Question

Find the minimum and maximum of the function z=8x−4y subject to:

y​≥1
y≤9
8x−3y≥13
8x+y≤65​

Answer 1

To find the minimum and maximum of the function z=8x-4y subject to given constraints, graph the inequalities to form a feasible region, find the vertices where the lines intersect, and evaluate the function at these points.

Step-by-step explanation:

To find the minimum and maximum of the function z=8x-4y given the constraints, we can use the method of linear programming. The constraints can be graphed on a coordinate plane to form a feasible region where the solution to the problem lies. The constraints given are y ≥ 1, y ≤ 9, 8x-3y ≥ 13, and 8x+y ≤ 65. Plotting these inequalities, we look for the vertices of the feasible region. The minimum and maximum values of the function z will occur at one of these vertices, which can be found by solving the system of equations created by the intersection of the lines. Once we have the coordinates of the vertices, we can substitute them back into the function z=8x-4y to determine the minimum and maximum values.