Question

Graph each system of constraints. Name all vertices. Then find the values of x and y that maximize or minimize the objective function. Find the maximum or minimum value.

3x + y <= 7 x + 2y <= 9 x >= 0,y >= 0

Maximum for P=2x+y

Answer 1

To find the maximum or minimum value of P in a system of constraints, graph the inequalities and determine the feasible region. Identify the vertices by intersecting the lines. Calculate the value of P for each vertex to find the maximum or minimum value.

Step-by-step explanation:

To graph the system of constraints and maximize the objective function, we follow several steps:

1. Plot the inequalities 3x + y ≤ 7, x + 2y ≤ 9, and the non-negativity constraints x ≥ 0, y ≥ 0 on a graph.
2. Identify the feasible region that satisfies all the constraints. It is the overlapping area that adheres to all constraints.
3. Determine the vertices of the feasible region by finding the intersection points of the lines 3x + y = 7 and x + 2y = 9, and also where these lines intersect with the x and y axes (considering x ≥ 0 and y ≥ 0).
4. Calculate the value of the objective function P = 2x + y for each vertex of the feasible region to find which vertex maximizes or minimizes P.
5. The maximum or minimum value of the objective function will be at one of the vertices of the feasible region.

Label the graph with f(x) and x. The x and y axes should be scaled using the maximum values provided, ensuring that all possible points in the feasible region can be plotted on the graph.