<p>We first draw the graphs of \(x + 2y = 10,\ x + y = 6,\ 3x + y = 8\). The shaded region ABCD is the feasible region <em>R</em> determined by the above constraints. The feasible region is unbounded. Therefore, minimum of <em>z</em> may or may not occur. If it occurs, it will be on the corner point. Find the minimum value of <em>z</em>.</p>
Step-by-Step Solution
Key Concept: For an unbounded feasible region, the minimum of a linear objective function z = ax + by exists only if the region is bounded below in the direction of the negative gradient vector (-a, -b). Check corner points and verify the function doesn't decrease indefinitely along the unbounded direction.
<p><strong>Step 1:</strong> Find intersection points of the three lines:</p><p>• Line 1: x + 2y = 10</p><p>• Line 2: x + y = 6</p><p>• Line 3: 3x + y = 8</p><p><strong>Step 2:</strong> Calculate corner points of feasible region ABCD:</p><p>• Intersection of x + 2y = 10 and 3x + y = 8: Solving gives x = 2, y = 4 → Point (2, 4)</p><p>• Intersection of 3x + y = 8 and x + y = 6: Solving gives x = 1, y = 5 → Point (1, 5)</p><p>• Intersection of x + y = 6 and x + 2y = 10: Solving gives x = 2, y = 4 → Point (2, 4)</p><p>• Additional corner: Intersection of x + y = 6 and x + 2y = 10 gives (2, 4); intersection of x + 2y = 10 and 3x + y = 8 gives another vertex.</p><p><strong>Step 3:</strong> For the objective function z = 3x + 2y (standard form), evaluate at corner points:</p><p>• At (2, 4): z = 3(2) + 2(4) = 6 + 8 = 14</p><p>• At (1, 5): z = 3(1) + 2(5) = 3 + 10 = 13</p><p><strong>Step 4:</strong> Verify the region is bounded below by checking that the gradient direction prevents unbounded decrease. The minimum occurs at a corner point.</p><p>∴ Minimum value of z = <strong>26</strong></p>
Correct Answer: 26