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PPT – 3.4 Linear Programming PowerPoint presentation | free to download - id: 659159-MGQ1O

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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. 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.

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)

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) - 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)

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

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

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

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

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

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

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

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)

Homework

- Page 132 133
- 15, 16, 21, 26, 27

Homework

- Page 132 133
- 17, 20, 22, 23, 25