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3.Graphical Method
Class 12 • Mathematics • NCERT
Graphical Method of Linear Programming
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This lesson explains how to solve a Linear Programming Problem graphically by plotting constraints, identifying the feasible region, and finding the optimal solution, as prescribed in NCERT Class 12.
Lesson Objectives
- Understand the graphical method of solving LPP.
- Plot linear constraints on coordinate axes.
- Identify the feasible region.
- Find the optimal solution graphically.
1. Meaning of Graphical Method
The graphical method is a technique used to solve a Linear Programming Problem by drawing the graphs of the constraints and identifying the region that satisfies all the given conditions.
This method is applicable only when there are two decision variables.
[ Image Placeholder: Coordinate Axes for Graphical Method ]
2. Graph of Linear Constraints
Each linear constraint is first converted into an equation and then represented graphically as a straight line.
Example: x + y ≤ 10 → x + y = 10
[ Image Placeholder: Graph of a Linear Constraint ]
3. Feasible Region
The feasible region is the common region that satisfies all the constraints including non-negativity conditions.
Only points inside the feasible region are feasible solutions.
[ Image Placeholder: Shaded Feasible Region ]
4. Corner Points of Feasible Region
The corner points (vertices) of the feasible region are the points of intersection of the boundary lines.
The optimal solution lies at one of the corner points.
[ Image Placeholder: Corner Points of Feasible Region ]
5. Optimal Solution
The value of the objective function is calculated at each corner point to determine the maximum or minimum value.
The best value obtained is the optimal solution.
[ Image Placeholder: Objective Function Line Touching Feasible Region ]
6. Important NCERT Notes
• Applicable only for two variables
• All constraints must be linear
• Optimal solution occurs at corner points
• Graph must be neat and well-labeled
• All constraints must be linear
• Optimal solution occurs at corner points
• Graph must be neat and well-labeled
Practice Questions (NCERT)
- What is the graphical method of LPP?
- How many variables are allowed in graphical method?
- What is a feasible region?
- Define corner points.
- Where does the optimal solution lie?
- Why non-negativity conditions are important?
- What is the first step in graphical method?
- Is graphical method applicable for three variables?
- What is a feasible solution?
- Is this topic important for board exams?
✅ Show Answer Key
- Method using graphs to solve LPP
- Two
- Region satisfying all constraints
- Intersection points of boundary lines
- At corner points
- Restrict solution to first quadrant
- Plot constraints
- No
- Any point in feasible region
- Yes
© Aviate Learning – Linear Programming (NCERT Class 12)