x1	 + 	x2	 − 	2x3	 = 	0
x1	 − 	2x2	 + 	x3	 = 	0
x1	 − 	x2	 − 	x3	 = 	−1

(x1, x2, x3) =

To solve the system of equations:

\[
\begin{aligned}
x_1 + x_2 - 2x_3 &= 0 \\
x_1 - 2x_2 + x_3 &= 0 \\
x_1 - x_2 - x_3 &= -1
\end{aligned}
\]

We can use various methods such as substitution, elimination, or matrix operations to find \( x_1 \), \( x_2 \), and \( x_3 \). I'll use the matrix method (Gaussian elimination) to solve it.

### Step 1: Write the system in matrix form
\[
\begin{bmatrix}
1 & 1 & -2 \\
1 & -2 & 1 \\
1 & -1 & -1
\end{bmatrix}
\begin{bmatrix}
x_1 \\
x_2 \\
x_3
\end{bmatrix}
=
\begin{bmatrix}
0 \\
0 \\
-1
\end{bmatrix}
\]

Let's proceed by applying Gaussian elimination to solve for \( x_1 \), \( x_2 \), and \( x_3 \).The solution to the system of equations is:

\[
(x_1, x_2, x_3) = (1, 1, 1)
\]

Would you like more details on how this solution was reached?

Here are some related questions:
1. How is Gaussian elimination applied to solve systems of equations?
2. What are other methods to solve systems of linear equations?
3. Can this method be applied to larger systems of equations?
4. How do matrix operations help in solving linear systems?
5. What is the condition for a system of equations to have a unique solution?

**Tip:** Always check the determinant of the coefficient matrix to ensure the system has a unique solution. If the determinant is zero, the system may have either no solution or infinitely many solutions.

x1 + x2 − 2x3 = 0, x1 − 2x2 + x3 = 0, x1 − x2 − x3 = −1. (x1, x2, x3) = ?

Learn how to solve a system of linear equations with three variables using Gaussian elimination. This problem involves matrix operations and fundamental linear algebra concepts. Suitable for students in Grades 10-12.

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