3.4 Linear Programming
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Transcript of 3.4 Linear Programming
3.4 Linear Programming
ConstraintsFeasible region
Bounded/ unboundVertices
Feasible Region
The area on the graph where all the answers of the system are graphed. This a bounded region.
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
Vertices of the region
Vertices are the points where the lines meet.
We need them for Linear Programming.
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) =
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.
2
4
5
yx
y
x
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 = - 3So the intersect is (5, - 3)
y ≤ 4 and x + y ≥ 2. would be x + 4 ≥ 2x = - 2
So its intersects is (- 2, 4)
Where is the feasible region?
Where is the feasible region?
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
To find the Maximum or Minimum we f( x, y) using the vertices
f( x, y) = 3x – 2y
( -2, 4) = 3(- 2) – 2(4) = - 14Min. of – 14 at ( - 2,4)
( 5, 4) = 3(5) – 2(4) = 7
(5, - 3) = 3(5) – 2( - 3) = 21Max. of 21 at ( 5, - 3)
Key concept
Step 1 Define the variablesStep 2 Write a system of inequalitiesStep 3 Graph the system of inequalitiesStep 4 Find the coordinates of the
vertices of the feasible regionStep 5 Write a function to be maximized
or minimizedStep 6 Substitute the coordinates of the
vertices into the functionStep 7 Select the greatest or least result.
Answer the problem
Key concept
Step 1 Define the variablesStep 2 Write a system of inequalitiesStep 3 Graph the system of inequalitiesStep 4 Find the coordinates of the
vertices of the feasible regionStep 5 Write a function to be maximized
or minimizedStep 6 Substitute the coordinates of the
vertices into the functionStep 7 Select the greatest or least result.
Answer the problem
Key concept
Step 1 Define the variablesStep 2 Write a system of inequalitiesStep 3 Graph the system of inequalitiesStep 4 Find the coordinates of the
vertices of the feasible regionStep 5 Write a function to be maximized
or minimizedStep 6 Substitute the coordinates of the
vertices into the functionStep 7 Select the greatest or least result.
Answer the problem
Key concept
Step 1 Define the variablesStep 2 Write a system of inequalitiesStep 3 Graph the system of inequalitiesStep 4 Find the coordinates of the
vertices of the feasible regionStep 5 Write a function to be maximized
or minimizedStep 6 Substitute the coordinates of the
vertices into the functionStep 7 Select the greatest or least result.
Answer the problem
Key concept
Step 1 Define the variablesStep 2 Write a system of inequalitiesStep 3 Graph the system of inequalitiesStep 4 Find the coordinates of the
vertices of the feasible regionStep 5 Write a function to be maximized
or minimizedStep 6 Substitute the coordinates of the
vertices into the functionStep 7 Select the greatest or least result.
Answer the problem
Key concept
Step 1 Define the variablesStep 2 Write a system of inequalitiesStep 3 Graph the system of inequalitiesStep 4 Find the coordinates of the
vertices of the feasible regionStep 5 Write a function to be maximized
or minimizedStep 6 Substitute the coordinates of the
vertices into the functionStep 7 Select the greatest or least result.
Answer the problem
Key concept
Step 1 Define the variablesStep 2 Write a system of inequalitiesStep 3 Graph the system of inequalitiesStep 4 Find the coordinates of the
vertices of the feasible regionStep 5 Write a function to be maximized
or minimizedStep 6 Substitute the coordinates of the
vertices into the functionStep 7 Select the greatest or least result.
Answer the problem
Find the maximum and minimum values of the functions
f( x, y) = 2x + 3y
Constraints
-x + 2y ≤ 2
x – 2y ≤ 4
x + y ≥ - 2
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)
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)
Off the
Graph.
No Max.
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)
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