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Section 2.1 Systems of Linear Equations: an Introduction

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Section 2.1 Systems of Linear Equations: an Introduction Chapter 2 Section 2.1 Systems of Linear Equations: An Introduction Definition A system of linear equations is a collection of multiple linear equations which are meant to be solved at the same time or simultaneously. Question What does it mean to solve a system of linear equations? Answer: To solve a system of equations means to find all for the unknowns that satisfy EVERY equation. To solve a system of linear equations, where all the equations are lines, means to find every that the share. We actually saw in the last section that solving a system of only two linear equations and two unknowns (or variables) is referred to as the intersection of two lines. Before continueing with solving systems of equations, we will first discuss how to setup a system of equations from a word problem. For the following three 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? Example 3: The management of Hartman Rent-A-Car has allocated $2.43 million to buy a fleet of new automobiles consisting of compact, intermediate-size, and full-size cars. Compacts cost $18, 000 each, intermediate-size cars cost $27, 000 each, and full-size cars cost $36, 000 each. If Hartman purchases twice as many compacts as intermediate-size cars and the total number of cars to be purchased is 100, determine how many cars of each type will be purchased. (Assume that the entire budget will be used.) " # of " compact ears purchased " " # of - sized y= intermediate cars purchased " z=u of # full-sized cars purchased [email protected] " + , , , |•o ± , X = Zy → X=2y Twree as many Compacts as intermediates 2 Summer 2018, Maya Johnson Let’s return to solutions of a system of equations. Only Three Possible Outcomes for a 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 − Infinitely Many Solutions: 2x +2y =4 4x +4y =8 3 Summer 2018, Maya Johnson No Solution: 2x +3y =6 2x 3y =2 − − Example 4: Determine whether the system of linear equations has one and only one solution, infinitely many solutions, or no solution. 5 2 4 x 3 y =6 1 − 5 4 x + 3 y =12 * 8. Y= -9+158187=6 - syloxisst ⇒ ⇒ rises .4FF8 X=8,y=6(8,=f - -3¥ . -3¥ Egx ,¥× Example 5: Determine the value of k for which the system of linear equations below has no solution. 3x y =3 − 9x + ky =6 4 Summer 2018, Maya Johnson Section 2.2 Systems of Linear Equations: Unique Solutions 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 24 8 22 − 2 3 85 27 3 − 10 7 33 6 − 7 4 5 Example 1: What value is in row 1, column 2 of the above matrix? Example 2: Find the augmented matrix for the following system of equations. 9x +5y 10z =11 − 4x 12y +17z =37 − x 2y =45 − Example 3: Find the system of equations for the following augmented matrix. 10 0 6 29 − 2 30 90 31 3 − 119 12 10 6 − 7 4 5 In order to solve the system, we need to “reduce” the matrix to a form where we can readily identify the solution. 5 Summer 2018, Maya Johnson 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 1intheupperrow. 4. If a column in the coefficient matrix has a leading 1, then the other entries in the column are zeros. Example 4: Which of the matrices below are in row-reduced form? 10 6 9 10 6 2 − − 2 01 8 1 3 2 00 0 0 3 00 0 0 01 12 6 6 7 6 − 7 4 5 4 5 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. Notation for Row Operations Letting Ri denote the ith row of a matrix, we write: Operation 1. R R Interchange row i with row j. i $ j Operation 2. cRi to mean: Replace row i with c times row i. Operation 3. Ri + aRj to mean: Replace row i with the sum of row i and a times row j. Unit Column Acolumninacoefficientmatrixiscalledaunit 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. 6 Summer 2018, Maya Johnson Example 5: Pivot the matrix below about the entry in row 1, column 1 3612 9 2 221 3 3 45 2 8 6 − − 7 4 5 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. You can determine the solution from the row-reduced matrix by turning it back into a system of equations. 7 Summer 2018, Maya Johnson Example 6: 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 − 8 Summer 2018, Maya Johnson c) 2x + x x =3 1 2 − 3 3x1 +2x2 + x3 =8 x1 +2xDiskin2 +2x3 =4 . EMI Example 7: 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 8: 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. " " # of one - bedroom units ⇐ " " of two - bedroom units y= # " units " - bedroom Z= # of three + z = 168 X + t Z = 168 Xty y + +2=0 - × = × =) y ytz = 0 - 3z X =3 Z X Hoiseth :eEn units - bedroom zgthme-bedroomun€)84oue-bedro•mun'=56 two 9 Summer 2018, Maya Johnson Section 2.3 Systems of Linear Equations: Underdetermined and Overdetermined Systems Infinitely Many Solutions: If an augmented coefficient matrix is in row-reduced form and there is at least one row which consists entirely of zeros, then,in most cases ,thesystemhasinfinitely many solutions and we use parameter t and/or s to write the solution. Note: The case when this assumption is not always true is when the system is overdetermined or underdetermined. Example 1: Solve the following system of equations x +2y 3z = 2 − − 3x y 2z =1 − − 2x +3y 5z = 3 − − No solution If an augmented coefficient matrix is in row-reduced form and there is at least one row which consists entirely of zeros to the left of the vertical line and a nonzero entry to the right of the line (the very last entry on that row), then the system has no solution.
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