Linear programming

Question

An objective function and a system of linear inequalities representing constraints are given. Complete parts a. through c.

Objective Function z = 2x + 7y

Constraints {x≥0, y≥0

{2x + y≤ 12

{ x + y≥ 6

a. Graph the system of inequalities representing the constraints

b. Find the value of the objective function at each corner of the graphed region.

c. Use the value in part (b) to determine the maximum value of the objective function and the value of x and y at which the maximum occurs.

Answer 1

Hi, Titus!

For this problem, you should have graph paper and four different colored pencils.

a. If we graph the constraints, we get one color going from the y-axis to the right for x≥ 0

and another color going from the x-axis up for y ≥ 0.

Next, we want to graph 2x + y ≤ 12.

If we solve 2x + y = 12 when x = 0 and when y = 0, we get the points, (0, 12) and (6, 0)

Draw that line and color under it.

Then, we want to graph x + y ≥ 6.

Again, solve the equation x + y = 6 when x = 0 and when y = 0, we get the points, (0, 6) and (6, 0).

Draw that line and color above it.

The only place where you have all four colors is a triangle with vertices at (6, 0), (0, 6) and (0, 12).

b. Plug your x and y values into the objective function to find the value at each corner.

z = 2x + 7y, so z = 2(6) + 7(0) = 12, z = 2(0) + 7(6) = 42 and z = 2(0) + 7(12) = 84

c. The maximum value of z is 84 and it occurs at (0, 12)

i hope this helps you understand this better!

Karen