, 2, x, minus, 8, y, minus, 3, z, equals, minus, 8
−2x−8y−3z=
−8
5, x, minus, 8, y, plus, 5, z, equals, 10
5x−8y+5z=
10

The system of linear equations you provided is:

\[
\begin{aligned}
-3x + 8y - z &= 2 \\
-2x - 8y - 3z &= -8 \\
5x - 8y + 5z &= 10
\end{aligned}
\]

We can solve this system using methods like substitution, elimination, or matrix methods (such as Gaussian elimination). I'll proceed with solving this system using Gaussian elimination.

### Step 1: Write the system as an augmented matrix.

\[
\begin{pmatrix}
-3 & 8 & -1 & | & 2 \\
-2 & -8 & -3 & | & -8 \\
5 & -8 & 5 & | & 10
\end{pmatrix}
\]

### Step 2: Apply Gaussian elimination to solve for \( x \), \( y \), and \( z \).

Let me calculate the solution.The solution to the system of equations is:

\[
x = -2, \quad y \approx 0, \quad z = 4
\]

The value of \( y \) is extremely close to zero, so it can be considered as \( y = 0 \).

Would you like more details on any of the steps or methods used in solving this system?

Here are 5 related questions you might find interesting:
1. How does Gaussian elimination work for solving systems of equations?
2. What are some alternative methods to solve a system of linear equations?
3. What happens when a system of equations has no solution or infinite solutions?
4. Can matrix operations always guarantee a solution for any system of equations?
5. How does rounding impact the precision of solutions in linear algebra?

**Tip:** When working with systems of equations, always check if the system is consistent (has a solution) or inconsistent (no solutions) before solving.

-3x + 8y - z = 2, -2x - 8y - 3z = -8, 5x - 8y + 5z = 10

Explore how to solve a system of linear equations using Gaussian elimination. This example features the system -3x + 8y - z = 2, -2x - 8y - 3z = -8, and 5x - 8y + 5z = 10, demonstrating how to apply matrix methods to find the values of x, y, and z. Suitable for students in Grades 9-12.

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