Maximize P = 7x + y Subject to the following constraints: 2x + y ≤ 150 4x + 3y ≤ 350 x + y ≥ 80 x, y ≥ 0

This problem is about optimizing a linear expression, a task often handled using a method called linear programming. Here, we aim to find the maximum value of the objective function P = 7 x + y subject to a number of constraints.
Let's break it down step-by-step:

Constraints Analysis:

The inequalities given are:
2 x + y ≤ 150
4 x + 3 y ≤ 350
x + y ≥ 80
x , y ≥ 0




Graphing:

Plot these inequalities on a graph to identify the feasible region. This region is where all constraints overlap. Since x and y must be non-negative, we consider only the first quadrant.


Finding the Intersection Points:

Solve the constraints in pairs to find the vertices of the feasible region, because these vertices will be potential candidates for maximizing P .


Evaluating the Objective Function:

Substitute the coordinates of each vertex into the objective function P = 7 x + y to compute P .


Identifying the Maximum Value:

Compare the values of P from each vertex.
The maximum value among these is the optimal solution.



Let's exemplify this:

Intersecting Points:

Solving 2 x + y = 150 and 4 x + 3 y = 350 simultaneously, we find one intersection.
Solving 2 x + y = 150 and x + y = 80 gives another intersection.
Solving 4 x + 3 y = 350 and x + y = 80 yields a third intersection.


Objective Function Evaluation:

Substitute each vertex into P to find which gives the highest value.



This process identifies the maximum value of P = 7 x + y within the constraints, ensuring the solution is both precise and useful.

Answered by SophiaElizab | 2025-07-21