Final answer:
To solve the linear programming problem, graph the inequalities, determine the feasible region, find the vertices, and calculate the objective function at each vertex to find the maximum z value.
Step-by-step explanation:
The question involves solving a linear programming problem with constraints to maximize the objective function z = 6x + 3y. The constraints for the problem are given as inequalities: 5x - y ≤ 15, 3x + y ≥ 12, x ≥ 3, and y ≤ 8. To solve this, we must first graph the inequalities to determine the feasible region and then find the vertices of this region. We can then test these vertices in the objective function to find which one gives the maximum value for z.
The first step is to graph each inequality on a coordinate plane:
- 5x - y ≤ 15 is a line, and we shade the area below it.
- 3x + y ≥ 12 is also a line, and we shade the area above it.
- x ≥ 3 is a vertical line passing through (3,0), and we shade to the right.
- y ≤ 8 is a horizontal line passing through (0,8), and we shade below.
The feasible region is where all shading overlaps. The vertices can be found by solving the system of equations at the intersections of the lines. Once these vertices are known, we calculate z for each of them to determine which gives the maximum value. The vertex that maximizes z represents the solution to the linear programming problem.