https://file.aiquickdraw.com/mathbot/file/1727182513571ki2ubmbr.jpg

Let's solve the system of equations using the elimination method as requested:

The system of equations from 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: Eliminate one variable (preferably \(z\)) from two equations.
To eliminate \(z\), we can subtract Equation 2 from Equation 1:
\[
(x + 3y - z) - (x + y - z) = 2 - 0
\]
Simplifying:
\[
2y = 2 \quad \Rightarrow \quad y = 1
\]

### Step 2: Substitute \(y = 1\) into two equations to eliminate \(y\) and solve for \(x\) and \(z\).
Substitute \(y = 1\) into Equation 2:
\[
x + (1) - z = 0 \quad \Rightarrow \quad x - z = -1 \quad \text{(Equation 4)}
\]

Substitute \(y = 1\) into Equation 3:
\[
3x + 2(1) - 3z = -1 \quad \Rightarrow \quad 3x + 2 - 3z = -1
\]
Simplifying:
\[
3x - 3z = -3 \quad \Rightarrow \quad x - z = -1 \quad \text{(Equation 5)}
\]

### Step 3: Solve for \(x\) and \(z\).
Equations 4 and 5 are identical, meaning they represent the same equation:
\[
x - z = -1 \quad \Rightarrow \quad x = z - 1
\]

Substitute this expression for \(x\) into one of the original equations to find \(z\). Let's use Equation 1:
\[
(z - 1) + 3(1) - z = 2
\]
Simplifying:
\[
z - 1 + 3 - z = 2 \quad \Rightarrow \quad 2 = 2
\]
This is a true statement, which means the system has infinitely many solutions along the line \(x = z - 1\), with \(y = 1\).

### Final Answer:
The solution is a set of parametric equations:
\[
x = z - 1, \quad y = 1, \quad z = z \quad (\text{where } z \in \mathbb{R})
\]

This describes infinitely many solutions.

Would you like further details, or do you have any questions?

Here are 5 related questions to explore this concept further:
1. What are the advantages of using the elimination method over substitution?
2. How do you identify whether a system has one solution, no solution, or infinitely many solutions?
3. Can elimination be applied to larger systems, such as 4x4 systems?
4. How would this system change if one equation were altered slightly?
5. What is the geometric interpretation of a system with infinitely many solutions?

**Tip:** Always simplify equations as early as possible to make the elimination method more efficient.

Solve the system using the elimination method: x + 3y - z = 2, x + y - z = 0, 3x + 2y - 3z = -1

Learn how to solve the system of equations x + 3y - z = 2, x + y - z = 0, and 3x + 2y - 3z = -1 using the elimination method. This problem covers key concepts from linear algebra and provides step-by-step guidance on finding infinitely many solutions. Suitable for high school students in Grades 10-12.

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