Math Problem Statement
Solve the following LPP by graphical method Max Z = 2x1 + 3x2 Subject to constraints x1 - x2 ≤ 2 and x1 + x2 ≥ 4 and x1 , x2 ≥ 0.
Solution
To solve the given Linear Programming Problem (LPP) using the graphical method, follow these steps:
Problem Statement:
- Objective Function: Maximize
- Constraints:
Step 1: Convert Inequalities to Equations
- For :
- Equation:
- For :
- Equation:
Step 2: Plot the Constraints on the Graph
-
Equation 1:
- Find intercepts:
- If : ⟹ (out of feasible region since )
- If : ⟹
- Points: and another point, for example if : ⟹
- Find intercepts:
-
Equation 2:
- Find intercepts:
- If : ⟹
- If : ⟹
- Points: and
- Find intercepts:
Step 3: Identify the Feasible Region
- The feasible region is the area where all constraints intersect while considering and .
- : This is the area below the line .
- : This is the area above the line .
Step 4: Find Intersection Points
- Intersection of and :
- Solve the system of equations:
- Add the equations:
- Substitute into one of the equations:
- Intersection Point:
Step 5: Evaluate the Objective Function at the Vertices
- Vertices of the Feasible Region: , , and
- Calculate at each vertex:
- At :
- At :
- At :
Step 6: Identify the Optimal Solution
- Maximum value of is 12 at the point .
Conclusion
The optimal solution to the LPP is:
- Optimal Point:
- Maximum Value:
Would you like more details on any step, or do you have any questions?
Relative Questions to Expand Understanding
- How do we interpret the feasible region in a linear programming problem?
- Why do we only evaluate the objective function at the vertices of the feasible region?
- What happens if the feasible region is unbounded?
- Can we use this graphical method for LPPs with more than two variables?
- What is the significance of having constraints and ?
Tip
When solving an LPP graphically, always ensure that the feasible region is accurately plotted and identify all vertices correctly, as the optimal solution lies at one of these vertices.
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Math Problem Analysis
Mathematical Concepts
Linear Programming
Graphical Method
Optimization
Formulas
Objective function: Z = 2x1 + 3x2
Constraint equations: x1 - x2 = 2, x1 + x2 = 4
Theorems
Fundamental theorem of linear programming (optimal solution lies at a vertex of the feasible region)
Suitable Grade Level
Grades 11-12 or early university (Introductory Linear Programming)
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