# 3.4 Linear Programming - PowerPoint PPT Presentation

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## 3.4 Linear Programming

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### 3.4 Linear Programming Constraints Feasible region Bounded/ unbound Vertices Feasible Region The area on the graph where all the answers of the system are graphed. – PowerPoint PPT presentation

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Title: 3.4 Linear Programming

1
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.

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.

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.

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.

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.

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.

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.

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 2y 2 - x 2y 2
• x y - 2 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 - 2 x y - 2 - 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