2 Chapter 12 A Preview Of College Algebra - Gocolumbia.edu

haL96885 ch12 001-049.qxd10/14/095:46 PMPage 7(12-7) 712.1 Solving Systems of Linear Equations by Using Augmented MatricesExample 5Solving a System of Linear Equations by Using an Augmented MatrixUse an augmented matrix and elementary row operations to solve e2x 3y 4f.x 4y 3Solutione2x 3y 4fx 4y 32 34 d1 4 31 4 3c d2 3414 3 dc0 5101 4 3 cd0 1 21 05 cd0 1 2x 0y 50x y 2cr r12 r r 2r221 r2 r2 5 r r 4r112 Answer:(5, 2)First form the augmented matrix.1 0 k1 d , interchange rows0 1 k21 and 2 to place a 1 in row 1, column 1.To work toward the reduced form cNext add 2r1 to r2 to produce a 0 in row 2, column 1.Then divide row 2 by 5 to produce a 1 in row 2, column 2.To complete the transformation to the form c10a 0 in row 1, column 2 by adding 4r2 to r1.0 k1 d , produce1 k2Write the equivalent system of equations for the reduced form.Does this answer check?Self-Check 5Use an augmented matrix and elementary row operations to solve e2x 9y 7f.x 5y 3Note that in Example 5, we first worked on column 1 and then on column 2. Thiscolumn-by-column strategy is recommended for transforming all augmented matrices toreduced form. This strategy also is used in Example 6.Example 6Solving a System of Linear Equations by Using an Augmented MatrixUse an augmented matrix and elementary row operations to solve e4x 3y 3f.2x 9y 4Solutione4x 3y 3f2x 9y 4r1 r14 r r 2r221 r2 2r215 c42 1 210103 3 d9 43 34 † 4§9 43 34 4 15 52 23 34 4 113First form the augmented matrix.1Then work on putting column 1 in the form c0r11means to multiply row 1 by .44 d.r2 2r1 means to subtract twice row 1 from row 2.Next work on column 2 to put the matrix in the reduced form c22r means to multiply row 2 by .15 215100 k1 d.1 k2

haL96885 ch12 001-049.qxd8(12-8)11/17/0911:16 AMChapter 12 A Preview of College Algebra3r1 r1 r24 Answer:Page 812 10 131x 0y 210x y 3101 1a , b2 333r1 r2 means to subtract r2 from row 1.44Write the equivalent system of equations for the reduced form.Does this answer check?Self-Check 6Use an augmented matrix and elementary row operations to solve e4x 3y 9f.2x 3y 01 0 k1 d always yields a unique solution for a consistent system0 1 k2of independent equations. For an inconsistent system with no solution or a consistent system of dependent equations with an infinite number of solutions, the reduced form is different. We now examine these two possibilities. Solving an inconsistent system by theaddition method produces an equation that is a contradiction. Note the matrix equivalentof this in Example 7.The reduced form cExample 7Solving an Inconsistent System by Using an Augmented MatrixUse an augmented matrix and elementary row operations to solve ex 2y 3f.5x 10y 11Solutionex 2y 3f5x 10y 11cr r 5r221 Answer:123 d5 10 1131 2 cd0 0 4x 2y 30 0 4First form the augmented matrix.1 d.0There is no need to proceed further because the last row in the matrixcorresponds to an equation that is a contradiction. Thus the originalsystem of equations is inconsistent and has no solution.Then work on putting column 1 in the form cThere is no solution.Self-Check 7Write the solution of the system of linear equations represented by c100 7 d.0 6We encourage you to compare this Example 7 to the Example 7 in Section 3.5 wherethe same problem was worked by the addition method. Example 8 examines a consistentsystem of dependent equations.

haL96885 ch12 001-049.qxd10/14/095:46 PMPage 912.1 Solving Systems of Linear Equations by Using Augmented MatricesExample 8(12-9) 9Solving a Consistent System of Dependent Equations by Usingan Augmented MatrixUse an augmented matrix and elementary row operations to solve e 4x 10y 2f.6x 15y 3Solutione 4x 10y 2f6x 15y 310 2d 1535 11 2 † 2§ 6 15 35 11 2† 2§ 00 051x y 220x 0y 015x y 22c1r1 r14 r2 r2 6r1 46Answer: There are infinitely many solutions,15all having the form a y , yb.22First form the augmented matrix.Then work on putting column 1 in the form c10 d.The last row corresponds to an equation that is an identity. Itcannot be used to simplify column 2 further.These two equations form a dependent system of equations withan infinite number of solutions.Because the coefficient of x in the first equation is 1, solve thisequation for x in terms of y. You can also solve this equation for12y in terms of x and express the answer in the form ax, x b.55You may wish to confirm that both of these represent the sameset of points. All the solutions are points on the same line,points that can be written in this form.Self-Check 8Write the solution of the system of linear equations represented by c10 1 4 d.0 0The general solution of a system of dependent linear equations describes all solutionsof the system and is given by indicating the relationship between the coordinates of thesolutions. The particular solutions obtained from the general solution contain only constant coordinates.51The general solution in Example 8 is a y , yb. By arbitrarily selecting y-values,22we can produce as many particular solutions as we wish. For y 0 and y 1, we obtain1two particular solutions a , 0b and (3, 1).2Self-Check Answers5 1 3 d3 284x 3y 12. ef2x 9y 43. (5, 6)1. c1 1 1 d32 121 1 9b. c d0 1 51 3 7c. c d0 1 34. a. c5. ( 8, 1)6. (1.5, 1)7. There is no solution; it is aninconsistent system.8. A dependent system with ageneral solution (y 4, y) ;three particular solutions are(4, 0), (5, 1), and (6, 2).

haL96885 ch12 001-049.qxd10(12-10)10/14/095:46 PMPage 10Chapter 12 A Preview of College AlgebraTechnology Self-Check Answer1.12.1Using the Language and Symbolism of Mathematics1. A matrix is aarray of numbers.2. The entries in an augmented matrix for a system oflinear equations consist of theandin the equations.3.systems of equations have the samesolution set.4. The notation r1 4 r2 denotes that rows 1 and 2 for amatrix are.15. The notation r1¿ r1 denotes that row2being replaced by multiplying the current rowby.12.16. The notation to represent that row 2 is being replacedby the sum of the current row 2 plus twice row 1 is.7. Thesolution of a system of dependentlinear equations describes all solutions of the systemand is given by indicating the relationship between thecoordinates of the solutions.8. Thesolutions obtained from a generalsolution contain only constant coordinates.isQuick Review1. Determine whether ( 2, 5) is a solution of the system4x 3y 7ef.5x 2y 12x 3y 42. If the x-coordinate of the solution of ef3x 2y 11is 5, determine the y-coordinate.12.1yx 1 are multiplied23by 6, the resulting equivalent equation is.3. If both sides of the equation3x 4y 8f by the substitution method.x 2y 22x 5y 15. Solve ef by the addition method.3x 4y 134. Solve eExercisesObjective 1 Use Augmented Matrices to Solve Systemsof Two Linear Equations with Two VariablesIn Exercises 1–6, write an augmented matrix for eachsystem of linear equations.1. 3x y 02. 5x y 32x y 54x 3y 293.4. 2x 7y 34x 123x 2y 1 3y 35. x 56. x 4y 6y 11In Exercises 7–12, write a system of linear equations in xand y that is represented by each augmented matrix.23 25 1 0 7. c8. cd d4 3 132 132 1 123 5 d9. c d10. c1 3 06 4 211. c1007 d1 812. 1023 41 50

haL96885 ch12 001-049.qxd11/17/0911:18 AMPage 1112.1 Solving Systems of Linear Equations by Using Augmented MatricesIn Exercises 13–20, use the given elementary rowoperations to complete each matrix.213. c11 1 d3 0 6 3 414. c d1 2 11 3 2 15. cd2 541 1 1 16. cd2 3 217. c36 dc r1 r2 dcr2 r2 2r11 3 2 c dr2 r2 2r11 1 1 c dr1 r21r1 r13 13 d4 6 c6 d4 61r1 r12312 d18. cdc 3 3 213 3 21r1 r1 5r21 5 16 d19. cdc0 1301 3r1 r1 4r21 4 10 d20. cd c01 201 2In Exercises 21–26, write the solution for the system of linearequations represented by each augmented matrix. If thematrix represents a consistent system of dependent equations,write the general solution and three particular solutions.1 0 51 0 4 21. c22. cd d0 190 1 61 4 31 35 d 23. c24. cd0 0 700 21 3 51 2 3 d d25. c26. c0 0 000 0In Exercises 27–34, label the elementary row operationused to transform the first matrix to the second. Use thenotation developed in this section.27. c236 81r1 ? d c7 10328. c235 9 d2 8129. c358 d2 230. c12131. c03 3 23132. c06 2 d3 133. c2 6 d1 110 8d78 d 934 d7 105 91?2 † 2 § 3 2 81 c0?58 d 13 26138 d0 9 91 28? d c01 31 6 2? † 1§ 0 131 0 4? d c0 1 1 c?34. 132† 1§130 ?10(12-11) 1101† 1§13In Exercises 35–50, use an augmented matrix andelementary row operations to solve each system of linearequations.35. x 3y 536. x 2y 92x y 53x 4y 737. x 3y 138. x 2y 73x 7y 74x 3y 339. 2x 5y 440. 2x 3y 174x 3y 64x y 1341. 4x 9y 542. 8x 3y 393x 12y 107x 2y 1143. 3x y 244. 3x 4y 02x y 62x 3y 1645. 6x 4y 1146. 5x y 710x 6y 172x 4y 547. 3x 4y 748. 4x 3y 26x 8y 1016x 12y 749. 2x y 550. 3x 6y 124x 2y 104x 8y 16Connecting Concepts to ApplicationsIn Exercises 51–60, write a system of linear equationsusing the variables x and y, and use this system to solve theproblem.51. Numeric Word Problem Find two numbers whosesum is 160 and whose difference is 4.52. Numeric Word Problem Find two numbers whosesum is 260 if one number is 3 times the other number.53. Complementary Angles The two angles shown arecomplementary, and one angle is 32 larger than theother. Determine the number of degrees in eachangle.xy54. Supplementary Angles The two angles shown aresupplementary, and one angle is 74 larger than theother. Determine the number of degrees in eachangle.xy

haL96885 ch12 001-049.qxd12(12-12)11/17/0911:19 AMPage 12Chapter 12 A Preview of College Algebra55. Fixed and Variable Costs A seamstress makes customcostumes for operas. One month the total of fixed andvariable costs for making 20 costumes was 3,200. Thenext month the total of the fixed and variable costs formaking 30 costumes was 4,300. Determine the fixedcost and the variable cost per costume.56. Rates of Two Bicyclists Two bicyclists depart at thesame time from a common location, traveling inopposite directions. One averages 5 km/h more than theother. After 2 hours, they are 130 km apart. Determinethe speed of each bicyclist.57. Rate of a River Current A small boat can go 30 kmdownstream in 1 hour, but only 14 km upstream in1 hour. Determine the rate of the boat and the rate ofthe current.58. Mixture of Two Disinfectants A hospital needs 100 Lof a 15% solution of disinfectant. How many liters of a40% solution and a 5% solution should be mixed toobtain this 15% solution?59. Mixture of a Fruit Drink A fruit concentrate is 15%water. How many liters of pure water and how manyliters of concentrate should be mixed to produce 100 Lof mixture that is 83% water?60. Basketball Scores During one game for the PhoenixSuns, Steve Nash scored 39 points on 17 field goals.How many of these field goals were 2-pointers and howmany were 3-pointers?61. Radius of a Train Wheel On a curve, the radius r1 ofthe inside train wheel is less than the radius r2 of theoutside wheel. This allows the outside wheel to travel agreater distance as the train goes around the curve. Onone curve, both radii are measured in inches, and theresult is r2 r1 0.25. The inside wheel covers12.1111.5 inches per revolution. Determine the radius of theinside wheel and the radius of the outside wheel.r1r2Group discussion questionsa1x b1y c1f for (x, y)a2x b2y c2in terms of a1, a2, b1, b2, c1, and c2. Assume thata1b2 a2b1 0.63. Discovery Questiona. Extend the augmented matrix notation given forsystems of two linear equations with two variablesto write an augmented matrix for this system.x 2y 3z 5 2x y z 1 ¶3x y z 4b. Write a system of linear equations that is representedby this augmented matrix. Use the variables x, y, and z.2 332 402 † 1 § 24 3 2c. Write a system of linear equations that is representedby this augmented matrix. Use the variables w, x, y,and z.3 41 111 4 24 1 2 45372 1 147Cumulative62. Challenge Question Solve eCumulative Review1. Write the first five terms of the arithmetic sequencedefined by an 2n 1.2. Write the 50th term of the arithmetic sequence definedby an 2n 1.3. Write the first five terms of the geometric sequencedefined by an 2n 1.4. Simplify(12a 3b) 0 (12a) 0 (3b) 0 12a0 3b0assuming all bases are nonzero.5. Simplify ( 1) 4 4 1 (1) 1/4.

haL96885 ch12 001-049.qxd10/23/098:50 PMPage 13(12-13) 1312.2 Systems of Linear Equations in Three VariablesSection 12.2Systems of Linear Equations in Three VariablesObjective:1. Solve a system of three linear equations in three variables.A first-degree equation in two variables of the form Ax By C is called a linear equation because its graph is a straight line if A and B are not both 0. Similarly, a first-degreeequation in three variables of the form Ax By Cz D also is called a linear equation.However, this name is misleading because if A, B, and C are not all 0, the graph ofAx By Cz D is not a line but a plane in three-dimensional space.The graph of a three-dimensional space on two-dimensional paper is limited in itsportrayal of the third dimension. Nonetheless, we can give the viewer a feeling for planesin a three-dimensional space by orienting the x-, y-, and z-axes as shown in the figure. Thisgraph illustrates the plane 2x 3y 4z 12, whose x-intercept is (6, 0, 0), whosey-intercept is (0, 4, 0), and whose z-intercept is (0, 0, 3). Drawing lines to connect theseintercepts gives the view of the plane in the region where all coordinates are positive.The plane defined by2x 3y 4z 12z543 5 4 31 2 1 5 4 3 2 112 136x5y1 2 3 4 5 3 4 51. Solve a System of Three Linear Equations in Three VariablesA system of three linear equations in three variables is referred to as a 3 3 (three-bythree) system. A solution of a system of equations with the three variables x, y, and z is anordered triple (x, y, z) that is a solution of each equation in the system.Example 1Determining Whether an Ordered Triple Is a Solution of a Systemof Linear Equationsx y z 6Determine whether (2, 3, 5) is a solution of 2x y z 12 ¶ .3x 2y 2z 3SolutionFirst EquationSecond EquationThird Equationx y z 62 ( 3) 5 ? 6 6 ? 6 is true.2x y z 122(2) ( 3) 5 ? 124 3 5 ? 1212 ? 12 is true.3x 2y 2z 33(2) 2( 3) 2(5) ?36 6 10 ?32 ? 3 is false.Answer:(2, 3, 5) is not a solution of this system.To be a solution of this system, the point must satisfy all three equations.Self-Check 1Determine whether (2, 3.25, 4.75) is a solution of the system in Example 1.The graph of each linear equation Ax By Cz D is a plane in three-dimensionalspace unless A, B, and C are all 0. A system of three linear equations in three variables canbe viewed geometrically as the intersection of a set of three planes. These planes mayintersect in one point, no points, or an infinite number of points. The illustrations in thefollowing box show some of the ways we can obtain these solutions. Can you sketch otherways of obtaining these solution sets?

haL96885 ch12 001-049.qxd14(12-14)10/14/095:47 PMPage 14Chapter 12 A Preview of College AlgebraTypes of Solution Sets for Linear Systems with Three EquationsA1x B1y C1z D1The linear system A2x B2y C2z D2 ¶ can haveA3x B3y C3z D3One SolutionIAn Infinite Numberof SolutionsNo SolutionIIIIIIIIIIIIIIIIIThe planes intersect at asingle point; the systemis consistent and theequations areindependent.The planes have no pointin common; the system isinconsistent.The planes intersect alonga line and thus have aninfinite number of commonpoints; the system isconsistent and theequations are dependent.Can I Use Graphs to Solve Systems of Three Linear Equationswith Three Variables?No, although the figures in the box can give us an intuitive understanding of the possiblesolutions to these systems it is not practical to actually solve these systems graphically.Thus we rely entirely on algebraic methods.In this section, we illustrate two methods for solving systems of three linear equationsin three variables (3 3 systems). Example 2 extends the substitution and addition methods from Sections 3.4 and 3.5 to solve a 3 3 system. Later in the section, we use augmented matrices, which were introduced in Section 12.1.Equivalent systems of equations have the same solution set. The general goal of eachstep of a solution process is to produce an equivalent system that is simpler than the previous step. By eliminating some of the variables, we can reduce a 3 3 system to a systemwith only two variables, and then we can eliminate another variable to produce an equation with only one variable. We can then back-substitute to obtain the values of the othertwo variables. This strategy is outlined in the following box.Strategy for Solving a 3 3 System of Linear Equations*Step 1. Write each equation in the general form Ax By Cz D.Step 2. Select one pair of equations and use the substitution method or the additionmethod to eliminate one of the variables.Step 3. Repeat step 2 with another pair of equations. Be sure to eliminate the samevariable as in step 2.Step 4. Eliminate another variable from the pair of equations produced in steps 2 and 3,and solve this 2 2 system of equations.Step 5. Back-substitute the values from step 4 into one of the original equations to solvefor the third variable.Step 6. Does this solution check in all three of the original equations?*If a contradiction is obtained in any of these steps, the system is inconsistent and has no solution. If an identity isobtained in any step, the system is either dependent with infinitely many solutions or inconsistent with no solution.

haL96885 ch12 001-049.qxd10/14/095:47 PMPage 1512.2 Systems of Linear Equations in Three VariablesExample 2(12-15) 15Solving a 3 3 System of Linear Equationsx y z 2Solve the system x y 2z 1 ¶ .x y z 0SolutionProduce a 2 2 System of Equations(1)x y z 2(2) x y 2z 1 ¶(3)x y z 0x y z 2(1)(2) x y 2z 12y z 32y z 3f.The 2 2 system of equations is e2y 3z 1(2) x y 2z 1x y z 0(3)2y

A Preview of College Algebra 12 CHAPTER r 1 r 2 haL96885_ch12_001-049.qxd 10/14/09 5:45 PM Page 1. 2 (12-2) Chapter 12 A Preview of College Algebra Section 12.1 Solving Systems of Linear Equations by Using Augmented Matrices The laws of mathematics can b