Consider the following constraints from a linear programming problem:

2X + Y ≤ 200
X + 2Y ≤ 200
X, Y ≥ 0

If these are the only constraints, which of the following points (X,Y) cannot be the optimal solution?

Question

Consider the following constraints from a linear programming problem:

2X + Y ≤ 200
X + 2Y ≤ 200
X, Y ≥ 0

If these are the only constraints, which of the following points (X,Y) cannot be the optimal solution?

BlackTom AI · Accepted Answer

We begin by listing and reading every option against the given constraints so we can test feasibility first.
Option 1: (0, 0). Check: 2X + Y = 0 ≤ 200 and X + 2Y = 0 ≤ 200, with X ≥ 0, Y ≥ 0. This point satisfies all constraints, so it is feasible. 
Option 2: (0, 200). Check: 2X + Y = 0 + 200 = 200 ≤ 200, which satisfies the first constraint. However, X + 2Y = 0 + 2×200 = 400 ≤ 200 is false. Therefore this point violates the second constraint and is not feasible. 
Option 3: (0, 100). Check: 2X + Y = 0 + 100 = 100 ≤ 200; X + 2Y = 0 + 2×100 = 200 ≤ 200. Both constraints hold, so this point is feasible. 
Option 4: (100, 0). Check: 2X + Y = 2×100 + 0 = 200 ≤ 200; X + 2Y = 100 + 0 = 100 ≤ 200. Both constraints hold, so this point is feasible.

Now, interpreting the question: which of these points cannot be the optimal solution under the given constraints? A point that is infeasible cannot be optimal by definition, since it does not lie in the feasible region the LP is optimizing over. Among the options, only (0, 200) fails to satisfy the constraints, specifically the second one, and thus cannot be the optimal solution.

The other three points are feasible (they satisfy all constraints), so depending on the (unspecified) objective function, any of them could potentially be the optimal solution; there is nothing restricting them from being optimal solely based on feasibility.

In summary, the key reason (0, 200) cannot be the optimal solution is that it violates the constraint X + 2Y ≤ 200, rendering it infeasible, whereas the remaining points satisfy all constraints and therefore remain candidates for optimality given an objective function.

QANT_620-VA1-2025SP-VR Quiz_2_LP

Consider the following constraints from a linear programming problem: 2X + Y ≤ 200 X + 2Y ≤ 200 X, Y ≥ 0 If these are the only constraints, which of the following points (X,Y) cannot be the optimal solution?

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