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2 Publisher: Richard Stratton Senior Sponsoring Editor: Cath Cantin Senior Marketing Manager: Jennifer Jones Discipline Product Manager: Gretchen Rice King Associate Editor: Janine Tangne Associate Editor: Jeannine Lawless Senior Project Editor: Kerr Falve Program Manager: Touraj Zadeh Senior Media Producer: Douglas Winicki Senior Content Manager: Maren Kunert Art and Design Manager: Jill Haber Cover Design Manager: Anne S. Katzeff Senior Photo Editor: Jennifer Meer Dare Senior Composition Buer: Chuck Dutton New Title Project Manager: Susan Peltier Manager of New Title Project Management: Pat O Neill Editorial Assistant: Am Haines Marketing Assistant: Michael Moore Editorial Assistant: Laura Collins Cover image: Carl Reader/age fotostock Copright 9 b Houghton Mifflin Harcourt Publishing Compan. All rights reserved. No part of this work ma be reproduced or transmitted in an form or b an means, electronic or mechanical, including photocoping and recording, or b an information storage or retrieval sstem without the prior written permission of Houghton Mifflin Harcourt Publishing Compan unless such coping is epressl permitted b federal copright law. Address inquiries to College Permissions, Houghton Mifflin Harcourt Publishing Compan, Berkele Street, Boston, MA Printed in the U.S.A. Librar of Congress Control Number: 797 Instructor s eamination cop ISBN-: ISBN-: For orders, use student tet ISBNs ISBN-: ISBN-: DOC- 9 8

3 Sstems of Linear Equations. Introduction to Sstems of Linear Equations. Gaussian Elimination and Gauss-Jordan Elimination. Applications of Sstems of Linear Equations CHAPTER OBJECTIVES Recognize, graph, and solve a sstem of linear equations in n variables. Use back-substitution to solve a sstem of linear equations. Determine whether a sstem of linear equations is consistent or inconsistent. Determine if a matri is in row-echelon form or reduced row-echelon form. Use elementar row operations with back-substitution to solve a sstem in row-echelon form. Use elimination to rewrite a sstem in row-echelon form. Write an augmented or coefficient matri from a sstem of linear equations, or translate a matri into a sstem of linear equations. Solve a sstem of linear equations using Gaussian elimination and Gaussian elimination with back-substitution. Solve a homogeneous sstem of linear equations. Set up and solve a sstem of equations to fit a polnomial function to a set of data points, as well as to represent a network.. Introduction to Sstems of Linear Equations HISTORICAL NOTE Carl Friedrich Gauss (777 8) is often ranked along with Archimedes and Newton as one of the greatest mathematicians in histor. To read about his contributions to linear algebra, visit college.hmco.com/pic/larsonela6e. Linear algebra is a branch of mathematics rich in theor and applications. This tet strikes a balance between the theoretical and the practical. Because linear algebra arose from the stud of sstems of linear equations, ou shall begin with linear equations. Although some material in this first chapter will be familiar to ou, it is suggested that ou carefull stud the methods presented here. Doing so will cultivate and clarif our intuition for the more abstract material that follows. The stud of linear algebra demands familiarit with algebra, analtic geometr, and trigonometr. Occasionall ou will find eamples and eercises requiring a knowledge of calculus; these are clearl marked in the tet. Earl in our stud of linear algebra ou will discover that man of the solution methods involve dozens of arithmetic steps, so it is essential to strive to avoid careless errors. A computer or calculator can be ver useful in checking our work, as well as in performing man of the routine computations in linear algebra.

4 Chapter Sstems of Linear Equations Linear Equations in n Variables Recall from analtic geometr that the equation of a line in two-dimensional space has the form a a b, a, a, and b are constants. This is a linear equation in two variables and. Similarl, the equation of a plane in three-dimensional space has the form a a a z b, a, a, a, and b are constants. Such an equation is called a linear equation in three variables,, and z. In general, a linear equation in n variables is defined as follows. Definition of a Linear Equation in n Variables A linear equation in n variables,,,..., n has the form a a a... a n n b. The coefficients a, a, a,..., a n are real numbers, and the constant term b is a real number. The number is the leading coefficient, and is the leading variable. a REMARK: Letters that occur earl in the alphabet are used to represent constants, and letters that occur late in the alphabet are used to represent variables. Linear equations have no products or roots of variables and no variables involved in trigonometric, eponential, or logarithmic functions. Variables appear onl to the first power. Eample lists some equations that are linear and some that are not linear. EXAMPLE Eamples of Linear Equations and Nonlinear Equations Each equation is linear. (a) 7 (c) Each equation is not linear. (a) z (c) sin (b) z (d) sin (b) e (d) e A solution of a linear equation in n variables is a sequence of n real numbers s, s, s,..., s n arranged so the equation is satisfied when the values s, s, s,..., n s n

5 Section. Introduction to Sstems of Linear Equations are substituted into the equation. For eample, the equation is satisfied when and. Some other solutions are and, and, and and. The set of all solutions of a linear equation is called its solution set, and when this set is found, the equation is said to have been solved. To describe the entire solution set of a linear equation, a parametric representation is often used, as illustrated in Eamples and. EXAMPLE SOLUTION Parametric Representation of a Solution Set Solve the linear equation. To find the solution set of an equation involving two variables, solve for one of the variables in terms of the other variable. If ou solve for in terms of, ou obtain. In this form, the variable is free, which means that it can take on an real value. The variable is not free because its value depends on the value assigned to. To represent the infinite number of solutions of this equation, it is convenient to introduce a third variable t called a parameter. B letting t, ou can represent the solution set as t, t, t is an real number. Particular solutions can be obtained b assigning values to the parameter t. For instance, t ields the solution and, and t ields the solution and. The solution set of a linear equation can be represented parametricall in more than one wa. In Eample ou could have chosen to be the free variable. The parametric representation of the solution set would then have taken the form s, s, s is an real number. For convenience, choose the variables that occur last in a given equation to be free variables. EXAMPLE SOLUTION Parametric Representation of a Solution Set Solve the linear equation z. Choosing and z to be the free variables, begin b solving for to obtain z z. Letting s and z t, ou obtain the parametric representation s t, s, z t

6 Chapter Sstems of Linear Equations where s and t are an real numbers. Two particular solutions are,, z and,, z. Sstems of Linear Equations A sstem of m linear equations in n variables is a set of m equations, each of which is linear in the same n variables: a a a... a n n b a a a... a n n b a a a... a n n b. a m a m a m... a mn n b m. REMARK: The double-subscript notation indicates a ij is the coefficient of j in the ith equation. A solution of a sstem of linear equations is a sequence of numbers s, s, s,..., s n that is a solution of each of the linear equations in the sstem. For eample, the sstem has and as a solution because both equations are satisfied when and. On the other hand, and is not a solution of the sstem because these values satisf onl the first equation in the sstem. Discover Graph the two lines in the -plane. Where do the intersect? How man solutions does this sstem of linear equations have? Repeat this analsis for the pairs of lines 6. In general, what basic tpes of solution sets are possible for a sstem of two equations in two unknowns?

7 Section. Introduction to Sstems of Linear Equations It is possible for a sstem of linear equations to have eactl one solution, an infinite number of solutions, or no solution. A sstem of linear equations is called consistent if it has at least one solution and inconsistent if it has no solution. EXAMPLE Sstems of Two Equations in Two Variables Solve each sstem of linear equations, and graph each sstem as a pair of straight lines. (a) (b) (c) SOLUTION (a) This sstem has eactl one solution, and. The solution can be obtained b adding the two equations to give, which implies and so. The graph of this sstem is represented b two intersecting lines, as shown in Figure.(a). (b) This sstem has an infinite number of solutions because the second equation is the result of multipling both sides of the first equation b. A parametric representation of the solution set is shown as t, t, 6 t is an real number. The graph of this sstem is represented b two coincident lines, as shown in Figure.(b). (c) This sstem has no solution because it is impossible for the sum of two numbers to be and simultaneousl. The graph of this sstem is represented b two parallel lines, as shown in Figure.(c). (a) Two intersecting lines: (b) Two coincident lines: (c) Two parallel lines: 6 Figure. Eample illustrates the three basic tpes of solution sets that are possible for a sstem of linear equations. This result is stated here without proof. (The proof is provided later in Theorem..)

8 6 Chapter Sstems of Linear Equations Number of Solutions of a Sstem of Linear Equations For a sstem of linear equations in n variables, precisel one of the following is true.. The sstem has eactl one solution (consistent sstem).. The sstem has an infinite number of solutions (consistent sstem).. The sstem has no solution (inconsistent sstem). Solving a Sstem of Linear Equations Which sstem is easier to solve algebraicall? z 9 z 7 z 9 z z The sstem on the right is clearl easier to solve. This sstem is in row-echelon form, which means that it follows a stair-step pattern and has leading coefficients of. To solve such a sstem, use a procedure called back-substitution. EXAMPLE Using Back-Substitution to Solve a Sstem in Row-Echelon Form Use back-substitution to solve the sstem. Equation Equation SOLUTION From Equation ou know that. B substituting this value of into Equation, ou obtain. Substitute. Solve for. The sstem has eactl one solution: and. The term back-substitution implies that ou work backward. For instance, in Eample, the second equation gave ou the value of. Then ou substituted that value into the first equation to solve for. Eample 6 further demonstrates this procedure. EXAMPLE 6 Using Back-Substitution to Solve a Sstem in Row-Echelon Form Solve the sstem. z 9 z z Equation Equation Equation

9 Section. Introduction to Sstems of Linear Equations 7 SOLUTION From Equation ou alread know the value of z. To solve for, substitute z into Equation to obtain. Substitute z. Solve for. Finall, substitute and z in Equation to obtain 9. The solution is,, and z. Substitute, z. Solve for. Two sstems of linear equations are called equivalent if the have precisel the same solution set. To solve a sstem that is not in row-echelon form, first change it to an equivalent sstem that is in row-echelon form b using the operations listed below. Operations That Lead to Equivalent Sstems of Equations Each of the following operations on a sstem of linear equations produces an equivalent sstem.. Interchange two equations.. Multipl an equation b a nonzero constant.. Add a multiple of an equation to another equation. Rewriting a sstem of linear equations in row-echelon form usuall involves a chain of equivalent sstems, each of which is obtained b using one of the three basic operations. This process is called Gaussian elimination, after the German mathematician Carl Friedrich Gauss (777 8). EXAMPLE 7 Using Elimination to Rewrite a Sstem in Row-Echelon Form Solve the sstem. z 9 z 7 SOLUTION Although there are several was to begin, ou want to use a sstematic procedure that can be applied easil to large sstems. Work from the upper left corner of the sstem, saving the in the upper left position and eliminating the other s from the first column. z 9 z z 7 z 9 z z Adding the first equation to the second equation produces a new second equation. Adding times the first equation to the third equation produces a new third equation.

10 8 Chapter Sstems of Linear Equations Now that everthing but the first has been eliminated from the first column, work on the second column. z 9 z 9 This is the same sstem ou solved in Eample 6, and, as in that eample, the solution is, z z z z, Each of the three equations in Eample 7 is represented in a three-dimensional coordinate sstem b a plane. Because the unique solution of the sstem is the point,, z,,, z. Adding the second equation to the third equation produces a new third equation. Multipling the third equation b produces a new third equation. the three planes intersect at the point represented b these coordinates, as shown in Figure.. z (,, ) Figure. Technolog Note Man graphing utilities and computer software programs can solve a sstem of m linear equations in n variables. Tr solving the sstem in Eample 7 using the simultaneous equation solver feature of our graphing utilit or computer software program. Kestrokes and programming snta for these utilities/programs applicable to Eample 7 are provided in the Online Technolog Guide, available at college.hmco.com /pic /larsonela6e.

11 Section. Introduction to Sstems of Linear Equations 9 Because man steps are required to solve a sstem of linear equations, it is ver eas to make errors in arithmetic. It is suggested that ou develop the habit of checking our solution b substituting it into each equation in the original sstem. For instance, in Eample 7, ou can check the solution,, and z as follows. Equation : Equation : Equation : 9 7 Substitute solution in each equation of the original sstem. Each of the sstems in Eamples, 6, and 7 has eactl one solution. You will now look at an inconsistent sstem one that has no solution. The ke to recognizing an inconsistent sstem is reaching a false statement such as 7 at some stage of the elimination process. This is demonstrated in Eample 8. EXAMPLE 8 An Inconsistent Sstem Solve the sstem. SOLUTION (Another wa of describing this operation is to sa that ou subtracted the first equation from the third equation to produce a new third equation.) Now, continuing the elimination process, add times the second equation to the third equation to produce a new third equation. Adding times the first equation to the second equation produces a new second equation. Adding times the first equation to the third equation produces a new third equation. Adding times the second equation to the third equation produces a new third equation. Because the third equation is a false statement, this sstem has no solution. Moreover, because this sstem is equivalent to the original sstem, ou can conclude that the original sstem also has no solution. As in Eample 7, the three equations in Eample 8 represent planes in a threedimensional coordinate sstem. In this eample, however, the sstem is inconsistent. So, the planes do not have a point in common, as shown in Figure. on the net page.

12 Chapter Sstems of Linear Equations Figure. This section ends with an eample of a sstem of linear equations that has an infinite number of solutions. You can represent the solution set for such a sstem in parametric form, as ou did in Eamples and. EXAMPLE 9 SOLUTION A Sstem with an Infinite Number of Solutions Solve the sstem. Begin b rewriting the sstem in row-echelon form as follows. The first two equations are interchanged. Adding the first equation to the third equation produces a new third equation. Adding times the second equation to the third equation eliminates the third equation. Because the third equation is unnecessar, omit it to obtain the sstem shown below. To represent the solutions, choose to be the free variable and represent it b the parameter t. Because and, ou can describe the solution set as t, t, t, t is an real number.

13 Section. Introduction to Sstems of Linear Equations Discover Graph the two lines represented b the sstem of equations. You can use Gaussian elimination to solve this sstem as follows. Graph the sstem of equations ou obtain at each step of this process. What do ou observe about the lines? You are asked to repeat this graphical analsis for other sstems in Eercises 9 and 9. SECTION. Eercises In Eercises 6, determine whether the equation is linear in the variables and sin 6. sin In Eercises 7, find a parametric representation of the solution set of the linear equation z In Eercises 6, use back-substitution to solve the sstem z. z z 6 z 6 z. 6. In Eercises 7, graph each sstem of equations as a pair of lines in the -plane. Solve each sstem and interpret our answer The smbol indicates an eercise in which ou are instructed to use a graphing utilit or a smbolic computer software program In Eercises 6, complete the following set of tasks for each sstem of equations. (a) Use a graphing utilit to graph the equations in the sstem. (b) Use the graphs to determine whether the sstem is consistent or inconsistent. (c) If the sstem is consistent, approimate the solution. (d) Solve the sstem algebraicall. (e) Compare the solution in part (d) with the approimation in part (c). What can ou conclude?

14 Chapter Sstems of Linear Equations In Eercises 7 6, solve the sstem of linear equations v. u v z 6 8. z z z 8 z z 8. z 8. z w 6 w z w z w 6. In Eercises 7 6, use a computer software program or graphing utilit to solve the sstem of linear equations The smbol indicates that electronic data sets for these eercises are available at college.hmco.com/pic/larsonela6e. These data sets are compatible with each of the following technologies: MATLAB, Mathematica, Maple, Derive, TI-8/TI-8 Plus, TI-8/TI-8 Plus, TI-86, TI-89, TI-9, and TI-9 Plus z.7w 8...8z.w 8...6z.w z.6w z z z z z z z z w w 7 6 z w 6 z w z w z w 6 z w z w In Eercises 6 68, state wh each sstem of equations must have at least one solution. Then solve the sstem and determine if it has eactl one solution or an infinite number of solutions. 6. 7z 66. z z 9z 8 z 67. z 68. z z z 9z True or False? In Eercises 69 and 7, determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the tet. If a statement is false, provide an eample that shows the statement is not true in all cases or cite an appropriate statement from the tet. 69. (a) A sstem of one linear equation in two variables is alwas consistent. (b) A sstem of two linear equations in three variables is alwas consistent. (c) If a linear sstem is consistent, then it has an infinite number of solutions.

15 Section. Introduction to Sstems of Linear Equations 7. (a) A sstem of linear equations can have eactl two solutions. (b) Two sstems of linear equations are equivalent if the have the same solution set. (c) A sstem of three linear equations in two variables is alwas inconsistent. 7. Find a sstem of two equations in two variables, and, that has the solution set given b the parametric representation t and t, where t is an real number. Then show that the solutions to our sstem can also be written as and t. t 7. Find a sstem of two equations in three variables,,, and, that has the solution set given b the parametric representation t, s, and s t, where s and t are an real numbers. Then show that the solutions to our sstem can also be written as s t, s, and t. In Eercises 7 76, solve the sstem of equations b letting A, B, and C z z z z 8 z z In Eercises 77 and 78, solve the sstem of linear equations for and. 77. cos sin 78. sin cos In Eercises 79 8, determine the value(s) of k such that the sstem of linear equations has the indicated number of solutions. 79. An infinite number of 8. An infinite number of solutions solutions k 6 k cos sin sin cos k 8. Eactl one solution 8. No solution k k k k 8. No solution 8. Eactl one solution kz 6 k k kz k 6 8 z z z 8. Determine the values of k such that the sstem of linear equations does not have a unique solution. kz k z k z 86. Find values of a, b, and c such that the sstem of linear equations has (a) eactl one solution, (b) an infinite number of solutions, and (c) no solution. z 6 z a bz c 87. Writing Consider the sstem of linear equations in and. a b c a b c a b c Describe the graphs of these three equations in the -plane when the sstem has (a) eactl one solution, (b) an infinite number of solutions, and (c) no solution. 88. Writing Eplain wh the sstem of linear equations in Eercise 87 must be consistent if the constant terms c, c, and c are all zero. 89. Show that if a b c for all, then a b c. 9. Consider the sstem of linear equations in and. a b e c d f Under what conditions will the sstem have eactl one solution? In Eercises 9 and 9, sketch the lines determined b the sstem of linear equations. Then use Gaussian elimination to solve the sstem. At each step of the elimination process, sketch the corresponding lines. What do ou observe about these lines?

16 Chapter Sstems of Linear Equations Writing In Eercises 9 and 9, the graphs of two equations are shown and appear to be parallel. Solve the sstem of equations algebraicall. Eplain wh the graphs are misleading Gaussian Elimination and Gauss-Jordan Elimination In Section., Gaussian elimination was introduced as a procedure for solving a sstem of linear equations. In this section ou will stud this procedure more thoroughl, beginning with some definitions. The first is the definition of a matri. Definition of a Matri If m and n are positive integers, then an m n matri is a rectangular arra a a... a n a a... a n... m rows a a a. a m a. a m a. a m... a n. a mn n columns in which each entr, a ij, of the matri is a number. An m n matri (read m b n ) has m rows (horizontal lines) and n columns (vertical lines). REMARK: The plural of matri is matrices. If each entr of a matri is a real number, then the matri is called a real matri. Unless stated otherwise, all matrices in this tet are assumed to be real matrices. The entr a ij is located in the ith row and the jth column. The inde i is called the row subscript because it identifies the row in which the entr lies, and the inde j is called the column subscript because it identifies the column in which the entr lies. A matri with m rows and n columns (an m n matri) is said to be of size m n. If m n, the matri is called square of order n. For a square matri, the entries a, a, a,... are called the main diagonal entries.

17 Section. Gaussian Elimination and Gauss-Jordan Elimination EXAMPLE Eamples of Matrices Each matri has the indicated size. (a) Size: (b) Size: (c) Size: (d) Size: e 7 One ver common use of matrices is to represent sstems of linear equations. The matri derived from the coefficients and constant terms of a sstem of linear equations is called the augmented matri of the sstem. The matri containing onl the coefficients of the sstem is called the coefficient matri of the sstem. Here is an eample. Sstem z z z 6 Augmented Matri 6 Coefficient Matri REMARK: Use to indicate coefficients of zero. The coefficient of in the third equation is zero, so a takes its place in the matri. Also note the fourth column of constant terms in the augmented matri. When forming either the coefficient matri or the augmented matri of a sstem, ou should begin b aligning the variables in the equations verticall. Given Sstem Align Variables 9 9 Augmented Matri 9 Elementar Row Operations In the previous section ou studied three operations that can be used on a sstem of linear equations to produce equivalent sstems.. Interchange two equations.. Multipl an equation b a nonzero constant.. Add a multiple of an equation to another equation.

18 6 Chapter Sstems of Linear Equations In matri terminolog these three operations correspond to elementar row operations. An elementar row operation on an augmented matri produces a new augmented matri corresponding to a new (but equivalent) sstem of linear equations. Two matrices are said to be row-equivalent if one can be obtained from the other b a finite sequence of elementar row operations. Elementar Row Operations. Interchange two rows.. Multipl a row b a nonzero constant.. Add a multiple of a row to another row. Although elementar row operations are simple to perform, the involve a lot of arithmetic. Because it is eas to make a mistake, ou should get in the habit of noting the elementar row operation performed in each step so that it is easier to check our work. Because solving some sstems involves several steps, it is helpful to use a shorthand method of notation to keep track of each elementar row operation ou perform. This notation is introduced in the net eample. EXAMPLE Elementar Row Operations (a) Interchange the first and second rows. (b) Multipl the first row b to produce a new first row. (c) Add times the first row to the third row to produce a new third row. Original Matri New Row-Equivalent Matri Notation R R Original Matri New Row-Equivalent Matri Notation 6 R R Original Matri New Row-Equivalent Matri Notation 8 R R R REMARK: Notice in Eample (c) that adding times row to row does not change row.

19 Section. Gaussian Elimination and Gauss-Jordan Elimination 7 Technolog Note Man graphing utilities and computer software programs can perform elementar row operations on matrices. If ou are using a graphing utilit, our screens for Eample (c) ma look like those shown below. Kestrokes and programming snta for these utilities/programs applicable to Eample (c) are provided in the Online Technolog Guide, available at college.hmco.com/pic/larsonela6e. In Eample 7 in Section., ou used Gaussian elimination with back-substitution to solve a sstem of linear equations. You will now learn the matri version of Gaussian elimination. The two methods used in the net eample are essentiall the same. The basic difference is that with the matri method there is no need to rewrite the variables over and over again. EXAMPLE Using Elementar Row Operations to Solve a Sstem Linear Sstem Associated Augmented Matri z 9 9 z 7 7 Add the first equation to the second Add the first row to the second row to equation. produce a new second row. z 9 z z R R R Add times the first equation to the Add times the first row to the third third equation. row to produce a new third row. z 9 z z 9 R R R Add the second equation to the third equation. z 9 z z Add the second row to the third row to produce a new third row. 9 R R R

20 8 Chapter Sstems of Linear Equations Multipl the third equation b. z 9 z z Multipl the third row b to produce a new third row. 9 R R Now ou can use back-substitution to find the solution, as in Eample 6 in Section.. The solution is,, and z. The last matri in Eample is said to be in row-echelon form. The term echelon refers to the stair-step pattern formed b the nonzero elements of the matri. To be in row-echelon form, a matri must have the properties listed below. Definition of Row-Echelon Form of a Matri A matri in row-echelon form has the following properties.. All rows consisting entirel of zeros occur at the bottom of the matri.. For each row that does not consist entirel of zeros, the first nonzero entr is (called a leading ).. For two successive (nonzero) rows, the leading in the higher row is farther to the left than the leading in the lower row. REMARK: A matri in row-echelon form is in reduced row-echelon form if ever column that has a leading has zeros in ever position above and below its leading. Technolog Note EXAMPLE Use a graphing utilit or a computer software program to find the reduced row-echelon form of the matri in part (f) of Eample. Kestrokes and programming snta for these utilities/programs applicable to Eample (f) are provided in the Online Technolog Guide, available at college.hmco.com/pic/ larsonela6e. Row-Echelon Form The matrices below are in row-echelon form. (a) (c) (b) (d) The matrices shown in parts (b) and (d) are in reduced row-echelon form. The matrices listed below are not in row-echelon form. (e) (f)

21 Section. Gaussian Elimination and Gauss-Jordan Elimination 9 It can be shown that ever matri is row-equivalent to a matri in row-echelon form. For instance, in Eample ou could change the matri in part (e) to row-echelon form b multipling the second row in the matri b. The method of using Gaussian elimination with back-substitution to solve a sstem is as follows. REMARK: For kestrokes and programming snta regarding specific graphing utilities and computer software programs involving Eample (f), please visit college.hmco.com/ pic/larsonela6e. Similar eercises and projects are also available on the website. Gaussian Elimination with Back-Substitution. Write the augmented matri of the sstem of linear equations.. Use elementar row operations to rewrite the augmented matri in row-echelon form.. Write the sstem of linear equations corresponding to the matri in row-echelon form, and use back-substitution to find the solution. Gaussian elimination with back-substitution works well as an algorithmic method for solving sstems of linear equations. For this algorithm, the order in which the elementar row operations are performed is important. Move from left to right b columns, changing all entries directl below the leading s to zeros. EXAMPLE SOLUTION Gaussian Elimination with Back-Substitution Solve the sstem. The augmented matri for this sstem is Obtain a leading in the upper left corner and zeros elsewhere in the first column The first two rows are interchanged. R R Adding times the first row to the third row produces a new third row. R R R

22 Chapter Sstems of Linear Equations Now that the first column is in the desired form, ou should change the second column as shown below. To write the third column in proper form, multipl the third row b 6 Similarl, to write the fourth column in proper form, ou should multipl the fourth row b. Multipling the fourth row b produces a new fourth row. R R The matri is now in row-echelon form, and the corresponding sstem of linear equations is as shown below. Using back-substitution, ou can determine that the solution is, 6, , Adding times the first row to the fourth row produces a new fourth row.. When solving a sstem of linear equations, remember that it is possible for the sstem to have no solution. If during the elimination process ou obtain a row with all zeros ecept for the last entr, it is unnecessar to continue the elimination process. You can simpl conclude that the sstem is inconsistent and has no solution.. Multipling the third row b produces a new third row. R R R Adding 6 times the second row to the fourth row produces a new fourth row. R 6R R R R

23 Section. Gaussian Elimination and Gauss-Jordan Elimination EXAMPLE 6 A Sstem with No Solution Solve the sstem. 6 SOLUTION The augmented matri for this sstem is 6. Appl Gaussian elimination to the augmented matri. R R R R R R 7 R R R 7 R R R Note that the third row of this matri consists of all zeros ecept for the last entr. This means that the original sstem of linear equations is inconsistent. You can see wh this is true b converting back to a sstem of linear equations.

24 Chapter Sstems of Linear Equations 7 Because the third equation is a false statement, the sstem has no solution. Discover Consider the sstem of linear equations Without doing an row operations, eplain wh this sstem is consistent. The sstem below has more variables than equations. Wh does it have an infinite number of solutions? Gauss-Jordan Elimination With Gaussian elimination, ou appl elementar row operations to a matri to obtain a (row-equivalent) row-echelon form. A second method of elimination, called Gauss-Jordan elimination after Carl Gauss and Wilhelm Jordan (8 899), continues the reduction process until a reduced row-echelon form is obtained. This procedure is demonstrated in the net eample. EXAMPLE 7 SOLUTION Gauss-Jordan Elimination Use Gauss-Jordan elimination to solve the sstem. z 9 z 7 In Eample, Gaussian elimination was used to obtain the row-echelon form 9. Now, rather than using back-substitution, appl elementar row operations until ou obtain a matri in reduced row-echelon form. To do this, ou must produce zeros above each of the leading s, as follows.

25 Now, converting back to a sstem of linear equations, ou have z. The Gaussian and Gauss-Jordan elimination procedures emplo an algorithmic approach easil adapted to computer use. These elimination procedures, however, make no effort to avoid fractional coefficients. For instance, if the sstem in Eample 7 had been listed as z 7 z 9 Section. Gaussian Elimination and Gauss-Jordan Elimination R R R R R R R 9R R both procedures would have required multipling the first row b which would have introduced fractions in the first row. For hand computations, fractions can sometimes be avoided b judiciousl choosing the order in which elementar row operations are applied., REMARK: No matter which order ou use, the reduced row-echelon form will be the same. The net eample demonstrates how Gauss-Jordan elimination can be used to solve a sstem with an infinite number of solutions. EXAMPLE 8 SOLUTION A Sstem with an Infinite Number of Solutions Solve the sstem of linear equations. The augmented matri of the sstem of linear equations is.

26 Chapter Sstems of Linear Equations Using a graphing utilit, a computer software program, or Gauss-Jordan elimination, ou can verif that the reduced row-echelon form of the matri is The corresponding sstem of equations is Now, using the parameter t to represent the nonleading variable t,.. t, t,, ou have where t is an real number. REMARK: Note that in Eample 8 an arbitrar parameter was assigned to the nonleading variable. You subsequentl solved for the leading variables and as functions of t. You have looked at two elimination methods for solving a sstem of linear equations. Which is better? To some degree the answer depends on personal preference. In real-life applications of linear algebra, sstems of linear equations are usuall solved b computer. Most computer programs use a form of Gaussian elimination, with special emphasis on was to reduce rounding errors and minimize storage of data. Because the eamples and eercises in this tet are generall much simpler and focus on the underling concepts, ou will need to know both elimination methods. Homogeneous Sstems of Linear Equations As the final topic of this section, ou will look at sstems of linear equations in which each of the constant terms is zero. We call such sstems homogeneous. For eample, a homogeneous sstem of m equations in n variables has the form a a a a a a a a a a a... m a m m a n n a n n a n n. a mn n. It is eas to see that a homogeneous sstem must have at least one solution. Specificall, if all variables in a homogeneous sstem have the value zero, then each of the equations must be satisfied. Such a solution is called trivial (or obvious). For instance, a homogeneous sstem of three equations in the three variables,, and must have,, and as a trivial solution.

27 Section. Gaussian Elimination and Gauss-Jordan Elimination EXAMPLE 9 Solving a Homogeneous Sstem of Linear Equations Solve the sstem of linear equations. SOLUTION Appling Gauss-Jordan elimination to the augmented matri ields the matri shown below. The sstem of equations corresponding to this matri is. Using the parameter t, the solution set is t, t, R R R R R R R R t, t is an real number. This sstem of equations has an infinite number of solutions, one of which is the trivial solution given b t. Eample 9 illustrates an important point about homogeneous sstems of linear equations. You began with two equations in three variables and discovered that the sstem has an infinite number of solutions. In general, a homogeneous sstem with fewer equations than variables has an infinite number of solutions. THEOREM. The Number of Solutions of a Homogeneous Sstem Ever homogeneous sstem of linear equations is consistent. Moreover, if the sstem has fewer equations than variables, then it must have an infinite number of solutions. A proof of Theorem. can be done using the same procedures as those used in Eample 9, but for a general matri.

28 6 Chapter Sstems of Linear Equations SECTION. Eercises In Eercises 8, determine the size of the matri In Eercises 9, determine whether the matri is in row-echelon form. If it is, determine whether it is also in reduced row-echelon form In Eercises, find the solution set of the sstem of linear equations represented b the augmented matri In Eercises 6, solve the sstem using either Gaussian elimination with back-substitution or Gauss-Jordan elimination z w. z z w 6 In Eercises 7, use a computer software program or graphing utilit to solve the sstem of linear equations z 8w z 6 z z 8z w z 6w z w z z z z

29 Section. Gaussian Elimination and Gauss-Jordan Elimination In Eercises 6, solve the homogeneous linear sstem corresponding to the coefficient matri provided Consider the matri A (a) If A is the augmented matri of a sstem of linear equations, determine the number of equations and the number of variables. (b) If A is the augmented matri of a sstem of linear equations, find the value(s) of k such that the sstem is consistent. (c) If A is the coefficient matri of a homogeneous sstem of linear equations, determine the number of equations and the number of variables. k. (d) If A is the coefficient matri of a homogeneous sstem of linear equations, find the value(s) of k such that the sstem is consistent. 8. Consider the matri A k 6. (a) If A is the augmented matri of a sstem of linear equations, determine the number of equations and the number of variables. (b) If A is the augmented matri of a sstem of linear equations, find the value(s) of k such that the sstem is consistent. (c) If A is the coefficient matri of a homogeneous sstem of linear equations, determine the number of equations and the number of variables. (d) If A is the coefficient matri of a homogeneous sstem of linear equations, find the value(s) of k such that the sstem is consistent. In Eercises 9 and, find values of a, b, and c (if possible) such that the sstem of linear equations has (a) a unique solution, (b) no solution, and (c) an infinite number of solutions. 9.. z z a b cz. The sstem below has one solution:,, and z. z 6 z 6 z z a b cz Equation Equation Equation Solve the sstems provided b (a) Equations and, (b) Equations and, and (c) Equations and. (d) How man solutions does each of these sstems have?. Assume the sstem below has a unique solution. a a a b Equation a a a b Equation a a a b Equation Does the sstem composed of Equations and have a unique solution, no solution, or an infinite number of solutions?

30 8 Chapter Sstems of Linear Equations In Eercises and, find the unique reduced row-echelon matri that is row-equivalent to the matri provided.... Writing Describe all possible reduced row-echelon matrices. Support our answer with eamples. 6. Writing Describe all possible reduced row-echelon matrices. Support our answer with eamples. True or False? In Eercises 7 and 8, determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the tet. If a statement is false, provide an eample that shows the statement is not true in all cases or cite an appropriate statement from the tet. 7. (a) A 6 matri has si rows. (b) Ever matri is row-equivalent to a matri in row-echelon form. (c) If the row-echelon form of the augmented matri of a sstem of linear equations contains the row, then the original sstem is inconsistent. (d) A homogeneous sstem of four linear equations in si variables has an infinite number of solutions. 8. (a) A 7 matri has four columns. (b) Ever matri has a unique reduced row-echelon form. (c) A homogeneous sstem of four linear equations in four variables is alwas consistent. (d) Multipling a row of a matri b a constant is one of the elementar row operations. In Eercises 9 and 6, determine conditions on a, b, c, and d such that the matri a b c d will be row-equivalent to the given matri In Eercises 6 and 6, find all values of (the Greek letter lambda) such that the homogeneous sstem of linear equations will have nontrivial solutions Writing Is it possible for a sstem of linear equations with fewer equations than variables to have no solution? If so, give an eample. 6. Writing Does a matri have a unique row-echelon form? Illustrate our answer with eamples. Is the reduced row-echelon form unique? 6. Writing Consider the matri a b c d. Perform the sequence of row operations. (a) Add times the second row to the first row. (b) Add times the first row to the second row. (c) Add times the second row to the first row. (d) Multipl the first row b. What happened to the original matri? Describe, in general, how to interchange two rows of a matri using onl the second and third elementar row operations. 66. The augmented matri represents a sstem of linear equations that has been reduced using Gauss-Jordan elimination. Write a sstem of equations with nonzero coefficients that is represented b the reduced matri. There are man correct answers. 67. Writing Describe the row-echelon form of an augmented matri that corresponds to a sstem of linear equations that is inconsistent. 68. Writing Describe the row-echelon form of an augmented matri that corresponds to a sstem of linear equations that has infinitel man solutions. 69. Writing In our own words, describe the difference between a matri in row-echelon form and a matri in reduced row-echelon form.

31 Section. Applications of Sstems of Linear Equations 9. Applications of Sstems of Linear Equations Sstems of linear equations arise in a wide variet of applications and are one of the central themes in linear algebra. In this section ou will look at two such applications, and ou will see man more in subsequent chapters. The first application shows how to fit a polnomial function to a set of data points in the plane. The second application focuses on networks and Kirchhoff s Laws for electricit. ( n, n ) (, ) (, ) (, ) Polnomial Curve Fitting Figure. Polnomial Curve Fitting Suppose a collection of data is represented b n points in the -plane,,,,,..., n, n and ou are asked to find a polnomial function of degree n p a a a... a n n whose graph passes through the specified points. This procedure is called polnomial curve fitting. If all -coordinates of the points are distinct, then there is precisel one polnomial function of degree n (or less) that fits the n points, as shown in Figure.. To solve for the n coefficients of p, substitute each of the n points into the polnomial function and obtain n linear equations in n variables a, a, a,..., a. n a a a... n a n a a a... n a n a a n a n.... n a n n n This procedure is demonstrated with a second-degree polnomial in Eample. EXAMPLE Polnomial Curve Fitting Determine the polnomial p a a a whose graph passes through the points,,,, and,. SOLUTION Substituting,, and into p and equating the results to the respective -values produces the sstem of linear equations in the variables a, a, and a shown below. p a a a a a a p a a a a a a Simulation p a a a a a 9a Eplore this concept further with an The solution of this sstem is a, a 8, and a 8, so the polnomial electronic simulation available on function is the website college.hmco.com/ pic/larsonela6e. p 8 8.

32 Chapter Sstems of Linear Equations The graph of p is shown in Figure.. 8 (, ) 8 (, ) (, ) 6 (, ) (, ) (, ) (, ) (, ) p() = p() = Figure. Figure.6 ( ) EXAMPLE SOLUTION Polnomial Curve Fitting Find a polnomial that fits the points,,,,,,,, and,. Because ou are provided with five points, choose a fourth-degree polnomial function p a a a a a. Substituting the given points into p produces the sstem of linear equations listed below. a a a 8a 6a a a a a a a a a a a a a a a 8a 6a The solution of these equations is a, a, which means the polnomial function is p a The graph of p is shown in Figure.6., a 8, a 7

33 Section. Applications of Sstems of Linear Equations The sstem of linear equations in Eample is relativel eas to solve because the -values are small. With a set of points with large -values, it is usuall best to translate the values before attempting the curve-fitting procedure. This approach is demonstrated in the net eample. EXAMPLE Translating Large -Values Before Curve Fitting Find a polnomial that fits the points, 6,, SOLUTION Because the given -values are large, use the translation z 8 to obtain z,,,, 7,, z,,, This is the same set of points as in Eample. So, the polnomial that fits these points is pz z z 8z 7z z z z 7 z. Letting z 8, ou have, 8,, z,,,, 9,, z,,,,,. z,,. p EXAMPLE An Application of Curve Fitting Find a polnomial that relates the periods of the first three planets to their mean distances from the sun, as shown in Table.. Then test the accurac of the fit b using the polnomial to calculate the period of Mars. (Distance is measured in astronomical units, and period is measured in ears.) (Source: CRC Handbook of Chemistr and Phsics) TABLE. Planet Mercur Venus Earth Mars Jupiter Saturn Mean Distance Period SOLUTION Begin b fitting a quadratic polnomial function p a a a to the points.87,.,.7,.6, and,. The sstem of linear equations obtained b substituting these points into p is

34 Chapter Sstems of Linear Equations Period.... (.,.88) Mars = p() Venus Earth (.,.) (.7,.6) Mercur (.87,.).... Mean distance from the sun Figure.7 a.87a.87 a. a.7a.7 a.6 a a a. The approimate solution of the sstem is a.6, which means that the polnomial function can be approimated b p Using p to evaluate the period of Mars produces p..96 ears. a.69, a. This estimate is compared graphicall with the actual period of Mars in Figure.7. Note that the actual period (from Table.) is.88 ears. Earth Venus Mercur Figure.8 ln Saturn Jupiter ln = ln Mars ln An important lesson ma be learned from the application shown in Eample : The polnomial that fits the given data points is not necessaril an accurate model for the relationship between and for -values other than those corresponding to the given points. Generall, the farther the additional points are from the given points, the worse the fit. For instance, in Eample the mean distance of Jupiter is.. The corresponding polnomial approimation for the period is. ears a poor estimate of Jupiter s actual period of.86 ears. The problem of curve fitting can be difficult. Tpes of functions other than polnomial functions often provide better fits. To see this, look again at the curve-fitting problem in Eample. Taking the natural logarithms of the distances and periods of the first si planets produces the results shown in Table. and Figure.8. TABLE. Planet Mercur Venus Earth Mars Jupiter Saturn Mean Distance Natural Log of Mean Period ) Natural Log of Period Now, fitting a polnomial to the logarithms of the distances and periods produces the linear relationship between ln and ln shown below. ln ln From this equation it follows that, or.

35 Section. Applications of Sstems of Linear Equations In other words, the square of the period (in ears) of each planet is equal to the cube of its mean distance (in astronomical units) from the sun. This relationship was first discovered b Johannes Kepler in 69. Network Analsis Networks composed of branches and junctions are used as models in man diverse fields such as economics, traffic analsis, and electrical engineering. In such models it is assumed that the total flow into a junction is equal to the total flow out of the junction. For eample, because the junction shown in Figure.9 has units flowing into it, there must be units flowing out of it. This is represented b the linear equation. Because each junction in a network gives rise to a linear equation, ou can analze the flow through a network composed of several junctions b solving a sstem of linear equations. This procedure is illustrated in Eample. Figure.9 EXAMPLE Analsis of a Network Set up a sstem of linear equations to represent the network shown in Figure., and solve the sstem. Figure.

36 Chapter Sstems of Linear Equations SOLUTION Each of the network s five junctions gives rise to a linear equation, as shown below. The augmented matri for this sstem is Junction Junction Junction Junction Junction. Gauss-Jordan elimination produces the matri. From the matri above, ou can see that,,, and. Letting t, ou have t, t, t, t, t where t is a real number, so this sstem has an infinite number of solutions. In Eample, suppose ou could control the amount of flow along the branch labeled. Using the solution from Eample, ou could then control the flow represented b each of the other variables. For instance, letting t would reduce the flow of and to zero, as shown in Figure.. Similarl, letting t would produce the network shown in Figure..

37 Section. Applications of Sstems of Linear Equations Figure. Figure. You can see how the tpe of network analsis demonstrated in Eample could be used in problems dealing with the flow of traffic through the streets of a cit or the flow of water through an irrigation sstem. An electrical network is another tpe of network where analsis is commonl applied. An analsis of such a sstem uses two properties of electrical networks known as Kirchhoff s Laws.. All the current flowing into a junction must flow out of it.. The sum of the products IR ( I is current and R is resistance) around a closed path is equal to the total voltage in the path. In an electrical network, current is measured in amps, resistance in ohms, and the product of current and resistance in volts. Batteries are represented b the smbol. The larger vertical bar denotes where the current flows out of the terminal. Resistance is denoted b the smbol. The direction of the current is indicated b an arrow in the branch. REMARK: A closed path is a sequence of branches such that the beginning point of the first branch coincides with the end point of the last branch.