medium difficulty9 min

Graphical Method and Corner Point Evaluation

The graphical method solves a two-variable LPP by drawing the constraint lines, shading the common feasible region, finding its corner points, and evaluating the objective function at those points.

In one line: Graph, shade, vertices, table, answer: GSTA keeps the solution organized.

Practice This Concept

Understand it

Each linear inequality represents a half-plane.
The feasible region is the common part satisfying all constraints.
If the feasible region is bounded, the maximum or minimum of a linear objective function occurs at a corner point.
For an unbounded region, corner point values alone may not be enough to decide the answer unless the direction of improvement is checked.

Example

For constraints x + y ≤ 6, x ≤ 4, y ≤ 5, x ≥ 0, y ≥ 0, the feasible region lies in the first quadrant below or on each boundary line. The objective value is tested at all corner points of the region.

Formula and rule

Corner point principle: For a bounded feasible region, maximum or minimum of Z = ax + by occurs at one or more vertices of the feasible region.

Boundary line method: replace each inequality by equality to draw its line, then choose the valid half-plane.

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

Maximize Z = 3x + 2y subject to x + y ≤ 4, x ≤ 3, y ≤ 2, x ≥ 0, y ≥ 0.

1Corner points are (0,0), (3,0), (3,1), (2,2), and (0,2).
2Evaluate: Z(0,0)=0, Z(3,0)=9, Z(3,1)=11, Z(2,2)=10, Z(0,2)=4.
3The maximum value is 11 at (3,1).

Method you can reuse

1Convert inequalities to boundary equations.
2Draw each line using intercepts or two points.
3Shade the side satisfying each inequality.
4Identify the common feasible region.
5List all corner points.
6Substitute each point in Z = ax + by.
7Choose the largest value for maximization or smallest value for minimization, then write the point and value together.

Sketch it in your answer

Draw x-axis and y-axis, plot each boundary line accurately, shade the common feasible region, label all vertices, and mark the direction of the feasible side for each constraint if needed.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

Why are corner points checked in the graphical method of linear programming?

Think first, then show answer

Because a linear objective function, when it has an optimum over a bounded feasible region, attains that optimum at a corner point.

Common wrong answer

Evaluating Z at intercepts of all lines instead of only the actual vertices of the feasible region leads to invalid answers.

Don't mix this up

A common error is shading the wrong side of a line after plotting it correctly, which changes the feasible region completely.

Exam tip

After drawing each boundary line, test a simple point such as (0, 0) when it is not on the line to decide the correct half-plane.

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

How CBSE asks it

Often asked as a complete long-answer problem requiring graph, feasible region, corner point table, optimum value, and final interpretation.

1
Enough for 1 mark
The graphical method solves a two-variable LPP by drawing the constraint lines, shading the common feasible region, finding its corner points, and evaluating the objective function at those points.
2
Add for 2 marks
Example: For constraints x + y ≤ 6, x ≤ 4, y ≤ 5, x ≥ 0, y ≥ 0, the feasible region lies in the first quadrant below or on each boundary line. The objective value is tested at all corner points of the region.
3
Add for 3 marks
Each linear inequality represents a half-plane.

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