3.4 Linear ProgrammingConstraints

Feasible regionBounded/ unbound

Vertices

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.

245

yxyx

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 conceptStep 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 conceptStep 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 conceptStep 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 conceptStep 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 conceptStep 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 conceptStep 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 conceptStep 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 + 3yConstraints-x + 2y ≤ 2x – 2y ≤ 4x + y ≥ - 2

Find the vertices

-x + 2y ≤ 2 - x + 2y = 2x – 2y ≤ 4 x – 2y = 4

0 = 0 Must not intersect-x + 2y ≤ 2 - x + 2y = 2x + y ≥ - 2 x + y = - 2

3y = 0y = 0 x + 0 = - 2Must intersect at ( - 2, 0)

x – 2y ≤ 4 x – 2y = 4 x – 2y = 4x + 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) = - 6Minimum - 6 at (0, - 2)

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