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Linear Programming Solving Systems of Equations with 3 Variables Inverses & Determinants of Matrices Cramer’s Rule 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 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 Testing Vertices Find the values of x and y that maximize and minimize P 3x 2 y P? 3 y x3 2 y x 7 x 0, y 0 What is the value of P at each vertex? 1. Graph the constraints 3 y 2 x3 y x 7 x 0, y 0 0,7 4,3 2. Find coordinates of each vertex 3. Evaluate P at each vertex P 3x 2 y P 3(0) 2(0) 0 P 3(2) 2(0) 6 0,0 2,0 P 3(0) 2(7) 14 P 3(4) 2(3) 18 when x=4 and y=3 P has a max value of 18 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 x y 120 30 x 60 30,90 40 y 100 60,60 P 150x 65y P 15030 6540 7100 P 15030 6590 10350 P 15060 6560 12900 P 15060 6540 11600 30,40 60,40 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. 1. Write an objective function and constraints for a linear program that models the problem. 2. 3. 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? Experie nced Teams Training Teams Total # of Teams x y x+y # of Rangers 2x y 30 # of Trainees 0 2y 16 # of trees planted 500x 200y 500x+200y Ranger Problem 1. 2. Write an objective function and constraints for a linear program that models the problem. 2 x y 30 2 y 16 x0 y 0 P 500x 200y 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? 15 experienced teams, 0 training teams 3. none 7500 trees Find a solution that uses all the trainees. How many trees will be planted in this case? 11 experienced teams, 8 training teams 7100 trees Homework due Wednesday Unit 3 Test on Tuesday 10/8 Solving Systems of Equations with 3 Variables We are going to focus on solving in two ways Solving by Elimination Solving by Substitution Elimination Ensure all variables in all equations are written in the same order Steps: 1. Pair the equations to eliminate a variable (ex: y) 2. Write the two new equations as a system and solve for final two variables (ex: x and z) 3. Substitute values for x and z into an original equation and solve for y Always write solutions as: (x,y,z) Example x 3 y 3 z 4 2 x 3 y z 15 4 x 3 y z 19 5,1,2 x 3 y 3z 4 2 x 3 y z 15 4 x 3 y z 19 2 x 3 y z 15 Practice x y 2z 3 2 x y 3z 7 x 2 y z 10 1,4,3 Substitution Choose one equation and solve for the variable 2. Substitute the expression for x into each of the other two equations 3. Write the two new equations as a system. Solve for y and x 4. Substitute the values for y and z into one of the original equations. Solve for x 1. Example x 2 y z 4 4 x y 2 z 1 2 x 2 y z 10 2,1,4 Practice x 4y z 6 2 x 5 y z 7 2x y z 1 4,1,6 Working with Matrices 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 1 0 0 0 1 0 Inverse Matrix 0 0 1 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 Verifying Inverses for 2x2 A= 2 3 1 2 3 1 2 B= 2 2 3 2 3 4 3 6 6 1 0 AB= = 1 2 1 2 2 2 3 4 0 1 Determinates for 2x2 a b Determinate of a 2x2 matrix c d is ad-bc Symbols: detA a b c d Ex: Find the determinate of 3 4 2 5 = -3*-5-(4*2) =15-8 =7 Inverse of a 2x2 Matrix a b Let A c d A-1= If det A≠0, then A has an inverse. 1 d b 1 d b det A c a ad bc c a If det A=0 then there is NOT a unique solution Ex: Determine if the matrix has an inverse. Find the inverse if it exists. 2 2 M 5 4 det M ad bc 2 4 2 5 8 10 2 Since det M does not equal 0 an inverse exists! 1 2 2 M 5 4 1 4 2 1 M det M 5 2 1 M 1 1 4 2 2 1 2 1 2 5 2 5 Systems with Matrices System of Equations x 2y 5 3x 5 y 14 Matrix equation 1 2 x 5 3 5 y 14 Coefficient matrix A Variable matrix X Constant matrix B Solving a System of Equations with Matrices Write the system as a matrix equation 1 2 x 5 3 5 y 14 2. Find A-1 1. 5 2 5 2 1 5 2 1 3 1 3 1 3 1 56 3. Solve for the variable matrix x 1 A B y x 5 2 5 y 3 1 14 x 3 y 1 Practice Problems P. 48 # 1, 4, 7, 11, 14, 17 p. 48 Check your answers!! 1 1 #1 1 / 2 1 #11 det=0 so no unique solution 1 3 #4 1 2 #14 det=-1 #17 det=-29 1 / 6 1 / 6 #7 1 / 8 1 / 8 Determinates for 3x3 Determinate of a 3x3 a1 b1 c1 a2 a3 b2 b3 c2 (a1b2 c3 a2b3c1 a3b1c2 ) (a1b3c2 a2b1c3 a3b2 c1 ) c3 On the calculator Enter the matrix 2nd => Matrix => MATH => det( => Matrix => Choose the matrix Verifying Inverses Multiply the matrices to ensure result is I If not then the two matrices are not inverses 3 4 1 A= 2 0 2 1 5 3 0 1 0 B= 1 0 1 0 1 0 3 4 1 0 1 0 AB= 2 0 2 1 0 1 = 1 5 3 0 1 0 4 4 4 AB= 0 0 0 5 4 5 0 4 0 3 0 1 0 4 0 0 0 0 2 0 2 0 0 0 0 5 0 1 0 3 0 5 0 Solving a System of Equations with Matrices 2 x y 3z 1 5 x y 2 z 8 x y 9z 5 (4, -10, 1) Practice Problems 1. 3x 5 y 0 x y 2 (5,-3) 2. x 4 z 2 y z 1 x 6 y z (5,0,1) 3. x y z 4 4x 5 y 4 y 3 z 9 (1,0,3) 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 x yz 9 2.45x 1.85 y .8 z 15 x y 2z 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 Homework P. 50 # 1, 2, 6, 9, 10, 11, 13, 14