Section 1.3: Gauss Elimination Method for Systems of Linear Equations In this section we will consider systems of equations with possibly more than two equations, more than two variables, or both. Below is an example of three equations with four variables. 2x 3y + z 4w = 12 x 2y + 5z 9w = 33 7x + 3y 4z + 6w = 23 Before learning a new technique for solving systems of equations, we will first discuss how to setup a system of equations from a word problem. For the following two examples, we will setup but not solve the resulting system of equations. Example 1: An insurance company has three types of documents to process: contracts, leases, and policies. Each contract needs to be examined for 2 hours by the accountant and for 3 hours by the attorney, each lease needs to be examined for 4 hours by the accountant and 1 hour by the attorney, and each policy needs to be examined for 2 hours by the accountant and 2 hour by the attorney. The company processes twice as many policies as contracts and leases combined. If the accountant has 40 hours and the attorney has 30 hours each week to spend working on these documents, how many documents of each type can they process each week? Note: ALWAYS define your variables when setting up a problem
Example 2: The Johnson Farm has 500 acres of land allotted for cultivating corn and wheat. The cost of cultivating corn and wheat (including seeds and labor) is $44 and $28 per acre, respectively. Jacob Johnson has $15, 600 available for cultivating these crops. If he wishes to use all the allotted land and his entire budget for cultivating these two crops, how many acres of each crop should he plant? Three Possible Outcomes for system of Linear Equations a) The system has one and only one solution. (Unique solution) b) The system has infinitely many solutions. c) The system has no solution. Unique Solution: 3x + 3y = 6 2x + y = 2 2 Spring 2018, Maya Johnson
Infinitely Many Solutions: 2x + 2y = 4 4x + 4y = 8 No Solution: 2x + 3y = 6 2x 3y = 2 A is an ordered rectangular array of numbers. Augmented Matrices The system of equations 2x + 4y 8z = 22 3x 8y + 5z = 27 x 7z = 33 can be represented as the following augmented matrix 2 4 8 3 8 5 1 0 7 22 27 33 Example 3: What value is in row 1, column 2 of the above matrix? 3 Spring 2018, Maya Johnson
Example 4: Find the augmented matrix for the following system of equations. 9x + 5y 10z = 11 4x 12y + 17z = 37 x 2y = 45 Example 5: Find the system of equations for the following augmented matrix. 10 0 6 29 30 9 0 31 1 19 12 10 In order to solve the system, we need to reduce the matrix to a form where we can readily identify the solution. A Matrix is in Row-Reduced Form when: 1. Each row consisting entirely of zeros lies below all rows having nonzero entries 2. The first nonzero entry in each (nonzero) row is a 1 (called a leading 1). 3. In any two successive (nonzero) rows, the leading 1 in the lower row lies to the right of the leading 1 in the upper row. 4. If a column in the coefficient matrix has a leading 1, then the other entries in the column are zeros. Example 6: Which of the matrices below are in row-reduced form? 1 0 6 9 0 1 8 1 0 0 0 0 1 0 6 2 0 0 0 0 0 1 12 6 Row Operations 1. Interchange any two rows. 2. Replace any row by a nonzero constant multiple of itself. 3. Replace any row by the sum of that row and a constant multiple of any other row. 4 Spring 2018, Maya Johnson
Notation for Row Operations Letting R i denote the ith row of a matrix, we write: Operation 1. R i R j Interchange row i with row j. Operation 2. cr i to mean: Replace row i with c times row i. Operation 3. R i + ar j to mean: Replace row i with the sum of row i and a times row j. Unit Column A column in a coefficient matrix is called a unit column if one of the entries is a 1 and the other entries are zeros. Note: If you transform a column in a coefficient matrix into a unit column then this is called pivotting on that column. Example 7: Pivot the matrix below about the entry in row 1, column 1 3 6 12 9 2 2 1 3 4 5 2 8 The Gauss-Jordan Elimination Method 1. Write the augmented matrix corresponding to the Linear system. 2. Begin by transforming the entry in row 1 column 1 into a 1. This is your first pivot element. 3. Next, transform every other entry in column 1 into a zero using the (3) row operations. (Make column 1 a unit column) 4. Choose the next pivot element (usually element in row 2 column 2) 5. Transform this 2nd pivot element into a 1, and every other entry in that column into a zero. 6. Continue until the final matrix is in row-reduced form. 5 Spring 2018, Maya Johnson
You can determine the solution from the row-reduced matrix by turning it back into a system of equations. Example 8: Solve the following system of linear equations using the Gauss-Jordan elimination method. a) 2x + 6y = 1 6x + 8y = 10 From this moment on, you may use the calculator function rref to perform the gauss-jordan elimination method to put a matrix into row-reduced form, and thus solve the system of equations!!! Calculator steps for using rref can be found in a link directly under these lecture notes on the course webpage. b) 2x + 2y = 4 3x + 6y = 5 c) 2x 1 + x 2 x 3 = 3 3x 1 + 2x 2 + x 3 = 8 x 1 + 2x 2 + 2x 3 = 4 6 Spring 2018, Maya Johnson
Example 9: A person has four times as many pennies as dimes. If the total face value of these coins is $1.26, how many of each type of coin does this person have? (Use gauss-jordan ) Example 10: Cantwell Associates, a real estate developer, is planning to build a new apartment complex consisting of one-bedroom units and two- and three-bedroom townhouses. A total of 168 units is planned, and the number of family units (two- and three-bedroom townhouses) will equal the number of one-bedroom units. If the number of one-bedroom units will be 3 times the number of three-bedroom units, find how many units of each type will be in the complex. 7 Spring 2018, Maya Johnson