While plotting constraints on a graph paper, terminal points on both the axes are connected by straight line because:
a) The Resources are limited in supply b) the objective function as a linear
function c) the constraints are linear equations or inequalities d) All of the above
The correct answer and explanation is :
Correct Answer: d) All of the above
Explanation (Approx. 300 words):
In linear programming, constraints are typically plotted on a graph to visually represent the feasible region — the set of all possible solutions that satisfy the given conditions. When plotting these constraints, the terminal points (intercepts) on the X-axis and Y-axis are connected by a straight line. This practice is based on several foundational principles of linear programming:
- Constraints are Linear Equations or Inequalities (Option c):
Constraints in linear programming are expressed in the form of linear equations or inequalities (e.g., ( ax + by \leq c )). The graph of a linear equation is always a straight line. Therefore, when two intercepts (such as when ( x = 0 ) and ( y = 0 )) are determined and plotted, connecting them with a straight line accurately represents the constraint. - The Objective Function is a Linear Function (Option b):
Though this doesn’t directly affect how the constraints are graphed, it plays a role in why the entire problem is linear. The linearity of the objective function ensures that the optimization (maximization or minimization) over a linear feasible region is meaningful and that the solution lies at the boundary or vertex of the feasible region — which is shaped by straight lines from linear constraints. - Resources are Limited in Supply (Option a):
This is the practical reason behind the constraints themselves. In real-world applications, resources like labor, materials, or time are finite. These limitations are mathematically expressed as inequalities, leading to the creation of constraint lines on a graph.
Conclusion:
All three options are interconnected and essential to understanding why terminal points are connected by a straight line when graphing constraints. Hence, Option d) All of the above is the most comprehensive and correct answer.
