⎧
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x+2y−z=4
2x−y+3z=−6
−x+y+2z=7
​
 


To solve the system of equations:

\[
\begin{cases}
x + 2y - z = 4 \\
2x - y + 3z = -6 \\
-x + y + 2z = 7
\end{cases}
\]

We can use the method of elimination or matrix operations. Here, we'll solve it using matrix operations.

1. **Write the system as an augmented matrix:**

\[
\begin{bmatrix}
1 & 2 & -1 & | & 4 \\
2 & -1 & 3 & | & -6 \\
-1 & 1 & 2 & | & 7
\end{bmatrix}
\]

2. **Perform row operations to convert the matrix to row-echelon form:**

First, we can use the first row to eliminate the \(x\)-terms from the second and third rows.

\[
R2 = R2 - 2R1
\]
\[
R3 = R3 + R1
\]

\[
\begin{bmatrix}
1 & 2 & -1 & | & 4 \\
0 & -5 & 5 & | & -14 \\
0 & 3 & 1 & | & 11
\end{bmatrix}
\]

Next, use the second row to eliminate the \(y\)-term from the third row.

\[
R3 = R3 + \frac{3}{-5}R2
\]

\[
\begin{bmatrix}
1 & 2 & -1 & | & 4 \\
0 & -5 & 5 & | & -14 \\
0 & 0 & 4 & | & 2.6
\end{bmatrix}
\]

3. **Back-substitution to find the solutions:**

From the third row, we get:
\[ 4z = 2.6 \implies z = \frac{2.6}{4} = 0.65 \]

From the second row, we get:
\[ -5y + 5(0.65) = -14 \implies -5y + 3.25 = -14 \implies -5y = -14 - 3.25 \implies y = \frac{-17.25}{-5} = 3.45 \]

From the first row, we get:
\[ x + 2(3.45) - 0.65 = 4 \implies x + 6.9 - 0.65 = 4 \implies x = 4 - 6.25 = -2.25 \]

So the solution to the system is:
\[ x = -2.25 \]
\[ y = 3.45 \]
\[ z = 0.65 \]

Would you like more details or have any questions?

Here are 8 related questions to expand on this topic:

1. How can you solve this system of equations using the substitution method?
2. What is the geometric interpretation of the solution to this system?
3. How can you use the determinant to verify if the system has a unique solution?
4. What is Cramer's Rule and how can it be applied to this system?
5. How can you convert this system to its reduced row-echelon form (RREF)?
6. What are some practical applications of solving systems of linear equations?
7. How does the solution change if one of the equations is altered slightly?
8. How would you solve this system using software tools like MATLAB or Python?

**Tip:** Always double-check your row operations to avoid simple arithmetic errors, which can significantly alter the solution.

Learn how to solve a system of equations using matrix operations and row echelon form with a detailed step-by-step solution. This problem covers linear algebra concepts such as systems of linear equations and matrix operations. Suitable for advanced high school students in Grades 11-12.

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