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?

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

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

Maximize P = 3x + y with the following constraints:

$$x+y\ \le300$$

$$y\ \ge100$$  

$$x\ \le150$$  

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

Which corner point will maximize the Objective Function
$$Z\ =\ 5x\ +\ 2y$$  

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?

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?

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?

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

Which inequality represents the situation?