Math Problem Statement

Solución del problema

Solution

The image contains a problem asking for the values of mm such that the following system of inequalities admits a solution:

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]$$. Thus, the correct answer is **A) [-3, 5]**. --- Would you like more details or have any questions? Here are 5 related questions to further explore the concept: 1. What happens if one of the inequalities $$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.

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Math Problem Analysis

Mathematical Concepts

Systems of Linear Equations
Inequalities
Non-negative Constraints

Formulas

-

Theorems

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Suitable Grade Level

Advanced High School

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