1. Get out your Vertices Worksheet!
You will be able to find the maximum and minimum values of a function over a region.
You will be able to solve real-world optimization problems using linear programming.
Today’s Objectives:

2. Warm Up
Graph the following system of inequalities. Find the coordinates of the THREE vertices.
𝑦 ≤ 3 – 𝑥
𝑦 ≥ 3
𝑥 ≥ –5
Solve the system of equations.
2. by Substitution 3. by Elimination
𝑦=𝑥− 𝑦+𝑥=6
4𝑦=−𝑥 𝑦−2𝑥=3
Jackie has a total of 50 nickels and pennies. She has 13 more pennies than nickels. How many of each does she have?

3. 3.3 Linear Programming New Vocabulary: Linear Programming-
Feasible Region-
Bounded-
Unbounded-
Optimize-

4. 3.3 Linear Programming
Linear Programming:

5. 3.3 Linear Programming
Feasible Region:

6. Bounded vs. Unbounded
The feasible region is enclosed, or bounded, by the constraints.
The maximum and minimum always occur at a vertex.

7. The feasible region is open, or unbounded, and can go on forever.
Bounded vs. Unbounded
The feasible region is open, or unbounded, and can go on forever.

8. Bounded Example 𝑦≥−3 𝑥≥−2 𝑦≤− 5 2 𝑥+7 𝑓 𝑥, 𝑦 =2𝑥−5𝑦
Graph the following system of inequalities. Name the coordinates and find the vertices of the feasible region. Find the maximum and minimum values of this region.
𝑦≥−3
𝑥≥−2
𝑦≤− 5 2 𝑥+7
𝑓 𝑥, 𝑦 =2𝑥−5𝑦
(𝒙, 𝒚)
𝟐𝒙−𝟓𝒚
𝒇(𝒙, 𝒚)

9. Unbounded Example 𝑦+3𝑥≤4 𝑦≥−3𝑥−4 𝑦≤8+𝑥 𝑓 𝑥, 𝑦 =4𝑦−3𝑥
Graph the following system of inequalities. Name the coordinates and find the vertices of the feasible region. Find the maximum and minimum values of this region.
𝑦+3𝑥≤4
𝑦≥−3𝑥−4
𝑦≤8+𝑥
𝑓 𝑥, 𝑦 =4𝑦−3𝑥
(𝒙, 𝒚)
𝟒𝐲−𝟑𝐱
𝒇(𝒙, 𝒚)

10. Examples 𝑑 ≥ 2 1 ≤𝑐 ≤ 5 𝑦 ≤𝑐 + 3 𝒅 𝑓(𝑐, 𝑑) = 3𝑐 – 2𝑑 𝒄
Graph the following system of inequalities. Name the coordinates and find the vertices of the feasible region. Find the maximum and minimum values of this region.
𝑑 ≥ 2
1 ≤𝑐 ≤ 5
𝑦 ≤𝑐 + 3
𝑓(𝑐, 𝑑) = 3𝑐 – 2𝑑 𝒅 𝒄

11. You Try One! 𝑦 ≥ –2 𝑦 ≥ 2𝑥 – 4 𝑥 – 2𝑦 ≥ –1 𝑓(𝑥, 𝑦) = 4𝑥 – 𝑦
Graph the following system of inequalities. Name the coordinates and find the vertices of the feasible region. Find the maximum and minimum values of this region.
𝑦 ≥ –2
𝑦 ≥ 2𝑥 – 4
𝑥 – 2𝑦 ≥ –1
𝑓(𝑥, 𝑦) = 4𝑥 – 𝑦

12. Linear Programming—Optimization
Optimize:

13. 7 Steps of Optimization Step 1: Define the variable
Step 2: Write a system of inequalities
Step 3: Graph the system of inequalities
Step 4: Find the coordinates of the vertices of the feasible region.
Step 5: Write a linear function to be maximized or minimized.
Step 6: Substitute the coordinates of the vertices into the function.
Step 7: Select the greatest or least result. Answer the problem.

14. Linear Programming Worksheet
Each week, Mackenzie can make 10 to 25 necklaces and 15 to 40 pairs of earrings. If she earns profits of $3 on each pair of earrings and $5 on each necklace, and she plan to sell at least 30 pieces of jewelry, how can she maximize profit?

15. Ticket Out The Door
Work through Ingrid’s optimization problem on the worksheet!
Turn in your worksheet on the back table.

16. HOMEWORK
Lesson 3.3
Pg.
#’s 14, 16, 17, 20, 23, and 24