https://file.aiquickdraw.com/mathbot/file/17266713499824xk9uzzv.jpg

The system of linear equations given in the image is:

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

### Step 1: Use the Elimination Method

We aim to eliminate variables by manipulating the equations.

#### Subtract Equation (2) from Equation (1):
\[
(x + 3y - z) - (x + y - z) = 2 - 0
\]
Simplifying this:
\[
(3y - y) = 2 \quad \Rightarrow \quad 2y = 2 \quad \Rightarrow \quad y = 1
\]

Now that we have \( y = 1 \), substitute this value back into both Equation (1) and Equation (2) to solve for \( x \) and \( z \).

#### Step 2: Substitute \( y = 1 \) into Equation (2):
\[
x + 1 - z = 0 \quad \Rightarrow \quad x - z = -1 \quad \Rightarrow \quad x = z - 1
\]

#### Step 3: Substitute \( y = 1 \) into Equation (1):
\[
x + 3(1) - z = 2 \quad \Rightarrow \quad x + 3 - z = 2 \quad \Rightarrow \quad x - z = -1
\]

This confirms \( x = z - 1 \) again, so the solution is consistent.

#### Step 4: Substitute \( x = z - 1 \) and \( y = 1 \) into Equation (3):
\[
3(z - 1) + 2(1) - 3z = -1
\]
Simplifying:
\[
3z - 3 + 2 - 3z = -1 \quad \Rightarrow \quad -1 = -1
\]

This is a true statement, so the system is consistent and dependent, meaning there are **infinitely many solutions**.

### Final Solution:
The general solution in terms of \( z \) is:
\[
(x, y, z) = (z - 1, 1, z)
\]

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Would you like more details or have any questions?

Here are 5 related questions:
1. How do you know if a system of equations has no solution?
2. Can the elimination method be used for any system of equations?
3. What are the conditions for a system to have infinitely many solutions?
4. How would you solve the same system using substitution instead of elimination?
5. How do the equations in a system change geometrically when they lead to no solution?

**Tip**: Always check for consistency in the equations after eliminating variables to confirm if a system has infinitely many solutions or exactly one.

Identify the number of solutions of the system of linear equations and solve it using elimination.

Learn how to solve a system of linear equations using the elimination method. This problem includes concepts like elimination, consistency, and dependent systems, with a solution indicating infinitely many solutions.

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