Maharashtra Board Class 12 Maths Part 1 Chapter 7 Linear Programming PDF Download

7 Linear Programming

Let's Study

7.1 Linear Inequations In Two Variables

7.1.1 Convex Sets

7.1.2 Graphical Representation Of Linear Inequations In Two Variables

7.1.3 Graphical Solution Of Linear Inequation

7.2 Linear Programming Problem (L.P.P.)

7.2.1 Meaning Of Linear Programming Problem

7.2.2 Mathematical Formulation Of L.P.P.

7.2.3 Solution Of L.P.P. By Graphical Methods

Let's Recall

A linear equation in two variables \(ax + by + c = 0\), where \(a, b, c \in \mathbb{R}\) and \(a\) and \(b\) are not zero simultaneously, represents a straight line. A straight line makes disjoint parts of the plane. The points lying on the straight line and two half planes on either side are represented by \(ax + by + c < 0\) and \(ax + by + c > 0\). We will now study the two half planes made by a line.

The sets of points \(\{(x, y) \mid ax + by + c < 0\}\) and \(\{(x, y) \mid ax + by + c > 0\}\) are two open half planes. The sets of points \(\{(x, y) \mid ax + by + c \leq 0\}\) and \(\{(x, y) \mid ax + by + c \geq 0\}\) are two half planes with common points. The sets \(\{(x, y) \mid ax + by + c \leq 0\}\) and \(\{(x, y) \mid ax + by + c \geq 0\}\) have the common boundary \(\{(x, y) \mid ax + by + c = 0\}\).

Linear Inequation In Two Variables

Definition: A linear inequation in two variables \(x, y\) is a mathematical expression of the form \(ax + by < c\) or \(ax + by > c\) where \(a \neq 0, b \neq 0\) simultaneously and \(a, b \in \mathbb{R}\).

Activity: Check which of the following ordered pairs is a solution of \(2x + 3y - 6 \leq 0\).

1. (1, -1)

2. (2, 1)

3. (-2, 1)

4. (1, -2)

5. (-3, 4)

Solution:

Graphical representation of linear inequation in two variables \(ax + by \leq\) or \(\geq c\) is a region on any one side of the straight line \(ax + by = c\) in the coordinate system, depending on the sign of inequality.

Teacher's Note

Linear inequations help us show areas on a graph, not just lines. Like a shop selling shirts and pants - we can show all combinations that cost less than Rs. 500.

Exam Trick

Remember: \(<\) or \(>\) means the line is dotted. \(\leq\) or \(\geq\) means the line is solid. This helps you draw the graph correctly in exams.

Points to Remember

A linear inequation has variables with power 1 only.
The solution is a region, not a single line.
We test a point (usually the origin) to find which side is the answer.
Open half planes do not include the line itself.
Closed half planes include the boundary line.

7.1.1 Convex Set

Definition: A set of points in a plane is said to be a convex set if line segment joining any two points of the set entirely lies within the set.

The Following Sets Are Convex Sets:

A circle with two points A and B connected by a line inside it.

A pentagon with two points A and B connected by a line inside it.

The Following Sets Are Not Convex Sets:

A V-shape with two points A and B, where the line between them goes outside.

A ring shape (circle inside a circle) with two points, where the line between them leaves the region.

Note:

(i) The convex sets may be bounded. Following are bounded convex sets.

A trapezoid shape.

A hexagon shape.

(ii) Convex sets may be unbounded.

An unbounded region below a line in the first quadrant.

An unbounded region above a line in the lower half.

Note:

1) Graphical representations of \(x \leq h\) and \(x \geq h\) on the Cartesian coordinate system. Draw the line \(x = h\) in XOY plane.

Teacher's Note

Convex sets are shapes where any two points inside can be connected by a straight line that stays inside. A pizza slice is convex, but a banana shape is not.

Exam Trick

Draw a line between any two points inside the shape. If the line always stays inside, it is convex. If it ever goes outside, it is not convex.

Points to Remember

A convex set is like a filled circle or square.
A non-convex set is like a crescent moon or ring.
Bounded convex sets have edges and corners.
Unbounded convex sets go on forever in some direction.
The common region of many inequations is always a convex set.

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