Final answer:
To find the minimum value of z=5x-7y with given constraints, graph the inequalities to identify the feasible region, then compute z at each vertex of this region.
Step-by-step explanation:
The objective is to find the minimum value of the objective function z = 5x - 7y, with the given constraints x + y ≤ 7, 2x - 3y + 6 ≥ 0, x ≥ 0, and y ≥ 0. To solve this linear programming problem, we need to identify the feasible region described by these inequalities, which is done by graphing them on the coordinate plane. Once the feasible region is graphed, we can find the vertices of this region. Since z is a linear function, its minimum and maximum values occur at the vertices of the feasible region. By evaluating z at each of the vertices, we can find the minimum value required.