Math Problem Statement
Min Z = 5x + 7y
S. T
x + y = 10
x ≥ 2
y ≥ 3
x ≥ 0
x ≥ 0
Solution
We are tasked with minimizing the objective function , subject to the following constraints:
x + y &= 10 \\ x &\geq 2 \\ y &\geq 3 \\ x &\geq 0 \\ y &\geq 0 \\ \end{aligned}$$ ### Steps to solve: 1. **Substitute from the equality constraint**: From $$x + y = 10$$, we can express $$y$$ in terms of $$x$$ as: $$y = 10 - x$$ 2. **Update the constraints**: Since $$y = 10 - x$$, we need to ensure that this satisfies the other constraints: $$x \geq 2 \quad \text{and} \quad y = 10 - x \geq 3$$ Solving $$10 - x \geq 3$$: $$10 - 3 \geq x \implies x \leq 7$$ Therefore, the feasible range for $$x$$ is: $$2 \leq x \leq 7$$ 3. **Express the objective function**: Substitute $$y = 10 - x$$ into the objective function $$Z = 5x + 7y$$: $$Z = 5x + 7(10 - x) = 5x + 70 - 7x = -2x + 70$$ 4. **Minimize $$Z$$**: Since the objective function is linear and decreasing in $$x$$, we minimize $$Z$$ by choosing the maximum value of $$x$$, which is $$x = 7$$. 5. **Calculate $$Z$$ at $$x = 7$$**: If $$x = 7$$, then: $$y = 10 - 7 = 3$$ Substituting into the objective function: $$Z = 5(7) + 7(3) = 35 + 21 = 56$$ ### Conclusion: The minimum value of $$Z$$ is **56**, which occurs at $$x = 7$$ and $$y = 3$$. --- Would you like more details or have any questions? Here are 5 related questions for further understanding: 1. What happens to $$Z$$ if $$x = 2$$ instead of 7? 2. How do we solve a maximization problem using the same approach? 3. Can you solve a minimization problem with more constraints? 4. How does the geometry of the feasible region affect the solution? 5. What if the constraint $$x + y = 10$$ were an inequality? **Tip:** In optimization problems, always check if the objective function is increasing or decreasing to identify boundary solutions effectively.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Programming
Optimization
Linear Inequalities
Formulas
Objective Function: Z = 5x + 7y
Substitution: y = 10 - x
Updated Objective Function: Z = -2x + 70
Theorems
Linear Programming Boundary Solution
Suitable Grade Level
Grades 10-12
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