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The image contains a problem asking for the values of \( m \) such that the following system of inequalities admits a solution:

\[
\begin{cases}
x + y + z = 5 \\
x - y + z = m \\
x, y, z \geq 0
\end{cases}
\]

The possible answer choices are:
A) \([-3, 5]\)  
B) \([-9, -3)\)  
C) \([1, -9, 5]\)  
D) \([-9, 3]\)  
E) \([5, 9]\)

### Step-by-step Analysis:

1. **First Equation: \( x + y + z = 5 \)**  
   Since \( x, y, z \geq 0 \), the sum \( x + y + z = 5 \) restricts the possible values each variable can take.

2. **Second Equation: \( x - y + z = m \)**  
   This equation depends on \( m \). To determine the valid range of \( m \), we'll explore the feasible values of \( x, y, z \) that satisfy the conditions.

3. **Non-Negative Constraints: \( x, y, z \geq 0 \)**  
   The system is further constrained by the non-negativity of each variable. We must ensure that all variables stay non-negative for any chosen value of \( m \).

### Determining the Range of \( m \):

To find the range of \( m \), we'll express \( m \) in terms of \( x \), \( y \), and \( z \) using the two equations:

- From the first equation: \( z = 5 - x - y \)
- Substitute into the second equation: \( m = x - y + (5 - x - y) \)  
  Simplifying, we get:  
  \( m = 5 - 2y \)

Given \( y \geq 0 \), the maximum value of \( m \) occurs when \( y = 0 \), giving \( m = 5 \). The minimum value of \( m \) occurs when \( y \) is maximized, subject to the constraint \( 5 - x - y \geq 0 \), leading to \( y = \frac{5}{2} \), giving \( m = -3 \).

### Conclusion:

The correct range for \( m \) is \([-3, 5]\).

Thus, the correct answer is **A) [-3, 5]**.

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Would you like more details or have any questions?

Here are 5 related questions to further explore the concept:
1. What happens if one of the inequalities \( x \geq 0 \) or \( y \geq 0 \) is removed?
2. How would the solution change if the first equation was \( x + y + z = 10 \)?
3. Can we solve this problem geometrically in three dimensions?
4. What would the solution be if the second equation was \( x - y - z = m \) instead?
5. How do the bounds on \( m \) change if \( z \geq 1 \) instead of \( z \geq 0 \)?

**Tip:** When solving systems of inequalities, always check how boundary conditions influence the feasible region.

Explore how to solve systems of linear inequalities with non-negative constraints using a step-by-step approach. This problem involves determining the range of values for a variable m given specific conditions in the system of inequalities. Suitable for advanced high school students and those interested in algebraic problem-solving.

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