Turn in your interims

1. Turn in your interims

2. Unit 3 Linear Programming Solving Systems of Equations with 3 Variables Inverses & Determinants of Matrices Cramer’s Rule

3. Linear Programming • What is it? • Technique that identifies the minimum or maximum value of a quantity • Objective function • Like the “parent function” • Constrains (restrictions) • Limits on the variables • Written as inequalities • What is the name of the region where our possible solutions lie? • Feasible region • Contains all of the points which satisfy the constraints

4. Vertex Principle of Linear Programming • If there is a max or a min value of the linear objective function, it occurs at one or more vertices of the feasible region

5. Testing Vertices • Find the values of x and y that maximize and minimize P? • What is the value of P at each vertex?

6. 1. Graph the constraints 2. Find coordinates of each vertex 3. Evaluate P at each vertex when x=4 and y=3 P has a max value of 18

7. Furniture Manufacturing • A furniture manufacturer can make from 30 to 60 tables a day and from 40 to 100 chairs a day. It can make at most 120 units in one day. The profit on a table is $150, and the profit on a chair is $65. How many tables and chairs should they make per day to maximize profit? How much is the maximum profit? • Define our variables: • X: number of tables • Y: number of chairs

9. Practice Problem • Teams chosen from 30 forest rangers and 16 trainees are planting trees. An experienced team consisting of two rangers can plant 500 trees per week. A training team consisting of one ranger and two trainees can plant 200 trees per week. • Write an objective function and constraints for a linear program that models the problem. • How many of each type of team should be formed to maximize the number of trees planted? How many trainees are used in this solution? How many trees are planted? • Find a solution that uses all the trainees. How many trees will be planted in this case?

10. Ranger Problem • Write an objective function and constraints for a linear program that models the problem. • How many of each type of team should be formed to maximize the number of trees planted? How many trainees are used in this solution? How many trees are planted? • Find a solution that uses all the trainees. How many trees will be planted in this case? 15 experienced teams, 0 training teams none 7500 trees 11 experienced teams, 8 training teams 7100 trees

11. Announcements Homework due Wednesday Unit 3 Test on Tuesday 10/8

12. Solving Systems of Equations with 3 Variables • We are going to focus on solving in two ways • Solving by Elimination • Solving by Substitution

13. Elimination • Ensure all variables in all equations are written in the same order • Steps: • Pair the equations to eliminate a variable (ex: y) • Write the two new equations as a system and solve for final two variables (ex: x and z) • Substitute values for x and z into an original equation and solve for y • Always write solutions as: (x,y,z)

14. Example

15. Practice

16. Substitution • Choose one equation and solve for the variable • Substitute the expression for x into each of the other two equations • Write the two new equations as a system. Solve for y and x • Substitute the values for y and z into one of the original equations. Solve for x

17. Example

18. Practice

19. Unit 4 Working with Matrices

20. Inverses and Determinates (2x2) • Square matrix • Same number of rows and columns • Identity Matrix (I) • Square matrix with 1’s along the main diagonal and 0’s everywhere else • Inverse Matrix • AA-1=I • If B is the multiplicative inverse of A then A is the inverse of B • To show they are inverses AB=I

21. Verifying Inverses for 2x2 • A= B= AB= =

22. Determinates for 2x2 • Determinate of a 2x2 matrix is ad-bc • Symbols: detA • Ex: Find the determinate of = -3*-5-(4*2) =15-8 =7

23. Inverse of a 2x2 Matrix • Let If det A≠0, then A has an inverse. • A-1= If det A=0 then there is NOT a unique solution

24. Ex: Determine if the matrix has an inverse. Find the inverse if it exists. Since det M does not equal 0 an inverse exists!

25. Systems with Matrices • System of Equations Matrix equation Coefficient matrix A Constant matrix B Variable matrix X

26. Solving a System of Equations with Matrices • Write the system as a matrix equation • Find A-1 • Solve for the variable matrix

27. Practice Problems • P. 48 # 1, 4, 7, 11, 14, 17

28. p. 48 Check your answers!! #11 det=0 so no unique solution #1 #14 det=-1 #4 #17 det=-29 #7

29. Determinates for 3x3 • Determinate of a 3x3 • On the calculator • Enter the matrix • 2nd => Matrix => MATH => det( => Matrix => Choose the matrix

30. Verifying Inverses • Multiply the matrices to ensure result is I • If not then the two matrices are not inverses • A= B= AB= = AB=

31. Solving a System of Equations with Matrices (4, -10, 1)

32. Practice Problems 2. 3. (5,-3) (5,0,1) (1,0,3)

33. Practice Solving Systems with Matrices • Suppose you want to fill nine 1-lb tins with a snack mix. You plan to buy almonds for $2.45/lb, peanuts for $1.85/lb, and raisins for $.80/lb. You have $15 and want the mix to contain twice as much of the nuts as of the raisins by weight. How much of each ingredient should you buy? • Let x represent almonds • Let y represent peanuts • Let z represent raisins

34. Calculator How To!! • To input a matrix: • 2nd, Matrix, Edit • Be sure to define the size of your matrix!! • To find the inverse of a matrix • 2nd, Matrix, 1, x-1, enter

35. Homework • P. 50 # 1, 2, 6, 9, 10, 11, 13, 14