Mathematics · Class 12 · Probability and Linear Programming · Term 2

Graphical Method of Solving Linear Programming Problems

Students will solve linear programming problems graphically, identifying feasible regions and optimal solutions.

• CBSE Learning Outcomes
• NCERT: Linear Programming - Class 12

About This Topic

The graphical method for solving linear programming problems requires students to plot linear inequalities on a coordinate plane, shade the feasible region where all constraints overlap, and identify corner points or vertices. They then evaluate the objective function at these points to find the maximum or minimum value, as per the corner point theorem from NCERT Class 12. This approach suits two-variable problems common in CBSE exams and connects to real-world applications like maximising profit or minimising cost in production.

In the Probability and Linear Programming unit, this topic strengthens skills in systems of inequalities and optimisation, building on Class 11 coordinate geometry. Students learn that the feasible region forms a convex polygon, and changes in constraints shift its boundaries, altering optimal solutions. This fosters analytical thinking essential for competitive exams like JEE.

Active learning suits this topic well because graphing activities make abstract constraints visible and interactive. When students sketch regions collaboratively and test objective values at corners, they grasp the method intuitively and spot errors in real time, leading to deeper retention and confidence in solving complex problems.

Key Questions

1. Analyze how the feasible region is determined by the system of linear inequalities.
2. Evaluate the corner point method for finding the optimal solution.
3. Predict the impact of changing a constraint on the feasible region and optimal solution.

Learning Objectives

• Identify the feasible region for a system of linear inequalities graphically.
• Calculate the coordinates of corner points of the feasible region.
• Evaluate the objective function at each corner point to determine the optimal solution.
• Analyze how changes in constraint coefficients affect the feasible region and optimal solution.
• Formulate a linear programming problem from a given word problem and solve it graphically.

Before You Start

Graphing Linear Equations in Two Variables

Why: Students need to be able to accurately plot lines on a coordinate plane to represent the constraints.

Solving Systems of Linear Inequalities

Why: Understanding how to find the region that satisfies multiple inequalities is fundamental to identifying the feasible region.

Basic Algebra: Evaluating Functions

Why: Students must be able to substitute values into the objective function to find its value at different points.

Key Vocabulary

Linear Inequality	An inequality involving linear expressions in two variables, defining a half-plane region on a graph.
Feasible Region	The region on a graph that satisfies all the constraints (linear inequalities) of a linear programming problem simultaneously.
Corner Point	A vertex of the feasible region, formed by the intersection of boundary lines of the constraints.
Objective Function	A linear function that represents the quantity to be maximized or minimized in a linear programming problem, such as profit or cost.
Optimal Solution	The point within the feasible region where the objective function achieves its maximum or minimum value.

Watch Out for These Misconceptions

Common Misconception

The feasible region is always a closed polygon.

What to Teach Instead

Feasible regions can be unbounded if constraints allow extension to infinity. Group graphing sessions help students test boundary lines and observe open regions, correcting this through visual comparison and peer explanations.

Common Misconception

The optimal solution lies anywhere in the feasible region.

What to Teach Instead

Per the corner point theorem, optima occur at vertices for linear functions. Hands-on evaluation at multiple points during pair activities reveals this pattern, building conviction over rote memorisation.

Common Misconception

Equality constraints are ignored in graphing.

What to Teach Instead

Equalities form boundaries of the feasible region. Collaborative plotting ensures students include them accurately, as group checks highlight omissions during shading steps.

Active Learning Ideas

Pair Graphing: Plotting Constraints

Pairs receive a set of linear inequalities and an objective function. They plot each line, shade the feasible region together, and mark corner points. Finally, they substitute points into the objective to identify the optimum.

30 min·Pairs

Small Group Optimisation Challenge

Distribute word problems on resource allocation. Groups graph constraints, locate feasible region, and find optimal corners. They present findings and verify with classmates using graph paper.

45 min·Small Groups

Whole Class Constraint Shift

Project a base feasible region. Alter one constraint as a class, redraw on mini whiteboards, and discuss shifts in optimum. Vote on predictions before revealing solutions.

35 min·Whole Class

Individual Verification Drill

Provide graphs with marked regions. Students independently evaluate objectives at corners and check for multiple optima or unbounded cases.

20 min·Individual

Real-World Connections

• Production managers in manufacturing plants use graphical methods to determine the optimal mix of products to produce to maximize profit, given limited resources like labour and raw materials.
• Logistics companies employ these techniques to plan delivery routes that minimize fuel consumption and delivery time, considering factors like distance and traffic constraints.
• Financial advisors might use linear programming to create investment portfolios that maximize returns while adhering to risk tolerance levels and diversification requirements.

Assessment Ideas

Quick Check

Present students with a system of two linear inequalities. Ask them to: 1. Graph the boundary lines. 2. Shade the correct region for each inequality. 3. Identify the corner points of the overlapping feasible region.

Discussion Prompt

Pose the question: 'Imagine a constraint is added to an existing linear programming problem. How might this new constraint change the shape and size of the feasible region? What are the possible impacts on the optimal solution?' Facilitate a class discussion where students share their predictions and reasoning.

Exit Ticket

Give each student a simple linear programming problem with an objective function and two constraints. Ask them to: 1. Write down the coordinates of the corner points of the feasible region. 2. State which corner point yields the maximum value for the given objective function.

Frequently Asked Questions

How do you determine the feasible region graphically?

Plot each inequality as a line, shade the half-plane satisfying it, and find the overlap. Test a point like the origin to confirm shading direction. This common region, often a polygon, contains all feasible solutions, preparing students for corner evaluations.

What is the corner point method in linear programming?

Identify vertices of the feasible region formed by intersection points of boundary lines. Substitute these into the objective function to compare values and select the maximum or minimum. This theorem guarantees optima at corners for bounded linear problems in two variables.

How can active learning help students master graphical LPP?

Activities like pair graphing and group challenges make constraints tangible through hands-on plotting and shading. Students predict optima, test corners collaboratively, and adjust for constraint changes, turning abstract theory into interactive discovery. This boosts engagement, error spotting, and exam readiness over passive lectures.

What happens if a constraint changes in LPP?

Tightening a constraint shrinks the feasible region, potentially shifting or eliminating the optimum. Loosening expands it, possibly introducing new corners. Class demonstrations with adjustable graphs help students visualise impacts, reinforcing sensitivity analysis for practical applications.