Math Problem Statement
Min Z = 5x + 7y S.T. x + y = 10 x ≥ 2 y ≥ 3 X ≥ 0 y ≥ 0
Solution
To solve the linear programming problem:
Minimize:
Subject to:
x + y &= 10 \\ x &\geq 2 \\ y &\geq 3 \\ x &\geq 0 \\ y &\geq 0 \end{aligned}$$ ### Steps to solve: 1. **Identify the constraints**: - The first constraint is $$x + y = 10$$, which represents a straight line in the xy-plane. - The second constraint $$x \geq 2$$ sets a minimum value for $$x$$. - The third constraint $$y \geq 3$$ sets a minimum value for $$y$$. - The non-negativity constraints $$x \geq 0$$ and $$y \geq 0$$ ensure that the solution lies in the first quadrant. 2. **Graph the constraints**: - $$x + y = 10$$ is a straight line that passes through the points (0, 10) and (10, 0). - $$x \geq 2$$ is a vertical line at $$x = 2$$, so only points to the right of this line are feasible. - $$y \geq 3$$ is a horizontal line at $$y = 3$$, so only points above this line are feasible. - The feasible region is bounded by these constraints. 3. **Find the feasible region**: The intersection of the lines forms a polygon, and the feasible region is where all the inequalities are satisfied: - $$x + y = 10$$ - $$x \geq 2$$ - $$y \geq 3$$ 4. **Determine the corner points**: The corner points of the feasible region occur where the lines intersect: - When $$x = 2$$, solve for $$y$$: $$2 + y = 10 \Rightarrow y = 8$$. So one point is $$(2, 8)$$. - When $$y = 3$$, solve for $$x$$: $$x + 3 = 10 \Rightarrow x = 7$$. So another point is $$(7, 3)$$. - A third point occurs at the intersection of $$x = 2$$ and $$y = 3$$, which gives the point $$(2, 3)$$. 5. **Evaluate Z at each corner point**: - At $$(2, 8)$$: $$Z = 5(2) + 7(8) = 10 + 56 = 66$$ - At $$(7, 3)$$: $$Z = 5(7) + 7(3) = 35 + 21 = 56$$ - At $$(2, 3)$$: $$Z = 5(2) + 7(3) = 10 + 21 = 31$$ 6. **Conclusion**: The minimum value of $$Z$$ is 31, which occurs at the point $$(2, 3)$$. ### Solution: The optimal solution is $$x = 2$$, $$y = 3$$, and the minimum value of $$Z$$ is **31**. --- Would you like more details on any step? Here are some related questions: 1. How can we graph the feasible region for this problem? 2. What are the typical steps in solving linear programming problems? 3. Can this method be applied to maximize an objective function? 4. How do the constraints affect the shape of the feasible region? 5. How can the Simplex method be used for linear programming? **Tip**: In linear programming, the minimum or maximum value will always occur at a corner point of the feasible region.Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Linear Programming
Optimization
Inequalities
Feasible Region
Formulas
Objective function: Z = 5x + 7y
Constraint equation: x + y = 10
Theorems
Corner-point theorem
Suitable Grade Level
Grades 10-12
Related Recommendation
Solve Linear Programming Problem: Minimize Z = 5x + 7y with Constraints
Minimize the Linear Objective Function Z = 5x + 7y with Constraints
Minimization of Z = 5x + 7y with Constraints x + y = 10
Linear Programming: Minimize Z = 5x + 7y Subject to Constraints
Maximize and Minimize Linear Programming Problem: z = 2x + 3y