9/8/16 In solving a system of equations, when will your answer be “no solution”? Identify the slope and y-intercept: 2𝑥+3𝑦=9 Solve for y: 3𝑥−4𝑦>12 Should the boundary in #3 be dotted or solid? Why? Is (1, -3) a solution to the inequality in #3? Why? A service club is selling copies of their holiday cookbook to raise funds for a project.  The printer’s set-up charge is $200, and each book costs $2 to print.  The cookbooks will sell for $6 each.  How many cookbooks must the members sell before they make a profit? Write an inequality that describes the cost of printing cookbooks. Write an inequality that describes the profit made by selling cookbooks.

2.2 Introduction to linear programming

What is linear programming? Situations often occur in business in which a company hopes to either maximize profit or minimize cost and many constraints need to be considered. We address these issues by creating linear inequalities and using them in linear programming.

A Few basic vocabulary words Linear programming: method for finding maximum or minimum values of a function over a given system of inequalities. Feasible region: the graph and vertices of a solution set. Objective equation: equation used to determine the minimum or maximum

Ex 1: Graph the system of inequalities Ex 1: 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 𝑃=4𝑥 −2𝑦

Ex 2: Graph the system of inequalities Ex 2: 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. −2≤𝑥≤6 1≤𝑦≤5 𝑦≤𝑥+3 𝑃=−5𝑥 +2𝑦

Steps for Linear Programming Define your variables. Identify what you would like to maximize or minimize and write your objective equation. Write a system of inequalities to describe your constraints. Find the coordinates of the vertices of the feasible region. Substitute these coordinates of the vertices into the objective equation. Select the greatest (maximum) or least (minimum) result. Answer the problem.

3. Each week, Mackenzie can make 10 – 25 necklaces and 15 – 40 pairs of earrings. If she earns profits of $3 on each pair of earrings and $5 on each necklace, and if she plans to sell at least 30 pieces of jewelry, how can she maximize profit? Define your variables. Identify what you would like to maximize or minimize and write your objective equation. Write a system of inequalities to describe your constraints. Find the coordinates of the vertices of the feasible region. Substitute these coordinates of the vertices into the objective function. Select the greatest (maximum) or least (minimum) result. Answer the problem.

4. A manufacturer of ski clothing makes ski pants and ski jackets 4. A manufacturer of ski clothing makes ski pants and ski jackets.  The profit on a pair of ski pants is $2.00 and on a jacket is $1.50.  Both pants and jackets require the work of sewing operators and cutters.  There are 60 minutes of sewing operator time and 48 minutes of cutter time available.  It takes 8 minutes to sew one pair of ski pants and 4 minutes to sew one jacket.  Cutters take 4 minutes on pants and 8 minutes on a jacket.  Find the maximum profit and the number of pants and jackets to maximize the profit. Define your variables. Write your objective equation. Write your system of inequalities. Find the coordinates of the vertices. Substitute these coordinates into the objective function. Select the max or min. Answer the problem.

5. A biologist is developing two new strains of bacteria 5. A biologist is developing two new strains of bacteria. Each sample of Type 1 bacteria produces 4 new viable bacteria and each sample of Type II produces 3 new viable bacteria. Altogether, at least 240 new viable bacteria must be produced. At least 30, but no more than 60, of the original samples must be Type 1. No more than 70 of the samples can be Type II. A sample of Type 1 costs $7 and a sample of type II costs $3. How many samples of each should be used to minimize the cost? What is the minimum cost? Define your variables. Write your objective equation. Write your system of inequalities. Find the coordinates of the vertices. Substitute these coordinates into the objective function. Select the max or min. Answer the problem.