1. Constraints Feasible region Bounded/ unbound Vertices
3.4 Linear Programming
Constraints
Feasible region
Bounded/ unbound
Vertices

2. Feasible Region
The area on the graph where all the answers of the system are graphed. This a bounded region.

3. Unbound Region
The area on the graph where all the answers of the system are graphed. This a unbounded region. It goes beyond the graph

4. Vertices of the region Vertices are the points where the lines meet.
We need them for Linear Programming.

5. After we have found the vertices
We place the x and y value a given function.
We are trying to find the maximum or minimum of the function,
written as f( x, y) =

6. The vertices come the system of equations called constraint.
For this problem
Given the constraints.
Here we find where the equations intersect by elimination or substitution.

7. Finding the vertices given the constraints
Take two the equations and find where they intersect.
x ≤ 5 and y ≤ 4 would be (5, 4)
x ≤ 5 and x + y ≥ 2, would be 5 + y ≥ 2
y = - 3
So the intersect is (5, - 3)
y ≤ 4 and x + y ≥ 2. would be x + 4 ≥ 2
x = - 2
So its intersects is (- 2, 4)

8. Where is the feasible region?

9. Where is the feasible region?

10. To find the Maximum or Minimum we f( x, y) using the vertices
f( x, y) = 3x – 2y
( -2, 4) = 3(- 2) – 2(4) = - 14
( 5, 4) = 3(5) – 2(4) = 7
(5, - 3) = 3(5) – 2( - 3) = 21

11. To find the Maximum or Minimum we f( x, y) using the vertices
f( x, y) = 3x – 2y
( -2, 4) = 3(- 2) – 2(4) = - 14
Min. of – 14 at ( - 2,4)
( 5, 4) = 3(5) – 2(4) = 7
(5, - 3) = 3(5) – 2( - 3) = 21
Max. of 21 at ( 5, - 3)

12. Key concept Step 1 Define the variables
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 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

13. Key concept Step 1 Define the variables
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 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. Key concept Step 1 Define the variables
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 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

15. Key concept Step 1 Define the variables
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 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

16. Key concept Step 1 Define the variables
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 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

17. Key concept Step 1 Define the variables
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 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

18. Key concept Step 1 Define the variables
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 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

19. Find the maximum and minimum values of the functions
f( x, y) = 2x + 3y
Constraints
-x + 2y ≤ 2
x – 2y ≤ 4
x + y ≥ - 2

20. Find the vertices -x + 2y ≤ 2 - x + 2y = 2 x – 2y ≤ 4 x – 2y = 4
0 = 0 Must not intersect
x + y ≥ x + y = - 2
3y = 0
y = 0 x + 0 = - 2
Must intersect at ( - 2, 0)

21. x – 2y ≤ 4 x – 2y = 4 x – 2y = 4
x + y ≥ x + y = x - y = 2
- 3y = 6
y = - 2
X + ( -2) = - 2 x = 0 (0, - 2)
The vertices are ( - 2,0) and (0,- 2)

22. Off the
Graph.
No Max.

23. Find the maximum and minimum values of the functions
f( x, y) = 2x + 3y
f( - 2, 0) = 2( - 2) + 3(0) = - 4
f( 0, - 2) = 2( 0) + 3( - 2) = - 6
Minimum - 6 at (0, - 2)

24. Homework
Page 132 – 133
# 15, 16, 21, 26, 27

25. Homework
Page 132 – 133
# 17, 20, 22, 23, 25