To graph the system of constraints, plot the lines for each inequality and find the vertices. To find the maximum or minimum values of the objective function, evaluate it at each vertex.
To graph the system of constraints, we first need to plot the lines that represent each inequality. Let's start by graphing the inequality x ≤ 5y. To do this, we can choose any two values for y and solve for the corresponding values of x. For example, when y = 0, x = 0. And when y = 1, x = 5. Plot these two points and draw a line connecting them.
Next, let's graph the inequality 5y ≤ 4x. Similarly, choose two values for x and solve for y. For instance, when x = 0, y = 0. And when x = 4, y = 5/4. Plot these two points and draw a line connecting them.
Finally, graph the line x ≥ 0 and y ≥ 0 by drawing two lines, one along the x-axis and the other along the y-axis, passing through the origin (0,0). This creates a triangle-shaped region in the first quadrant.
The vertices of the feasible region are the points where the lines intersect. To find the values of x and y that maximize or minimize the objective function P = 3x + 2y within this region, we can evaluate the function at each vertex and determine which yields the maximum or minimum value.
The probable question may be:
Graph each system of constraints. Name all vertices. Then find the values of x and y that maximize or minimize the objective function.
x≤5
y≤4
x≥0,y≥0
Maximum for P = 3x + 2y