https://file.aiquickdraw.com/mathbot/file/172974557796816dyoovp.jpg

The system of equations given in the image is:

\[
\begin{aligned}
1. & \quad x + y - z = 0 \\
2. & \quad x - y - z = 0 \\
3. & \quad x + y + 2z = 0
\end{aligned}
\]

Let's solve this system of linear equations.

### Step 1: Subtract equation 2 from equation 1

\[
(x + y - z) - (x - y - z) = 0
\]
This simplifies to:
\[
2y = 0 \quad \Rightarrow \quad y = 0
\]

### Step 2: Substitute \( y = 0 \) into equations 1 and 3

- From equation 1:
\[
x + 0 - z = 0 \quad \Rightarrow \quad x = z
\]

- From equation 3:
\[
x + 0 + 2z = 0 \quad \Rightarrow \quad x = -2z
\]

### Step 3: Equating the values of \( x \)

Since \( x = z \) from one equation and \( x = -2z \) from another:
\[
z = -2z \quad \Rightarrow \quad 3z = 0 \quad \Rightarrow \quad z = 0
\]

### Step 4: Substitute \( z = 0 \) into \( x = z \)

\[
x = 0
\]

Thus, the solution to the system is:
\[
x = 0, \quad y = 0, \quad z = 0
\]

The system has only the trivial solution.

---

Do you have any questions or would you like more details on a specific step? Here are five related questions:

1. How do you verify the solution of a system of equations?
2. Can this system of equations be solved using matrix methods like Gaussian elimination?
3. What are homogeneous systems of linear equations, and how are they related to this system?
4. Could a system like this have non-trivial solutions?
5. How can graphical methods be used to interpret solutions to systems of equations?

**Tip:** Always check the consistency of equations after solving to ensure accuracy.

Solve the following system of equations: x + y - z = 0, x - y - z = 0, x + y + 2z = 0

Learn how to solve a system of linear equations involving x, y, and z using substitution and simplification methods. The system results in a trivial solution, demonstrating key concepts in algebra. Suitable for Grades 9-12.

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