Find the minimum of P = 7x + 4y subject to

3x + 5y ≥ 20
3x + y ≤ 16
−2x + y ≤ 3
x ≥ 0, y ≥ 0

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http://www.zweigmedia.com/RealWorld/LPGrapher/lpg.html

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To find the minimum of the function P = 7x + 4y, subject to the given constraints, we can use the method of linear programming.

Step 1: Graph the feasible region.
To do this, we will graph the inequalities and identify the region that satisfies all the constraints.

The inequality 3x + 5y ≥ 20 can be rewritten as y ≥ (20 - 3x)/5.
The inequality 3x + y ≤ 16 can be rewritten as y ≤ 16 - 3x.
The inequality -2x + y ≤ 3 can be rewritten as y ≤ 3 + 2x.

Plotting the lines y = (20 - 3x)/5, y = 16 - 3x, and y = 3 + 2x on a graph, we get:

(0,4) (4,0)
|-------------------|
|-------------------|
|-------------------|
|--------| |
| --------------|

The feasible region is the shaded region above the line y = (20 - 3x)/5, below the line y = 16 - 3x, and below the line y = 3 + 2x.

Step 2: Identify the corner points of the feasible region.
The corner points are the vertices of the feasible region. These are the points where the lines intersect.

From the graph, we can see that the corner points are:
A: (4, 0)
B: (0, 4)
C: (4, 4)
D: (3, 5)
E: (0, 0)

Step 3: Evaluate P at each corner point.
Substitute the x and y coordinates of each corner point into the function P = 7x + 4y, and calculate the value of P.

A: P = 7(4) + 4(0) = 28
B: P = 7(0) + 4(4) = 16
C: P = 7(4) + 4(4) = 44
D: P = 7(3) + 4(5) = 41
E: P = 7(0) + 4(0) = 0

Step 4: Determine the minimum value of P.
Compare the values of P at each corner point and identify the minimum value.

From the calculations above, we can see that the minimum value of P is 16, which occurs at point B: (0, 4).

To find the minimum of P = 7x + 4y subject to the given constraints, we can use linear programming. Linear programming is a mathematical method used to optimize (maximize or minimize) a linear objective function, subject to a set of linear equality and inequality constraints.

Step 1: Identify the variables and the objective function:
Variables: x, y
Objective function: P = 7x + 4y

Step 2: Write the constraints in standard form:
3x + 5y ≥ 20
3x + y ≤ 16
-2x + y ≤ 3
x ≥ 0, y ≥ 0

Step 3: Graph the feasible region:
To visualize the feasible region, we can plot the inequalities on a graph and shade in the region that satisfies all the constraints. In this case, we will have a triangular feasible region.

Step 4: Identify the vertices of the feasible region:
The vertices of the feasible region are the corner points of the shaded region.

Step 5: Evaluate the objective function at each vertex:
Plug in the values of x and y for each vertex into the objective function P = 7x + 4y, and calculate the corresponding value of P.

Step 6: Compare the objective function values:
Identify the vertex that yields the minimum value of P. This will be the optimal solution.

By following these steps, you should be able to find the minimum value of P = 7x + 4y subject to the given constraints.