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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.
    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 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
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