Linear Programming and Systems of Inequalities

Solve each of the following systems of linear inequalities a graphing approach. Verify using a graphing calculator. Which of following are the corner points?

• (2, 0), (4, 5), (6, 0)

• (2, 0), (3, 5), (0, 6)

• (0, 2), (3.5, 4.5), (6, 2)

• (0, 0), (3, 5), (6, 0)

It requires that each variables to be greater than or equal to zero

• maximization

• minimization

• inequality

• non-negativity constraints

It is a method for finding a maximum or minimum value of some quantity, given a set of constraints.

• Operations research

• Maximization

• Minimization

• Linear programming

• x = 0, y = 300

• x = 150, y = 150

• x = 100, y 150

• x = 150, y = 100

Graph the system of inequalities. Name the coordinates of the vertices of the feasible region. Find the maximum and minimum values of the function for this region.
3 ≤ y ≤ 6
y ≤ 3x + 12
y ≤ -2x + 6
f(x , y) = 4x - 2y

• The maximum value is 0 at (0, 6). The minimum value is - 20 at (-3, 3).

• The maximum value is 0 at (-3, 3). The minimum value is - 20 at (0, 6).

• The maximum value is 0 at (-2, 6). The minimum value is - 20 at (1.5, 3).

• The maximum value is 0 at (1.5, 3). The minimum value is - 20 at (-2, 6).

• (7,4)

• (3,10)

• (10,1)

• (8,5)

A farmer can plant up to 6 acres of land with soybeans and corn. Her use of a necessary pesticide is limited by federal regulations to 15 gallons for her entire 6 acres. Soybeans require 2 gallons of pesticide for every acre planted and corn requires 3 gallons per acre. The profit the farmer makes by earning $4,000 for every acre of soybeans he plants and $3,000 for every acre he plants with barley can be modeled by P=4000x+3000y .  If x represents acres of soybeans and y represents acres of corns, which inequalities represent the possible solutions to her situation?

• x≥0
  y≥0
  x+y≥6 
  3x+2y≥15

• x≥0 
  y≥0 
  x+y≤6
  2x+3y≤15

• x≥0 
  y≥0 
  4000x+y≤6
  2x+600y≤15

• P=4000x+3000y

A farmer can plant up to 6 acres of land with soybeans and corn. Her use of a necessary pesticide is limited by federal regulations to 15 gallons for her entire 6 acres. Soybeans require 2 gallons of pesticide for every acre planted and corn requires 3 gallons per acre. The profit the farmer makes by earning $4,000 for every acre of soybeans he plants and $3,000 for every acre he plants with barley can be modeled by P=4000x+3000y . What is the maximum profit that she can earn?

• $15,000

• $21,000

• $18,000

• $25,000

A farmer can plant up to 6 acres of land with soybeans and corn. Her use of a necessary pesticide is limited by federal regulations to 15 gallons for her entire 6 acres. Soybeans require 2 gallons of pesticide for every acre planted and corn requires 3 gallons per acre. The profit the farmer makes by earning $4,000 for every acre of soybeans he plants and $3,000 for every acre he plants with barley can be modeled by P=4000x+3000y . What combination of soybeans and corn will maximize profit?

• 4 acres of soybeans and 2 acres of corn

• 6 acres of soybeans and 0 acres of corn

• 3 acres of soybeans and 3 acres of corn

• 0 acres of soybeans and 5 acres of corn

Sarah makes small purses (x) and big purses (y). She can make no more than 8 purses a week.

Which inequality represents the situation?

• x + y ≤ 8

• x + y > 8

• 2x + 3y ≤ 6

• x + y ≤ 10