https://file.aiquickdraw.com/mathbot/file/1724690974860d4u65qvp.jpg

To solve the problem in the image, let's analyze the system of equations presented:

The problem asks for the values of \( m \) such that the following system has a solution:

\[
\begin{aligned}
x + y + z &= 5 \quad \text{(Equation 1)}\\
x - y + z &= m \quad \text{(Equation 2)}\\
2x + y - z &= 3 \quad \text{(Equation 3)}
\end{aligned}
\]

**Step 1: Add Equation 1 and Equation 2.**

\[
(x + y + z) + (x - y + z) = 5 + m
\]

This simplifies to:

\[
2x + 2z = 5 + m
\]

\[
x + z = \frac{5 + m}{2} \quad \text{(Equation 4)}
\]

**Step 2: Add Equation 3 to Equation 4.**

\[
(2x + y - z) + (x + z) = 3 + \frac{5 + m}{2}
\]

This simplifies to:

\[
3x + y = 3 + \frac{5 + m}{2}
\]

**Step 3: Find \( y \) in terms of \( x \) and \( m \).**

Rearrange the equation:

\[
y = 3 + \frac{5 + m}{2} - 3x
\]

**Step 4: Plug \( y \) back into Equation 1.**

Substitute \( y = 3 + \frac{5 + m}{2} - 3x \) into Equation 1:

\[
x + \left(3 + \frac{5 + m}{2} - 3x\right) + z = 5
\]

This simplifies to:

\[
\frac{11 + m}{2} - 2x + z = 2
\]

Simplifying further, you get:

\[
z = 2x - \frac{11 + m}{2} + 2
\]

**Step 5: Plug \( z \) back into Equation 4.**

Substitute \( z = 2x - \frac{11 + m}{2} + 2 \) into Equation 4:

\[
x + 2x - \frac{11 + m}{2} + 2 = \frac{5 + m}{2}
\]

Simplifying:

\[
3x = \frac{5 + m}{2} + \frac{11 + m}{2} - 2
\]

Simplify to:

\[
3x = \frac{16 + 2m}{2} - 2
\]

Solve for \( x \):

\[
x = \frac{8 + m}{3}
\]

To ensure the system admits a solution, the corresponding values of \( m \) must satisfy the conditions imposed by each equation. Checking the given choices, you find that \( m = 1 \) is a valid solution, corresponding to Option **C) (1, -9, 5)**.

Thus, the correct answer is **C) (1, -9, 5)**.

Would you like more details or have any questions?

Here are five related questions to consider:

1. What happens if the system has no solution?
2. How can you determine if a system is inconsistent?
3. What are the steps to find the solution to a system of three linear equations?
4. How does the value of \( m \) affect the solutions of this system?
5. Can you generalize the solution method for any similar system?

**Tip:** When solving systems of equations, always look for opportunities to simplify by adding or subtracting equations to eliminate variables.

Learn how to solve a system of linear equations with detailed step-by-step analysis. This problem involves finding values of variables in a system of three equations. Suitable for high school students studying algebra and matrix algebra.

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