Question

For the feasible region shown below, the non-trivial constraints of the linear programming problem are 

• A. $x+y\le6,\; x+3y\le9$
• B. $x+y\le6,\; x+3y\ge9$
• C. $x+y\ge6,\; x+3y\le9$
• D. $x+y\ge6,\; 3x+y\le9$

Hint:

In graph-based Linear Programming questions, first determine the line equations from intercepts and then check which side of the line contains the shaded feasible region.

Answer 1

The Correct Option is A

To determine the non-trivial constraints of the linear programming problem for the given feasible region, we need to analyze the graph and identify the lines forming the boundaries. 

The feasible region is the shaded area, which is bounded by the lines and the axes. We need to determine which of the given inequalities correspond to these boundaries.

1. Examine the line configurations in the graph:
   • The line with endpoints on the axes suggests equations like \(x + y = c\) or \(x + 3y = c\).
2. Identify the intersections with axes:
   • The line \(x + y = 6\) intersects the x-axis at \(x = 6, y = 0\) and the y-axis at \(x = 0, y = 6\).
   • The line \(x + 3y = 9\) intersects the x-axis at \(x = 9, y = 0\) and the y-axis at \(x = 0, y = 3\).
3. Determine the inequalities:
   • The region is below both lines, indicating:
     • \(x + y \le 6\)
     • \(x + 3y \le 9\)
4. Verify the feasible region:
   • The shaded area is the common region of these inequalities satisfied. It confirms both lines are boundaries with the correct inequalities.

Thus, the non-trivial constraints defining the feasible region are:

\(x+y\le6,\; x+3y\le9\)

This solution validates the correct answer and ensures that the feasible region is correctly bounded by these constraints.