MATH 105: Finite Mathematics 3-1: The Inverse of a Matrix Prof. - - PowerPoint PPT Presentation

Introduction to Linear Programming Systems of Linear Inequalities Regions in the Plane Conclusion

MATH 105: Finite Mathematics 3-1: The Inverse of a Matrix

• Prof. Jonathan Duncan

Walla Walla College

Winter Quarter, 2006

Introduction to Linear Programming Systems of Linear Inequalities Regions in the Plane Conclusion

Outline

1

Introduction to Linear Programming

2

Systems of Linear Inequalities

3

Regions in the Plane

4

Conclusion

Introduction to Linear Programming Systems of Linear Inequalities Regions in the Plane Conclusion

Outline

1

Introduction to Linear Programming

2

Systems of Linear Inequalities

3

Regions in the Plane

4

Conclusion

Introduction to Linear Programming Systems of Linear Inequalities Regions in the Plane Conclusion

Systems of Linear Equations

In the last few sections, we have solving systems of linear

• equations. In doing this, we’ve been answering questions like:

Sample Questions:

1 How do we use up exactly 45 lemons and 30 oranges? 2 How many days should cog factory A, B, and C run to

produce exactly 550 basic, 725 standard, and 480 deluxe cogs?

3 How many servings of chicken, potatoes, and spinach should

be served to yield 30 calories from carbs, 25 from protein, and 15 from fat?

Introduction to Linear Programming Systems of Linear Inequalities Regions in the Plane Conclusion

Systems of Linear Equations

In the last few sections, we have solving systems of linear

• equations. In doing this, we’ve been answering questions like:

Sample Questions:

1 How do we use up exactly 45 lemons and 30 oranges? 2 How many days should cog factory A, B, and C run to

produce exactly 550 basic, 725 standard, and 480 deluxe cogs?

3 How many servings of chicken, potatoes, and spinach should

be served to yield 30 calories from carbs, 25 from protein, and 15 from fat?

Introduction to Linear Programming Systems of Linear Inequalities Regions in the Plane Conclusion

Systems of Linear Equations

In the last few sections, we have solving systems of linear

• equations. In doing this, we’ve been answering questions like:

Sample Questions:

1 How do we use up exactly 45 lemons and 30 oranges? 2 How many days should cog factory A, B, and C run to

produce exactly 550 basic, 725 standard, and 480 deluxe cogs?

3 How many servings of chicken, potatoes, and spinach should

be served to yield 30 calories from carbs, 25 from protein, and 15 from fat?

Introduction to Linear Programming Systems of Linear Inequalities Regions in the Plane Conclusion

Systems of Linear Equations

In the last few sections, we have solving systems of linear

• equations. In doing this, we’ve been answering questions like:

Sample Questions:

1 How do we use up exactly 45 lemons and 30 oranges? 2 How many days should cog factory A, B, and C run to

produce exactly 550 basic, 725 standard, and 480 deluxe cogs?

3 How many servings of chicken, potatoes, and spinach should

be served to yield 30 calories from carbs, 25 from protein, and 15 from fat?

Introduction to Linear Programming Systems of Linear Inequalities Regions in the Plane Conclusion

Systems of Linear Equations

In the last few sections, we have solving systems of linear

• equations. In doing this, we’ve been answering questions like:

Sample Questions:

1 How do we use up exactly 45 lemons and 30 oranges? 2 How many days should cog factory A, B, and C run to

produce exactly 550 basic, 725 standard, and 480 deluxe cogs?

3 How many servings of chicken, potatoes, and spinach should

be served to yield 30 calories from carbs, 25 from protein, and 15 from fat?

Introduction to Linear Programming Systems of Linear Inequalities Regions in the Plane Conclusion

Systems of Linear Equations

In the last few sections, we have solving systems of linear

• equations. In doing this, we’ve been answering questions like:

Sample Questions:

1 How do we use up exactly 45 lemons and 30 oranges? 2 How many days should cog factory A, B, and C run to

produce exactly 550 basic, 725 standard, and 480 deluxe cogs?

3 How many servings of chicken, potatoes, and spinach should

be served to yield 30 calories from carbs, 25 from protein, and 15 from fat? In each of these cases, we are focusing on finding a unique solution which uses up or produces exactly a given amount of resources. Is this realistic?

Introduction to Linear Programming Systems of Linear Inequalities Regions in the Plane Conclusion

Linear Programming

Consider making tart and sweet drinks from lemons and oranges. Suppose that the sweet drinks sold for more than the tart drinks. How many batches of each type of drink would we want to make? Example There are several approaches you could take:

1 Make only sweet drink since it sells for more. 2 Make only whichever type of drink you can make the most of. 3 Try to find the best way to divide up our fruit between both

sweet and tart to make as much money as possible.

Introduction to Linear Programming Systems of Linear Inequalities Regions in the Plane Conclusion

Linear Programming

Consider making tart and sweet drinks from lemons and oranges. Suppose that the sweet drinks sold for more than the tart drinks. How many batches of each type of drink would we want to make? Example There are several approaches you could take:

1 Make only sweet drink since it sells for more. 2 Make only whichever type of drink you can make the most of. 3 Try to find the best way to divide up our fruit between both

sweet and tart to make as much money as possible.

Introduction to Linear Programming Systems of Linear Inequalities Regions in the Plane Conclusion

Linear Programming

Consider making tart and sweet drinks from lemons and oranges. Suppose that the sweet drinks sold for more than the tart drinks. How many batches of each type of drink would we want to make? Example There are several approaches you could take:

1 Make only sweet drink since it sells for more. 2 Make only whichever type of drink you can make the most of. 3 Try to find the best way to divide up our fruit between both

sweet and tart to make as much money as possible.

Introduction to Linear Programming Systems of Linear Inequalities Regions in the Plane Conclusion

Linear Programming

Consider making tart and sweet drinks from lemons and oranges. Suppose that the sweet drinks sold for more than the tart drinks. How many batches of each type of drink would we want to make? Example There are several approaches you could take:

1 Make only sweet drink since it sells for more. 2 Make only whichever type of drink you can make the most of. 3 Try to find the best way to divide up our fruit between both

sweet and tart to make as much money as possible.

Introduction to Linear Programming Systems of Linear Inequalities Regions in the Plane Conclusion

Linear Programming

Consider making tart and sweet drinks from lemons and oranges. Suppose that the sweet drinks sold for more than the tart drinks. How many batches of each type of drink would we want to make? Example There are several approaches you could take:

1 Make only sweet drink since it sells for more. 2 Make only whichever type of drink you can make the most of. 3 Try to find the best way to divide up our fruit between both

sweet and tart to make as much money as possible.

Introduction to Linear Programming Systems of Linear Inequalities Regions in the Plane Conclusion

Linear Programming

Consider making tart and sweet drinks from lemons and oranges. Suppose that the sweet drinks sold for more than the tart drinks. How many batches of each type of drink would we want to make? Example There are several approaches you could take:

1 Make only sweet drink since it sells for more. 2 Make only whichever type of drink you can make the most of. 3 Try to find the best way to divide up our fruit between both

sweet and tart to make as much money as possible. Trying to find the distribution of resources which makes us the most profit (or the lest cost) is called linear programming (when the re- strictions are linear expressions).

Introduction to Linear Programming Systems of Linear Inequalities Regions in the Plane Conclusion

Solving Linear Programming Problems

Solving a linear programming problem involves finding and using two types of information. Parts of a Linear Programming Problem The two parts of a linear programming problem are:

1 Objective Function:

The number to be maximized or minimized.

2 Constraints:

Limits on a resource or other part of the problem.

Introduction to Linear Programming Systems of Linear Inequalities Regions in the Plane Conclusion

Solving Linear Programming Problems

Solving a linear programming problem involves finding and using two types of information. Parts of a Linear Programming Problem The two parts of a linear programming problem are:

1 Objective Function:

The number to be maximized or minimized.

2 Constraints:

Limits on a resource or other part of the problem.

Introduction to Linear Programming Systems of Linear Inequalities Regions in the Plane Conclusion

Solving Linear Programming Problems

Solving a linear programming problem involves finding and using two types of information. Parts of a Linear Programming Problem The two parts of a linear programming problem are:

1 Objective Function:

The number to be maximized or minimized.

2 Constraints:

Limits on a resource or other part of the problem.

Introduction to Linear Programming Systems of Linear Inequalities Regions in the Plane Conclusion

Solving Linear Programming Problems

Solving a linear programming problem involves finding and using two types of information. Parts of a Linear Programming Problem The two parts of a linear programming problem are:

1 Objective Function:

The number to be maximized or minimized.

2 Constraints:

Limits on a resource or other part of the problem.

Introduction to Linear Programming Systems of Linear Inequalities Regions in the Plane Conclusion

Solving Linear Programming Problems

Solving a linear programming problem involves finding and using two types of information. Parts of a Linear Programming Problem The two parts of a linear programming problem are:

1 Objective Function:

The number to be maximized or minimized.

2 Constraints:

Limits on a resource or other part of the problem. In this section we will focus on the second item. But first, let’s see an example of setting up a problem.

Introduction to Linear Programming Systems of Linear Inequalities Regions in the Plane Conclusion

A Linear Programming Problem

Example A Mexican restaurant sells two kinds of salsa: hot and mild. The main ingredients are tomatoes and jalape˜

• nos. They have 100 lbs.
• f jalape˜

nos and 400 lbs. of tomatoes on hand. It takes 1 lb. of jalape˜ nos and 10 lbs. of tomatoes to make a batch of mild salsa. It takes 3 lbs. of jalape˜ nos and 8 lbs. of tomatoes to make a batch

• f hot salsa. If they make a profit of $3.00 on each batch of mild

salsa and $7.00 on each batch of hot salsa, how much of each type should be made to maximize their profit?

Introduction to Linear Programming Systems of Linear Inequalities Regions in the Plane Conclusion

A Linear Programming Problem

Example A Mexican restaurant sells two kinds of salsa: hot and mild. The main ingredients are tomatoes and jalape˜

• nos. They have 100 lbs.
• f jalape˜

nos and 400 lbs. of tomatoes on hand. It takes 1 lb. of jalape˜ nos and 10 lbs. of tomatoes to make a batch of mild salsa. It takes 3 lbs. of jalape˜ nos and 8 lbs. of tomatoes to make a batch

• f hot salsa. If they make a profit of $3.00 on each batch of mild

salsa and $7.00 on each batch of hot salsa, how much of each type should be made to maximize their profit? Let x = batches of mild salsa y = batches of hot salsa

Introduction to Linear Programming Systems of Linear Inequalities Regions in the Plane Conclusion

A Linear Programming Problem

Example A Mexican restaurant sells two kinds of salsa: hot and mild. The main ingredients are tomatoes and jalape˜

• nos. They have 100 lbs.
• f jalape˜

nos and 400 lbs. of tomatoes on hand. It takes 1 lb. of jalape˜ nos and 10 lbs. of tomatoes to make a batch of mild salsa. It takes 3 lbs. of jalape˜ nos and 8 lbs. of tomatoes to make a batch

• f hot salsa. If they make a profit of $3.00 on each batch of mild

salsa and $7.00 on each batch of hot salsa, how much of each type should be made to maximize their profit? Let x = batches of mild salsa y = batches of hot salsa Maximize: 3x + 7y Subject to: 10x + 8y ≤ 400 x + 3y ≤ 100

Introduction to Linear Programming Systems of Linear Inequalities Regions in the Plane Conclusion

Outline

1

Introduction to Linear Programming

2

Systems of Linear Inequalities

3

Regions in the Plane

4

Conclusion

Introduction to Linear Programming Systems of Linear Inequalities Regions in the Plane Conclusion

Linear Inequalities

The last two expressions in the previous problem are examples of linear inequalities. Linear Inequalities A linear inequality is an expression of the form Ax + By C where is one of ≤, ≥, <, or >. Graphing Linear Inequalities Graph each of the linear inequalities below by: (a) graphing the associated line, and (b) shading the correct side of the line.

1 x + 2y ≤ 8 2 7x + 4y ≥ 28

Introduction to Linear Programming Systems of Linear Inequalities Regions in the Plane Conclusion

Linear Inequalities

The last two expressions in the previous problem are examples of linear inequalities. Linear Inequalities A linear inequality is an expression of the form Ax + By C where is one of ≤, ≥, <, or >. Graphing Linear Inequalities Graph each of the linear inequalities below by: (a) graphing the associated line, and (b) shading the correct side of the line.

1 x + 2y ≤ 8 2 7x + 4y ≥ 28

Introduction to Linear Programming Systems of Linear Inequalities Regions in the Plane Conclusion

Linear Inequalities

The last two expressions in the previous problem are examples of linear inequalities. Linear Inequalities A linear inequality is an expression of the form Ax + By C where is one of ≤, ≥, <, or >. Graphing Linear Inequalities Graph each of the linear inequalities below by: (a) graphing the associated line, and (b) shading the correct side of the line.

1 x + 2y ≤ 8 2 7x + 4y ≥ 28

Introduction to Linear Programming Systems of Linear Inequalities Regions in the Plane Conclusion

Systems of Inequalities

A system of linear inequalities, like a system of equations, is a collection of inequalities. A solution to the system is a point (x, y) which makes each inequality in the system true. Example If the previous two inequalities are considered a system, which of the following points are solutions to that system?

1 (2, 0) 2 (5, 6) 3 (6, −2)

Introduction to Linear Programming Systems of Linear Inequalities Regions in the Plane Conclusion

Systems of Inequalities

A system of linear inequalities, like a system of equations, is a collection of inequalities. A solution to the system is a point (x, y) which makes each inequality in the system true. Example If the previous two inequalities are considered a system, which of the following points are solutions to that system?

1 (2, 0) 2 (5, 6) 3 (6, −2)

Introduction to Linear Programming Systems of Linear Inequalities Regions in the Plane Conclusion

Systems of Inequalities

A system of linear inequalities, like a system of equations, is a collection of inequalities. A solution to the system is a point (x, y) which makes each inequality in the system true. Example If the previous two inequalities are considered a system, which of the following points are solutions to that system?

1 (2, 0) 2 (5, 6) 3 (6, −2)

Introduction to Linear Programming Systems of Linear Inequalities Regions in the Plane Conclusion

Systems of Inequalities

A system of linear inequalities, like a system of equations, is a collection of inequalities. A solution to the system is a point (x, y) which makes each inequality in the system true. Example If the previous two inequalities are considered a system, which of the following points are solutions to that system?

1 (2, 0) 2 (5, 6) 3 (6, −2)

Introduction to Linear Programming Systems of Linear Inequalities Regions in the Plane Conclusion

Systems of Inequalities

A system of linear inequalities, like a system of equations, is a collection of inequalities. A solution to the system is a point (x, y) which makes each inequality in the system true. Example If the previous two inequalities are considered a system, which of the following points are solutions to that system?

1 (2, 0) 2 (5, 6) 3 (6, −2)

Introduction to Linear Programming Systems of Linear Inequalities Regions in the Plane Conclusion

Graphing a Region in the Plane

Example Graph the region in the plane defined by the following system of linear inequalities.    x + 3y ≤ 6 x ≥ y ≥

Introduction to Linear Programming Systems of Linear Inequalities Regions in the Plane Conclusion

Graphing a Region in the Plane

Example Graph the region in the plane defined by the following system of linear inequalities.    x + 3y ≤ 6 x ≥ y ≥

Introduction to Linear Programming Systems of Linear Inequalities Regions in the Plane Conclusion

Graphing a Region in the Plane

Example Graph the region in the plane defined by the following system of linear inequalities.    x + 3y ≤ 6 x ≥ y ≥

• ✁

Introduction to Linear Programming Systems of Linear Inequalities Regions in the Plane Conclusion

Outline

1

Introduction to Linear Programming

2

Systems of Linear Inequalities

3

Regions in the Plane

4

Conclusion

Introduction to Linear Programming Systems of Linear Inequalities Regions in the Plane Conclusion

Properties of Regions in the Plane

The last two regions we have seen have a very basic difference. Bounded and Unbounded Regions A region in the plane is called bounded if it can be completely enclosed in a circle. A region which is not bounded is called unbounded. Another important part of a region in the plane is the corner points of the graph. That is, where two lines which border the region intersect each other. These can be found by solving the appropriate system of equations.

Introduction to Linear Programming Systems of Linear Inequalities Regions in the Plane Conclusion

Properties of Regions in the Plane

The last two regions we have seen have a very basic difference. Bounded and Unbounded Regions A region in the plane is called bounded if it can be completely enclosed in a circle. A region which is not bounded is called unbounded. Another important part of a region in the plane is the corner points of the graph. That is, where two lines which border the region intersect each other. These can be found by solving the appropriate system of equations.

Introduction to Linear Programming Systems of Linear Inequalities Regions in the Plane Conclusion

Properties of Regions in the Plane

The last two regions we have seen have a very basic difference. Bounded and Unbounded Regions A region in the plane is called bounded if it can be completely enclosed in a circle. A region which is not bounded is called unbounded. Another important part of a region in the plane is the corner points of the graph. That is, where two lines which border the region intersect each other. These can be found by solving the appropriate system of equations.

Introduction to Linear Programming Systems of Linear Inequalities Regions in the Plane Conclusion

An Example

Example Graph the system of inequalities, find the corner points of the region defined, and determine if the region is bounded or unbounded.

           x + y ≥ 2 2x + 3y ≤ 12 3x + y ≤ 12 x ≥ y ≥

Introduction to Linear Programming Systems of Linear Inequalities Regions in the Plane Conclusion

An Example

Example Graph the system of inequalities, find the corner points of the region defined, and determine if the region is bounded or unbounded.

           x + y ≥ 2 2x + 3y ≤ 12 3x + y ≤ 12 x ≥ y ≥ Lines Involved Intersection x = 0 and 2x + 3y = 12 (0, 4) x = 0 and x + y = 2 (0, 2) y = 0 and x + y = 2 (2, 0) y = 0 and 3x + y = 12 (4, 0) 2x + 3y = 12 and 3x + y = 12 ( 24

7 , 12 7 )

Introduction to Linear Programming Systems of Linear Inequalities Regions in the Plane Conclusion

An Example

Example Graph the system of inequalities, find the corner points of the region defined, and determine if the region is bounded or unbounded.

           x + y ≥ 2 2x + 3y ≤ 12 3x + y ≤ 12 x ≥ y ≥ Lines Involved Intersection x = 0 and 2x + 3y = 12 (0, 4) x = 0 and x + y = 2 (0, 2) y = 0 and x + y = 2 (2, 0) y = 0 and 3x + y = 12 (4, 0) 2x + 3y = 12 and 3x + y = 12 ( 24

7 , 12 7 )

The region is bounded as it can be enclosed in a circle.

Introduction to Linear Programming Systems of Linear Inequalities Regions in the Plane Conclusion

Outline

1

Introduction to Linear Programming

2

Systems of Linear Inequalities

3

Regions in the Plane

4

Conclusion

Introduction to Linear Programming Systems of Linear Inequalities Regions in the Plane Conclusion

Important Concepts

Things to Remember from Section 3.1

1 Graphing Linear Inequalities 2 Graphing a System of Linear Inequalities 3 Locating Corner Points in a Region Defined by Linear

Inequalities

4 Identifying Regions as Bounded or Unbounded

Introduction to Linear Programming Systems of Linear Inequalities Regions in the Plane Conclusion

Important Concepts

Things to Remember from Section 3.1

1 Graphing Linear Inequalities 2 Graphing a System of Linear Inequalities 3 Locating Corner Points in a Region Defined by Linear

Inequalities

4 Identifying Regions as Bounded or Unbounded

Introduction to Linear Programming Systems of Linear Inequalities Regions in the Plane Conclusion

Important Concepts

Things to Remember from Section 3.1

1 Graphing Linear Inequalities 2 Graphing a System of Linear Inequalities 3 Locating Corner Points in a Region Defined by Linear

Inequalities

4 Identifying Regions as Bounded or Unbounded

Introduction to Linear Programming Systems of Linear Inequalities Regions in the Plane Conclusion

Important Concepts

Things to Remember from Section 3.1

1 Graphing Linear Inequalities 2 Graphing a System of Linear Inequalities 3 Locating Corner Points in a Region Defined by Linear

Inequalities

4 Identifying Regions as Bounded or Unbounded

Introduction to Linear Programming Systems of Linear Inequalities Regions in the Plane Conclusion

Important Concepts

Things to Remember from Section 3.1

1 Graphing Linear Inequalities 2 Graphing a System of Linear Inequalities 3 Locating Corner Points in a Region Defined by Linear

Inequalities

4 Identifying Regions as Bounded or Unbounded

Introduction to Linear Programming Systems of Linear Inequalities Regions in the Plane Conclusion

Next Time. . .

In the next section we will practice setting up linear programming

• problems. These problems ask us to find the “best” solution to a

problem with constraints. The constraints are linear inequalities and the “best” solution is given by a linear expression. For next time Read Section 3.2 Review for Quiz

Introduction to Linear Programming Systems of Linear Inequalities Regions in the Plane Conclusion

Next Time. . .

In the next section we will practice setting up linear programming

• problems. These problems ask us to find the “best” solution to a

problem with constraints. The constraints are linear inequalities and the “best” solution is given by a linear expression. For next time Read Section 3.2 Review for Quiz