Linear Programming Problem

Step 1: Plot the constraints
The given constraints are: : The line passes through and .
: The line passes through and .
and : Restricts the feasible region to the first quadrant.
The feasible region is the intersection of all these constraints, forming a polygon bounded by the vertices.
Step 2: Find corner points
The corner points of the feasible region are determined by solving the equations pairwise: : Intersection of and .
: Intersection of and .
: Intersection of and .
: Intersection of and .
Step 3: Evaluate the objective function at corner points
Substitute the coordinates of the vertices into :
Step 4: Find the maximum value of
The maximum value of is , which occurs at .
Conclusion: The maximum value of is when and .

Step 1: Define constraints in linear programming
Constraints are the conditions or restrictions imposed on decision variables (e.g., ) in a linear programming problem.
They typically represent limitations on resources or other requirements.
Step 2: Identify from options
(A) Feasible solutions: These are solutions satisfying all constraints, but they are not the constraints themselves.
(B) Constraints: These are restrictions on decision variables, and this is the correct answer.
(C) Optimal solutions: These maximize or minimize the objective function but are not constraints.
(D) Infeasible solutions: These do not satisfy all constraints.
Step 3: Conclude the result
The restrictions are called constraints.

Step 1: Identify constraint lines.
Convert inequalities to equations for plotting:

Step 2: Find intersection points.
Solving for intersection points:
1. Solve and .
2. Solve and .
3. Solve and .

Step 3: Identify feasible region.
Graph all lines and shade the feasible region satisfying constraints.

Step 4: Compute Z-values at corner points.
Evaluate at each intersection point to find the minimum.

Final Answer: Minimum value occurs at .

The feasible region of a Linear Programming Problem (LPP) is determined by the intersection of the inequalities.
The feasible region is the set of points that satisfy all the constraints.
- Plot the given constraints on a coordinate plane:
1. is a straight line.
2. is another straight line.
3. represents the first quadrant.
- The feasible region will be bounded by the lines and will be the area that satisfies all these constraints.
From the diagram, the feasible region is the region enclosed by the points , and the correct region is .

- The feasible region is the area that satisfies all the given constraints for the Linear Programming Problem (LPP).
This region is bounded by the lines representing the constraints. The graph shows this feasible region as the shaded portion. Therefore, Assertion (A) is correct.
- The objective function is . The given condition has a minimum value of 380 at the point .
- If , this means the values of and are outside the feasible region, as the region representing less than 380 does not intersect the feasible region. Thus, Reason (R) correctly explains why the region does not intersect the feasible region.
Hence, both Assertion (A) and Reason (R) are correct, and Reason (R) is the correct explanation for Assertion (A).