1. Section 3.4 - Linear Programming
ALGEBRA TWO
CHAPTER THREE: SYSTEMS OF LINEAR EQUATIONS AND INEQUALITIES
Section Linear Programming

2. LEARNING GOALS Goal One - Solve linear programming problems.
Goal Two - Use linear programming to solve real-life problems.

3. VOCABULARY
Optimization means finding the maximum or minimum value of some quantity.
Linear programming is the process of optimizing a linear objective function subject to a system of linear inequalities called constraints. 1

4. VOCABULARY
The graph of the system of constraints is called the feasible region. If an objective function has a maximum or a minimum value, then it must occur at a vertex of the feasible region. Moreover, the objective function will have both a maximum and a minimum value if the feasible region is bounded. 1

5. Solving a Linear Programming Problem
Find the minimum and maximum values of the objective function C = 3x + 2y subject to the following constraints
3x + 4y < 20 3x - y < 5
x > 0 y > 0 √
STEP 1: Graph all the inequalities.
3x + 4y < 20
4y < -3x + 20
y < (-3/4)x + 5 1

6. Solving a Linear Programming Problem
Find the minimum and maximum values of the objective function C = 3x + 2y subject to the following constraints
3x + 4y < 20 3x - y < 5
x > 0 y > 0 √ √
STEP 1: Graph all the inequalities.
3x - y < 5
-y < -3x + 5
y > 3x - 5 1

7. Solving a Linear Programming Problem
Find the minimum and maximum values of the objective function C = 3x + 2y subject to the following constraints
3x + 4y < 20 3x - y < 5
x > 0 y > 0 √ √ √
STEP 1: Graph all the inequalities.
x > 0 1

8. Solving a Linear Programming Problem
Find the minimum and maximum values of the objective function C = 3x + 2y subject to the following constraints
3x + 4y < 20 3x - y < 5
x > 0 y > 0 √ √ √ √
STEP 1: Graph all the inequalities.
y > 0 1

9. Solving a Linear Programming Problem
Find the minimum and maximum values of the objective function C = 3x + 2y subject to the following constraints
3x + 4y < 20 3x - y < 5
x > 0 y > 0 √ √ √ √
STEP 1: Graph all the inequalities.
Now you must color in the area where the shading overlaps. 1

10. Solving a Linear Programming Problem
Find the minimum and maximum values of the objective function C = 3x + 2y subject to the following constraints
3x + 4y < 20 3x - y < 5
x > 0 y > 0 √ √ √ √
STEP 1: Graph all the inequalities.
Identify the ordered pairs at each vertex.
(0, 5)
(8/3, 3)
(0, 0)
(5/3, 0) 1

11. Solving a Linear Programming Problem
Find the minimum and maximum values of the objective function C = 3x + 2y subject to the following constraints
3x + 4y < 20 3x - y < 5
x > 0 y > 0 √ √ √ √
The feasible region determined by the constraints is shown. The three vertices (0, 5), (0, 0), and (5/3, 0) are intercepts. The fourth vertex (8/3, 3) is found by solving the intersecting lines. 1

12. Solving a Linear Programming Problem
Find the minimum and maximum values of the objective function C = 3x + 2y subject to the following constraints
3x + 4y < 20 3x - y < 5
x > 0 y > 0 √ √ √ √
To find the minimum and maximum values of C, evaluate C = 3x + 2y at each of the four vertices. 1

13. Solving a Linear Programming Problem
Find the minimum and maximum values of the objective function C = 3x + 2y subject to the following constraints
3x + 4y < 20 3x - y < 5
x > 0 y > 0 √ √ √ √
C = 3x + 2y
C = 3(0) + 2(5) = 10
C = 3(0) + 2(0) = 0
Minimum
C = 3(5/3) + 2(0) = 5
C = 3(8/3) + 2(3) = 14
Maximum
Minimum = 0; Maximum = 14 1

14. ASSIGNMENT
READ & STUDY: pg
WRITE: pg #13, #15, #17, #19, #21, #37, #39