Solve the linear programming problem: maximize P = 3x + 4y subject to x + y ≤ 7, x ≥ 0, and y ≥ 0.
To solve this linear programming problem, we will first graph the constraints and then find the vertices of the feasible region. Finally, we will evaluate the objective function at each vertex to determine the maximum value.

1. Graph the constraints:

The constraint x + y ≤ 7 can be rewritten as y ≤ -x + 7. This is a line with slope -1 and y-intercept 7. We shade below the line since y must be less than or equal to -x + 7.

The constraints x ≥ 0 and y ≥ 0 indicate that we are only considering the first quadrant (where both x and y are non-negative).

2. Find the vertices of the feasible region:

The feasible region is bounded by the lines x = 0, y = 0, and y = -x + 7. The vertices of the feasible region are where these lines intersect: (0, 0), (0, 7), and (7, 0).

3. Evaluate the objective function at each vertex:

P(0, 0) = 3(0) + 4(0) = 0
P(0, 7) = 3(0) + 4(7) = 28
P(7, 0) = 3(7) + 4(0) = 21

The maximum value of P occurs at the vertex (0, 7), so the maximum value of P is 28.