Final answer:
To maximize P = 5x + 7y with given constraints, a graph of the inequalities is plotted to find the feasible region. The maximum of P is found by testing the vertices of this region by substituting into the objective function.
Step-by-step explanation:
The student is asking how to maximize the objective function P = 5x + 7y given the constraints x ≥ 0, y ≥ 0, 2x + 3y ≤ 18, and 5x + 2y ≤ 23. This is a problem of linear programming where the goal is to find the maximum value of the objective function within the feasible region defined by the constraints.
To solve this, we first graph the inequalities to find the feasible region. Each inequality represents a half-plane, and their intersection is the feasible region where we look for the solution. Since both x and y must be non-negative, we are limited to the first quadrant of the coordinate system.
We also calculate the vertices of the feasible region by finding the points of intersection of the lines formed by turning the inequalities into equalities. These points are often found where the constraint lines intersect each other or with the axes. We then test these vertices by substituting them into the objective function to determine which one gives the highest value for P.