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3.4 Linear Programming

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. This a bounded region. Unbound Region.

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3.4 Linear Programming

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  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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