Final answer:
To solve the given LP problem, graph the constraints to identify the feasible region and calculate the corner points. If the region is bounded, evaluate the objective function at each point to find the maximum. If the region is empty or the objective function is unbounded, no optimal solution exists.
Step-by-step explanation:
To solve the given linear programming (LP) problem, we need to maximize the objective function p = x + 2y subject to the constraints x + 6y ≤ 15, 3x + y ≤ 11, and x ≥ 0, y ≥ 0. First, we graph these inequalities to determine the feasible region, which is where all constraints are satisfied simultaneously. If a feasible region is found, we then check the corner points of this region to find the maximum value of p.
We find the corner points by the intersection of the lines:
and then along with the intercepts on the axes. The corner points typically give us the potential candidates for the optimum solution in a LP problem.
Once we find these points, we plug them into the objective function to see which one gives us the highest value of p. If the region is bounded and a maximum value exists, this point is our optimum solution. If there is no bounded feasible region (meaning it's empty) or if the objective function increases without bound (as we move further out in the feasible region), we conclude that there's no optimal solution.
In case the LP problem has an empty feasible region or if the objective function is unbounded, that will be the answer to the problem. However, if we do find a bounded feasible region with maximum value, p, x, and y will have specific numerical values.