3Systems of Linear Equations31273.1 Solving Systems of Linear Equations by Graphing3.2 Solving Systems of Equations by Using the SubstitutionMethod3.3 Solving Systems of Equations by Using the Addition Method3.4 Applications of Systems of Linear Equations in TwoVariables3.5 Systems of Linear Equations in Three Variables andApplications3.6 Solving Systems of Linear Equations by Using MatricesmiL06198_ch03_127-1701/18/0704:17 PMPage 127 indesign Disk_1:Desktop Folder:Sanjay2_18/1/07:128 Chapter 3 Systems of Linear Equationssection 3.1 Solving Systems of Linear Equations by Graphing1. SolutionstoSystemsofLinearEquationsA linear equation in two variables has an infinite number of solutions that forma line in a rectangular coordinate system. Two or more linear equations form asystemoflinearequations. Forexample:A solution to a system of linear equations is an ordered pair that is a solu-tiontoeach individuallinearequation.example1 Determining Solutions to a System of Linear EquationsDeterminewhethertheorderedpairsaresolutionstothesystem.a. b.Solution:a. Substitute the ordered pairinto both equations: True TrueBecausetheorderedpairisasolutiontoeachequation, itisasolutiontothesystem ofequations.b. Substitute the ordered pairinto both equations: TrueFalseBecause the ordered pairis not a solution to the second equation,itisnot asolutiontothesystemofequations.A solution to a system of two linear equations maybeinterpretedgraphicallyasapointofintersectionbetweenthetwolines. GraphingthelinesfromEx-ample 1, wehaveNotice that the lines intersect at(Figure 3-1). 12, 423x y 2 x y 610, 623x y 2 3102 162 2 x y 6 102 162 610, 6212, 423x y 2 3122 142 2 x y 6 122 142 612, 4210, 62 12, 42 3x y 2 x y 6 2x 4y 10 x 3y 5Objectives 1. Solutions to Systems ofLinear Equations2. Dependent andInconsistent Systems ofLinear Equations3. Solving Systems of LinearEquations by Graphing4 5 4 5 3 1 2 3213454511 2yx32x y 6(2, 4)3x y 2Figure 3-1See e-ProfessorSolutions to Systems of LinearEquations by GraphingmiL06198_ch03_127-17012/19/0609:37 AMPage 128 Nishant Nishant:Desktop Folder:prakash 19/12/06:CONFIRMING PAGESSection 3.1 Solving Systems of Linear Equations by Graphing 129Objective 1: ExercisesFor Exercises 12, determine which points are solutions to the given system.1. 2. y 3x 7 y 4x 3 2x 7y 30 y 8x 5(2, 11) 11, 12, 11, 132,, , 11, 42 a32, 5b 10, 3022. DependentandInconsistentSystemsofLinearEquationsWhentwolinesaredrawninarectangularcoordinatesystem, threegeometricrelationshipsarepossible.1. Twolinesmayintersectatexactlyonepoint.2. Twolinesmayintersectatnopoint. Thisoccursifthelinesareparallel.3. Twolinesmayintersectatinfinitelymanypoints alongtheline. Thisoccursiftheequationsrepresentthesameline(thelinesarecoinciding).Solutions to Systems of Linear Equations in Two VariablesOneuniquesolution Nosolution InfinitelymanysolutionsOnepointofintersection Parallellines CoincidinglinesSystemisconsistent. Systemisinconsistent. Systemisconsistent.Systemisindependent. Systemisindependent. Systemisdependent.Objective 2: ExercisesFor Exercises 13, the graph of a system of linear equations is given.a. Identify whether the system is consistent or inconsistent.b. Identify whether the system is dependent or independent.c. Identify the number of solutions to the system.miL06198_ch03_127-17012/19/0609:37 AMPage 129 Nishant Nishant:Desktop Folder:prakash 19/12/06:CONFIRMING PAGESSolution:Tographeachequation, writetheequationinslope-interceptformFirstequation: Secondequation:Slope:Slope:From their slope-intercept forms, we see that the lines have different slopes,indicating that the lines must intersect at exactly one point. We can graph thelinesbyusingtheslopeandy-intercepttofindthepointofintersection (Figure3-2).23y 23x 2 3y3 2x363 3y 2x 6 2x 3y 6 32y 32x 12y mx b.130 Chapter 3 Systems of Linear EquationsFigure 3-23. SolvingSystemsofLinearEquationsbyGraphingexample2 Solving a System of Linear Equations by GraphingSolvethesystembygraphingbothlinearequationsandfindingthepoint(s)ofintersection.2x 3y 6y 32x 124 5 4 5 3 1 2 3213454511 2yx32(3, 4)Point of intersection2x 3y 6y x 32121. 2. 3. x 3y 6 4x 2y 2 3x y 1 y 13x 2 2x y 4 y x 34 5 4 5 3 1 2 313454511 2yx3224 5 4 5 3 1 2 313454511 2yx3224 5 4 5 3 1 2 313454511 2yx322See e-ProfessorSolving Systems of LinearEquations by GraphingmiL06198_ch03_127-17012:23:0605:39Page 130CONFIRMING PAGESSection 3.1 Solving Systems of Linear Equations by Graphing 131Thepointappearstobethepointofintersection. Thiscanbeconfirmedbysubstitutingandintobothequations. True TrueThesolutionis13, 42. 2x 3y 6 2132 3142 66 12 6y 32x 124 32132 124 9212y 4 x 313, 42Tip: In Example 2, the lines could also have been graphed by using the x- and y-intercepts or by using a table of points. However, the advantage of writing the equationsin slope-intercept form is that we can compare the slopes and y-intercepts of each line.1. If the slopes differ, the lines are different and nonparallel and must cross in exactlyone point.2. If the slopes are the same and the y-intercepts are different, the lines are paralleland do not intersect.3. If the slopes are the same and the y-intercepts are the same, the two equations rep-resent the same line.example3 Solving a System of Equations by GraphingSolvethesystembygraphing.Solution:Tographtheline, writeeachequationinslope-interceptform.Becausethelineshavethesameslopebutdifferenty-intercepts, theyareparallel(Figure3-3). Twoparallellinesdonotintersect, whichimpliesthatthesystemhasnosolution. Thesystemisinconsistent. y 13x 1 y 13x 2 6y62x6663y3 x363 3y x 6 6y 2x 6 x 3y 6 6y 2x 6 x 3y 6Figure 3-34 5 4 5 3 1 2 313454511 2yx322 y x 113 y x 213See e-ProfessorSolving Systems of LinearEquations by GraphingmiL06198_ch03_127-17012/19/0609:37 AMPage 131 Nishant Nishant:Desktop Folder:prakash 19/12/06:CONFIRMING PAGES132 Chapter 3 Systems of Linear EquationsSolution:Write the first equation in slope-intercept form. The second equation is alreadyinslope-interceptform.Firstequation: Secondequation:Notice that the slope-intercept forms of thetwo lines are identical. Therefore, the equa-tionsrepresentthesameline(Figure3-4).Thesystemisdependent, andthesolutiontothesystemofequationsisthesetofallpointsontheline.Because not all the ordered pairs in the solution set can be listed, we canwritethesolutioninset-buildernotation. Furthermore, theequationsandrepresentthesameline. Therefore, thesolutionsetmaybewrittenasor5 1x, y20 x 4y 86. 5 1x, y20 y 14x 26y 14x 2 x 4y 8 y 14x 2 4y4x484 4y x 8y 14x 2 x 4y 8Figure 3-44 5 4 5 3 1 2 313454511 2yx322 y x 214Objective 3: ExercisesFor Exercises 15, solve the systems of equations by graphing.1. 2. 3.y 3 y 5x 4 x 2y 1 3x 2y 6 y 2x 3 2x y 44 5 4 5 3 1 2 3213454511 2yx324 5 4 5 3 1 2 3213454511 2yx324 5 4 5 3 1 2 3213454511 2yx32example4 Solving a System of Linear Equations by GraphingSolvethesystembygraphing. y 14x 2 x 4y 8miL06198_ch03_127-17012/19/0609:37 AMPage 132 Nishant Nishant:Desktop Folder:prakash 19/12/06:CONFIRMING PAGESSection 3.1 Solving Systems of Linear Equations by Graphing 133CalculatorConnectionsThesolutiontoasystemofequationscanbefoundbyusingeitheraTracefeatureoranIntersect featureonagraphingcalculatortofindthepointofintersectionbetweentwocurves.Forexample, considerthesystem 5x y 1 2x y 64. 5. 4x 2 2y 2x 3y 3 2x y 4 y 23x 14 5 4 5 3 1 2 3213454511 2yx324 5 4 5 3 1 2 3213454511 2yx32Figure 3-5 Figure 3-6Firstgraphtheequationstogetheronthesameviewingwindow. Recallthatto enter the equations into the calculator, the equations must be written withthey-variableisolated.Isolate y.By inspection of the graph, it appears that the solution isThe Traceoption on the calculator may come close tobut may not show the ex-actsolution(Figure3-5). However, anIntersect featureonagraphingcalcu-lator may provide the exact solution (Figure 3-6). See your users manual forfurtherdetails.UsingTrace UsingIntersect11, 4211, 42. 5x y 1y 5x 1 2x y 6 y 2x 6miL06198_ch03_127-17012/19/0609:38 AMPage 133 Nishant Nishant:Desktop Folder:prakash 19/12/06:CONFIRMING PAGES134 Chapter 3 Systems of Linear Equationssection 3.1 Practice Exercises PracticeProblems e-Professors Self-Tests Videos NetTutor1. Define the key terms.a. System of linear equations b. Solution to a system of linear equationsc. Consistent system d. Inconsistent systeme. Dependent system f. Independent systemMixed Exercises2. Determine if the ordered pair is a solution to the system.a. (2, 2) b.For Exercises 35, answer true or false.3. An inconsistent system has one solution.4. Parallel lines form an inconsistent system.5. Lines with different slopes intersect in one point.For Exercises 68, solve the system by graphing.6. 7. 8.4x 8y 8 3x y 2 y x 5y 12x 2 6x 2y 4 y 2x 712, 22 y 12x 1 5x 7y 44 5 4 5 3 1 2 3213454511 2yx324 5 4 5 3 1 2 3213454511 2yx324 5 4 5 3 1 2 3213454511 2yx32Also see MathZone IQ Chapter 3, Section 1.GoonlinetoformiL06198_ch03_127-17012/19/0609:38 AMPage 134 Nishant Nishant:Desktop Folder:prakash 19/12/06:CONFIRMING PAGESSection 3.2 Solving Systems of Equations by Using the Substitution Method 135section 3.2 Solving Systems of Equations by Using theSubstitution Method1. TheSubstitutionMethodInthissectionandSection3.3, wewillpresenttwoalgebraicmethodstosolveasystemofequations. Thefirstiscalledthesubstitutionmethod.Objectives 1. The Substitution Method2. Solving InconsistentSystems and DependentSystemsSolving a System of Equations by the Substitution Method1. Isolateoneofthevariablesfromoneequation.2. Substitutethequantityfoundinstep1intotheother equation.3. Solvetheresultingequation.4. Substitutethevaluefoundinstep3backintotheequationinstep1tofindthevalueoftheremainingvariable.5. Checkthesolutioninbothequations, andwritetheanswerasanor-deredpair.example1 Using the Substitution Method to Solve a Linear SystemSolvethesystembyusingthesubstitutionmethod.Solution:The y variable in the second equation is the easiest variable to isolate becauseitscoefficientis1.Step 1: Solve the secondequation for y.Step 3: Solve for x.x 1315x1551515x 515x 12 73x 12x 12 73x 216x 62 76x y 6 y 6x 63x 2y 7 6x y 6 3x 2y 7Avoiding Mistakes:Do not substitute into the same equation fromwhich it came. This mistakewill result in an identity:6 66x 6x 6 66x 16x 62 66x y 6y 6x 6Step 2: Substitute thequantity for y in the otherequation.6x 6See e-ProfessorThe Substitution MethodmiL06198_ch03_127-17012/19/0609:38 AMPage 135 Nishant Nishant:Desktop Folder:prakash 19/12/06:CONFIRMING PAGES136 Chapter 3 Systems of Linear Equations Thesolutionis113, 42.2 4 6 1 8 76 a13b 4 6 3 a13b 2142 76x y 6 3x 2y 7y 4y 2 6y 6a13b 6y 6x 6 Step 4: Substitute into the expressiony 6x 6.x 13Step 5: Check the orderedpairin eachoriginal equation.113, 42Objective 1: Exercises For Exercises 14, solve by using the substitution method.1. 2. 3. 4. x 3y 42x 3y 512x 2y 07x y 1x 10y 34 7x y 314x 12y 4 y 5x 112. SolvingInconsistentSystemsandDependentSystemsexample2 Using the Substitution Method to Solve a Linear SystemSolvethesystembyusingthesubstitutionmethod.Solution:Step 1: The x variable is already isolated.Step 2: Substitute the quantity into the other equation.Step 3: Solve for y.The equation reduces to a contradiction,indicating that the system has no solution.Thereisnosolution. Hencethelinesneverintersectandmust Thesystemisinconsistent. beparallel. Thesystemisinconsistent.8 64y 8 4y 6x 2y 4 212y 42 4y 62x 4y 6x 2y 4 2x 4y 6x 2y 4See e-ProfessorSolving Inconsistent Systemsand Dependent SystemsmiL06198_ch03_127-17012/19/0609:38 AMPage 136 Nishant Nishant:Desktop Folder:prakash 19/12/06:CONFIRMING PAGESexample3 Solving a Dependent SystemSolvebyusingthesubstitutionmethod.Solution:Step 1: Solve for one of the variables.Step 2: Substitute the quantityfory in the other equation.Step 3: Solve for x. Apply the distributiveproperty to clear the parentheses.The system reduces to the identityTherefore, the original two equa-tions are equivalent, and the system is dependent. The solution consists of allpointsonthecommonline. Becausetheequationsandrepresentthesameline, thesolutionmaybewrittenasor 5 1x, y20y 3 2x6 5 1x, y204x 2y 66y 3 2x4x 2y 66 6. 6 6 4x 4x 6 62x 3 4x 212x 32 6y 2x 3 y 3 2x4x 2y 6y 3 2x 4x 2y 6Section 3.2 Solving Systems of Equations by Using the Substitution Method 137Tip: The answer to Example 2 can be verified by writing each equation in slope-intercept form and graphing the equations.Equation 1 Equation 2Notice that the equations have the same slope, but different y-intercepts; therefore,the lines must be parallel. There is no solution to this system of equations.y 12x 32y 12x 24y42x4642y2x2 424y 2x 6 2y x 4 2x 4y 6 x 2y 4yx3 4 5 4 3 1 223412345112x 2y 42x 4y 65vTip: We can confirm the results of Example 3 by writing each equation in slope-intercept form. The slope-intercept forms are identical, indicating that the lines are thesame.y 2x 3 y 3 2xy 2x 3 2y 4x 6 4x 2y 6slope-intercept formmiL06198_ch03_127-17012/19/0609:38 AMPage 137 Nishant Nishant:Desktop Folder:prakash 19/12/06:CONFIRMING PAGES138 Chapter 3 Systems of Linear EquationsObjective 2: ExercisesFor Exercises 14, solve the systems.1. 2. 3. 4. x 4y 112y 3x 35x y 102y 10x 5y 17x 3x 7y 42x 6y 2x 3y 1section 3.2 Practice Exercises PracticeProblems e-Professors Self-Tests Videos NetTutorReview ExercisesFor Exercises 12, using the slope-intercept form of the lines, a. determine whether the system is consistent orinconsistent and b. determine whether the system is dependent or independent.1. 2.Mixed ExercisesFor Exercises 37, solve the systems by using the substitution method.3. 4. 5.6. 7. y 15x 2155x y 3x 16y 536x y 5x 14y 74y 311x 611x 2y 64x y 7 3x 11y 9 y 34x 48x 4y 1 10x 15y 526x 3y 8 4x 6y 1section 3.3 Solving Systems of Equations by Using theAddition Method1. The AdditionMethodThenextmethodwepresenttosolvesystemsoflinearequationsistheadditionmethod (sometimes called the elimination method).Objectives 1. The Addition Method2. Solving InconsistentSystems and DependentSystemsAlso see MathZone IQ Chapter 3, Section 2.GoonlinetoformiL06198_ch03_127-17012/19/0609:38 AMPage 138 Nishant Nishant:Desktop Folder:prakash 19/12/06:CONFIRMING PAGESSection 3.3 Solving Systems of Equations by Using the Addition Method 139Solving a System of Equations by the Addition Method1. Write both equations in standard form:2. Clear fractions or decimals (optional).3. Multiplyoneorbothequationsbynonzeroconstantstocreateoppositecoefficients for one of the variables.4. Add the equations from step 3 to eliminate one variable.5. Solve for the remaining variable.6. Substitute the known value found in step 5 into one of the original equa-tions to solve for the other variable.7. Check the ordered pair in both equations.Ax By Cexample1 Solving a System by the Addition MethodSolve the system by using the addition method.Solution:Wemaychoosetoeliminateeithervariable. Toeliminatex, changethecoeffi-cients to 12 and 12.The solution is (3, 2).x 34x 124x 10 24x 5122 2 4x 5y 2 y 2 y 2 12x 16y 4 12x 15y 612x 16y 4 3x 4y 112x 15y 6 4x 5y 2 3x 4y 1 3x 1 4y4x 5y 2 4x 5y 2 3x 1 4y 4x 5y 2Step 1: Write both equations instandard form. There are nofractions or decimals.Step 3: Multiply the firstequation by 3.Multiply thesecond equationby 4.Step 4: Add theequations.Step 5: Solve for y.Step 6: Substitute back into one ofthe originalequations andsolve for x.Step 7: Check theordered pair (3, 2) in bothoriginal equations.y 2See e-ProfessorThe Addition MethodMultiply by 3.Multiply by 4.miL06198_ch03_127-17012/19/0609:38 AMPage 139 Nishant Nishant:Desktop Folder:prakash 19/12/06:CONFIRMING PAGESexample2 Solving a System of Equations by the Addition MethodSolve the system by using the addition method.Solution:Step 7: Check the ordered pair (0, 2) in each original equation.The solution is (0, 2).0.05y 0.02x 0.100.051220.02102 0.100.10 0.10 x 2y 6 y102 21226 1224 4 x 0 x 4 4 x 2122 6 122x 2y 6 yy 2 y 2 2x 5y 10 2x 5y 102x 6y 12 x 3y 62x 5y 10 0.02x 0.05y 0.10 x 3y 6 0.05y 0.02x 0.100.02x 0.05y 0.10 x 2y 6 y x 3y 6 0.05y 0.02x 0.10 x 2y 6 y140 Chapter 3 Systems of Linear EquationsTip: To eliminate the x variable in Example 1, both equations were multiplied byappropriate constants to create 12x and 12x. We chose 12 because it is the leastcommon multiple of 4 and 3.We could have solved the system by eliminating the y-variable. To eliminate y,we would multiply the top equation by 4 and the bottom equation by 5. This wouldmake the coefficients of the y-variable 20 and 20, respectively.15x 20y 5 3x 4y 116x 20y 8 4x 5y 2Multiply by 4.Multiply by 5.Step 1: Writeboth equations instandard form.Step 2: Cleardecimals.Step 3: Createoppositecoefficients.Step 4: Add theequations.Step 5: Solvefor y.Step 6: To solvefor x, substituteinto oneof the originalequations.y 2Multiply by 100.Multiply by 2.miL06198_ch03_127-17012/19/0609:38 AMPage 140 Nishant Nishant:Desktop Folder:prakash 19/12/06:CONFIRMING PAGES2. SolvingInconsistentSystemsandDependentSystemsexample3 Solving a System of Equations by the Addition MethodSolve the system by using the addition method.Solution:Notice that both variables were eliminated. The system of equations is reducedtotheidentityTherefore, thetwooriginalequationsareequivalentandthesystemisdependent. Thesolutionsetconsistsofaninfinitenumberofor-dered pairs (x, y) that fall on the common line of intersection or equivalentlyThe solution set can be written in set notation asore 1x, y2`15x 12y 1f 51x, y204x 10y 20615x 12y 1.4x 10y 20,0 0.0 0 4x 10y 204x 10y 202x 5y 104x 10y 20 4x 10y 202x 5y 10 10a15x 12yb 10 1 4x 10y 20 15x12y 14x 10y 2015x 12y 1Section 3.3 Solving Systems of Equations by Using the Addition Method 141Objective 1: ExercisesFor Exercises 14, solve the system by the addition method.1. 2.3. 4. 0.8x 1.4y 3 7y 4x 11 1.2x 0.6y 3 3x 10y 13 5x 3y 84 3x 4y 14 3x 7y 20 3x y 1Step 1:Equations are instandard form.Step 2:Clear fractions.Step 3:Multiply the firstequation by 2.Step 4:Add the equations.Multiply by 2.Tip: The system in Example 3 was dependent because eliminating the variablesresulted in an identity. If both variables are eliminated and the resulting equation is acontradiction, the system is inconsistent and has no solution.miL06198_ch03_127-17012/19/0609:38 AMPage 141 Nishant Nishant:Desktop Folder:prakash 19/12/06:CONFIRMING PAGESReview ExercisesUse the slope-intercept form of the lines to determine the number of solutions for the system of equations.1.Mixed ExercisesFor Exercises 24, solve the systems by using the addition method.2. 3. 4. 31x y2 y 2x 2x 13y 133x 4y 10 x 2y 4 3x y 1 4x 3y 5 y 12x 1 y 12x 4142 Chapter 3 Systems of Linear EquationsObjective 2: ExercisesFor Exercises 13, solve the systems.1. 2. 3.x 2y 4.5 0.2x 0.3y 0.7 6x 4y 2 12x y 766y 14 4x 3x 2y 1section 3.3 Practice Exercises PracticeProblems e-Professors Self-Tests Videos NetTutorFor Exercises 510, solve by using either the addition method or the substitution method.5. 6. 7.8. 9. 10.Expanding Your SkillsUse the addition method first to solve for x. Then repeat the addition method again, using the original system ofequations, this time solving for y.11. 5x 3y 2 6x 2y 5 6x 3y 3 4y 1014x 16y 24x 7y 16 y 2x 1 2x y 9 2x 5y 9 2x 4y 8 y 43x 1 16x 15y 4 4x 3 14x 5y 9Also see MathZone IQ Chapter 3, Section 3.GoonlinetoformiL06198_ch03_127-17012/19/0609:39 AMPage 142 Nishant Nishant:Desktop Folder:prakash 19/12/06:CONFIRMING PAGESsection 3.4 Applications of Systems of Linear Equations in Two Variables1. ApplicationsInvolvingCostIn Chapter 1 we solved numerous application problems using equations thatcontainedonevariable. However, whenanapplicationhasmorethanoneunknown, sometimesitismoreconvenienttousemultiplevariables. Inthissection, wewillsolveapplicationscontainingtwounknowns. Whentwovariablesarepresent, thegoalistosetupasystemoftwoindependentequations.example1 Solving a Cost ApplicationAtanamusementpark, fivehotdogsandonedrinkcost$16. Twohotdogsandthreedrinkscost$9. Findthecostperhotdogandthecostperdrink.Solution:Let h represent the cost per hot dog. Label the variables.Let d represent the cost per drink.Write two equations.Thissystemcanbesolvedbyeitherthesubstitutionmethodortheaddi-tionmethod. Wewillsolvebyusingthesubstitutionmethod. Thed-variableinthefirstequationistheeasiestvariabletoisolate.Becausethe cost per hot dog is $3.00.Becausethe cost per drink is $1.00. d 1,h 3,d 5132 16 d 1h 313h 3913h 48 92h 15h 48 92h 315h 162 92h 3d 95h d 16 d 5h 16aCost of 2hot dogsb acost of 3drinks b $9 2h 3d 9aCost of 5hot dogsb acost of 1drink b $16 5h d 16Section 3.4 Applications of Systems of Linear Equations in Two Variables 143Objectives 1. Applications Involving Cost2. Applications InvolvingMixtures3. Applications InvolvingPrincipal and Interest4. Applications InvolvingDistance, Rate, and Time5. Applications InvolvingGeometrySolve for d in the firstequation.Substitute the quantityfor d in thesecond equation.Clear parentheses.Solve for h.Substitutein theequation d 5h 16.h 35h 16See e-ProfessorApplications Involving CostmiL06198_ch03_127-17012/19/0609:39 AMPage 143 Nishant Nishant:Desktop Folder:prakash 19/12/06:CONFIRMING PAGES2. ApplicationsInvolvingMixturesexample2 Solving an Application Involving ChemistryOne brand of cleaner used to etch concrete is 25% acid. A stronger industrial-strength cleaneris50%acid. Howmanygallonsofeachcleanershouldbemixedtoproduce20galofa40%acidsolution?Solution:Let x represent the amount of 25% acid cleaner.Let y represent the amount of 50% acid cleaner.144 Chapter 3 Systems of Linear EquationsObjective 1: Exercises1. A group of four golfers pays $150 to play a round of golf. Of these four, one is a member of the club andthree are nonmembers. Another group of golfers consists of two members and one nonmember and pays atotal of $75. What is the cost for a member to play a round of golf, and what is the cost for a nonmember?2. Sam has a pocket full of change consisting of dimes and quarters. The total value is $3.15. There are 7 morequarters than dimes. How many of each coin are there?25% Acid 50% Acid 40% AcidNumber of gallons of solution x y 20Number of gallons of pure acid 0.25x 0.50y 0.40(20), or 8Fromthefirstrowofthetable, wehaveFromthesecondrowofthetablewehaveTherefore, 8galof25%acidsolutionmustbeaddedto12galof50%acidsolutiontocreate20galofa40%acidsolution.x 8x 12 20x y 20y 1225y 30025x 50y 800 25x 50y 800 y 20 Multiply by 25. 25x 25y 500 x 0.25x 0.50y 825x 50y 800 x y 20x y 20 0.25x 0.50y 8Amount ofpure acid in25% solution amount ofpure acid in50% solution amount ofpure acid inresulting solutionaAmount of 25% solutionb aamount of50% solutionb atotal amountof solution b x y 20Multiply by 100to clear decimals.Create oppositecoefficients of x.Add theequations toeliminate x.Substitute back into one ofthe originalequations.y 12miL06198_ch03_127-17012/19/0609:39 AMPage 144 Nishant Nishant:Desktop Folder:prakash 19/12/06:CONFIRMING PAGES3. ApplicationsInvolvingPrincipalandInterestexample3 Solving a Mixture Application Involving FinanceSerenainvestedmoneyintwoaccounts: asavingsaccountthatyields4.5%simple interest and a certificate of deposit that yields 7% simple interest. Theamountinvestedat7%wastwicetheamountinvestedat4.5%. Howmuchdid Serena invest in each account if the total interest at the end of 1 year was$1017.50?Solution:Letx representtheamountinvestedinthesavingsaccount(the4.5%account).Lety representtheamountinvestedinthecertificateofdeposit(the7%account).Section 3.4 Applications of Systems of Linear Equations in Two Variables 145Objective 2: Exercises1. A jar of one face cream contains 18% moisturizer, and another type contains 24% moisturizer. How manyounces of each should be combined to get 12 oz of a cream that is 22% moisturizer?2. How much pure bleach must be combined with a solution that is 4% bleach to make 12 oz of a 12% bleachsolution?4.5% Account 7% Account TotalPrincipal x yInterest 0.045x 0.07y 1017.50Because the amount invested at 7% was twice the amount invested at 4.5%,wehaveFromthesecondrowofthetable, wehaveMultiply by 1000 to clear decimals.Because the y-variable in the first equation isisolated, we will use the substitution method.Substitute the quantity 2x into the secondequation.45x 7012x2 1,017,50045x 70y 1,017,500y 2xInterestearned from4.5% account interestearned from7% account atotalinterestb0.045x 0.07y 1017.50Amountinvestedat 7% 2 amountinvestedat 4.5%y 2xSee e-ProfessorApplications InvolvingPrincipal and InterestmiL06198_ch03_127-17012/19/0609:39 AMPage 145 Nishant Nishant:Desktop Folder:prakash 19/12/06:CONFIRMING PAGESSolve for x.Substituteinto the equationto solve for y.Becausethe amount invested in the savings account is $5500.Becausethe amount invested in the certificate of deposit is$11,000.y 11,000,x 5500,y 11,000y 2xx 5500 y 2155002y 2xx 5500x 1,017,500185185x 1,017,50045x 140x 1,017,500146 Chapter 3 Systems of Linear EquationsObjective 3: Exercises1. Alex invested $27,000 in two accounts: one that pays 2% simple interest and one that pays 3% simpleinterest. At the end of the first year, his total return was $685. How much was invested in each account?2. A credit union offers 5.5% simple interest on a certificate of deposit (CD) and 3.5% simple interest on asavings account. If Mr. Roderick invested $200 more in the CD than in the savings account and the totalinterest after the first year was $245, how much was invested in each account?4. ApplicationsInvolvingDistance, Rate, and Timeexample4 Solving a Distance, Rate, and Time ApplicationAplaneflies660mifrom AtlantatoMiamiin1.2hrwhentravelingwithatailwind. The return flight against the same wind takes 1.5 hr. Find the speedoftheplaneinstillairandthespeedofthewind.Solution:Let p represent the speed of the plane in still air.Let w represent the speed of the wind.The speed of the plane with the wind: (Planes still airspeed) (windspeed):The speed of the plane against the wind: (Planes still airspeed) (windspeed):Setupacharttoorganizethegiveninformation:p wp wDistance Rate TimeWith a tailwind 660 1.2Against a head wind 660 1.5 p wp wSee e-ProfessorApplications InvolvingDistance, Rate and TimemiL06198_ch03_127-17012/19/0609:39 AMPage 146 Nishant Nishant:Desktop Folder:prakash 19/12/06:CONFIRMING PAGESTwoequationscanbefoundbyusingtherelationship Add the equations.Substituteinto the equation Solve for w.Thespeedoftheplaneinstillairis495mph, andthespeedofthewindis55mph.55 w550 p w. p 495 550 14952 wp 495 990 2p 440 p w 550 p w 660 1p w2 11.52Divide by 1.5.6601.51p w21.51.5440 p w 660 1p w2 11.22Divide by 1. 2.6601.21p w21.21.2550 p w660 1p w2 11.52660 1p w2 11.22 Distanceagainstwind speedagainstwind timeagainstwind660 1p w2 11.52Distancewithwind speedwith wind timewithwind 660 1p w2 11.221distance rate time2.d rtSection 3.4 Applications of Systems of Linear Equations in Two Variables 147Notice that the first equation may be divided by 1.2and still leave integer coefficients. Similarly, thesecond equation may be simplified by dividing by 1.5.Objective 4: Exercises1. It takes a boat 2 hr to go 16 mi downstream with the current and 4 hr to return against the current. Find thespeed of the boat in still water and the speed of the current.2. A plane flew 720 mi in 3 hr with the wind. It would take 4 hr to travel the same distance against the wind.What is the rate of the plane in still air and the rate of wind?See e-ProfessorApplications InvolvingGeometry5. ApplicationsInvolvingGeometryexample5 Solving a Geometry ApplicationThe sum of the two acute angles in a right triangle is 90. The measure of oneangle is 6 less than 2 times the measure of the other angle. Find the measureofeachangle.miL06198_ch03_127-17012/19/0609:39 AMPage 147 Nishant Nishant:Desktop Folder:prakash 19/12/06:CONFIRMING PAGESSolution:Let x represent one acute angle.Let y represent the other acute angle.The sum of the two acute angles is 90:One angle is 6 less than 2 times the other angle:Because one variable is already isolated, wewill use the substitution method.Substituteinto the firstequation.To find x, substituteinto theequation Thetwoacuteanglesinthetrianglemeasure32 and58.x 58x 64 6x 21322 6x 2y 6.y 32 x 2y 6y 323y 963y 6 90x 2y 6 12y 62 y 90x 2y 6x y 90x 2y 6x y 90148 Chapter 3 Systems of Linear EquationsxyObjective 5: ExercisesFor Exercises 12, solve the applications involving geometry. If necessary, refer to the geometry formulas listed inthe inside front cover of the text.1. In a right triangle, one acute angle measures 6 more than 3 times the other. If the sum of the measures ofthe two acute angles must equal 90, find the measures of the acute angles.2. One angle measures 3 more than twice another. If the two angles are complementary, find the measures ofthe angles.Expanding Your SkillsSolve the business application.3. A rental car company rents a compact car for $20 a day, plus $0.25 per mile. A midsize car rents for $30 aday, plus $0.20 per mile.a. Write a linear equation representing the cost to rent the compact car.b. Write a linear equation representing the cost to rent a midsize car.c. Find the number of miles at which the cost to rent either car would be the same.miL06198_ch03_127-17012/19/0609:39 AMPage 148 Nishant Nishant:Desktop Folder:prakash 19/12/06:CONFIRMING PAGESSection 3.5 Systems of Linear Equations in Three Variables and Applications 149section 3.4 Practice Exercises PracticeProblems e-Professors Self-Tests Videos NetTutorReview ExercisesFor Exercises 12, state which method you would prefer to use to solve the system. Then solve the system.1. 2. 5x 2y 62x y 3y 9 2x3x y 16Mixed Exercises3. Antonio invested twice as much money in an account paying 5% simple interest as he did in an accountpaying 3.5% simple interest. If his total interest at the end of 1 year is $303.75, find the amount he investedin the 5% account.4. A school carnival sold tickets to ride on a Ferris wheel. The charge was $1.50 for an adult and $1.00 forstudents. If 54 tickets were sold for a total of $70.50, how many of each type of ticket were sold?5. How many liters of 20% saline solution must be mixed with 50% saline solution to produce 16 L of a31.25% saline solution?6. It takes a pilothr to travel with the wind to get from Jacksonville, Florida, to Myrtle Beach, South Carolina.Her return trip takes 2 hr flying against the wind. What is the speed of the wind and the speed of the plane instill air if the distance between Jacksonville and Myrtle Beach is 280 mi?7. Two angles are complementary. One angle measures 6 more than 5 times the measure of the other. What arethe measures of the two angles?Also see MathZone IQ Chapter 3, Section 4.134section 3.5 Systems of Linear Equations inThree Variables and Applications1. SolutionstoSystemsofLinearEquationsin Three VariablesA linear equation in three variables can be written in the form where A, B, and C are not all zero. For example, the equationisalinearequationinthreevariables. Solutionstothisequationareorderedtriplesoftheform(x, y, z)thatsatisfytheequation. Somesolutionstotheequationare 2x 3y z 6z 6 2x 3y Cz D, Ax By Objectives 1. Solutions to Systems ofLinear Equations in ThreeVariables2. Solving Systems of LinearEquations in ThreeVariables3. Applications of LinearEquations in ThreeVariablesGoonlinetoformiL06198_ch03_127-17012/19/0609:39 AMPage 149 Nishant Nishant:Desktop Folder:prakash 19/12/06:CONFIRMING PAGESSolution: Check:TrueTrueInfinitelymanyorderedtriplesserveassolutionstotheequation Thesetofallorderedtriplesthataresolutionstoalinearequationinthreevariables may be represented graphically by a plane in space.Asolutiontoasystemoflinearequationsinthreevariablesisanorderedtriplethatsatisfieseach equation. Geometrically, asolutionisapointofinter-sectionoftheplanesrepresentedbytheequationsinthesystem.A system of linear equations in three variables may have one unique solution,infinitelymanysolutions, ornosolution.z 6.2x 3y 12, 0, 22 2122 3102 122 611, 1, 12 2112 3112 112 6150 Chapter 3 Systems of Linear EquationsObjective 1: Exercises1. How many solutions are possible when solving a system of three equations with three variables?2. Which of the following points are solutions to the system?3. Which of the following points are solutions to the system?x y 4z 6x 3y z 14x y z 4112, 2, 22, 14, 2, 12, 11, 1, 12 2x y z 104x 2y 3z 10x 3y 2z 812, 1, 72, 13, 10, 62, 14, 0, 22 2. SolvingSystemsofLinearEquationsin Three VariablesTosolveasysteminvolvingthreevariables, thegoalistoeliminateonevariable.This reduces the system to two equations in two variables. One strategy for elimi-nating a variable is to pair up the original equations two at a time.Solving a System of Three Linear Equations in Three Variables1. Write each equation in standard form 2. Chooseapairofequations, andeliminateoneofthevariablesbyusingthe addition method.3. Choose a different pair of equations and eliminate the same variable.4. Oncesteps2and3arecomplete, youshouldhavetwoequationsintwovariables. Solve this system by using the methods from Sections 3.2 and 3.3.5. Substitute the values of the variables found in step 4 into any of the threeoriginalequationsthatcontainthethirdvariable. Solveforthethirdvariable.6. Check the ordered triple in each of the original equations.Ax By Cz D.miL06198_ch03_127-17012/19/0609:39 AMPage 150 Nishant Nishant:Desktop Folder:prakash 19/12/06:CONFIRMING PAGESexample1 Solving a System of Linear Equations in Three VariablesSolve the system.Solution:Step 2: Eliminate the y-variable from equationsand .Step 3: Eliminate the y-variable again, this time from equationsand.Step 4: Now equationsandcan be paired up to form a linear systemin two variables. Solve this system.Onceonevariablehasbeenfound, substitutethisvalueintoeitherequationinthe two-variable system, that is, either equationor.Substituteinto equation. D z 27x 5z 37x 5122 37x 10 37x 7x 1DE Dz 257z 11428x 20z 1228x 77z 1267x 5z 34x 11z 18DEE D6x 3y 9z 212x 3y 2z 34x 11z 182x y 3z 72x 3y 2z 3ACC A4x 2y 6z 143x 2y z 117x 5z 32x y 3z 73x 2y z 11 ABB A2x y 3z 73x 2y z 112x 3y 2z 3 ABC2x y 3z 73x 2y z 112x 3y 2z 3 Section 3.5 Systems of Linear Equations in Three Variables and Applications 151Step 1: The equations are already in standardform. It is often helpful to label the equations. The y-variable can be easily eliminated from equationsandand from equations and. This is accomplished by creatingopposite coefficients for the y-terms and thenadding the equations.C AB AMultiply by 2.Multiply by 3.Multiply by 4.EDMultiply by 7.ETip: It is important to notethat in steps 2 and 3, thesame variable is eliminated.miL06198_ch03_127-1701/4/0704:11 PMPage 151 pinnacle HardDisk_2:Desktop Folder:Sanjay2_4/1/07:The solution is (1, 3, 2).Check: True True True2x y 3z 7 2112 132 3122 73x 2y z 11 3112 2132 122 112x 3y 2z 3 2112 3132 2122 32 y 6 7y 4 7y 32x y 3z 72112 y 3122 7A152 Chapter 3 Systems of Linear EquationsStep 5: Now that two variables are known,substitute these values (x and z) intoany of the original three equations tofind the remaining variable y.Substituteandintoequation.Step 6: Check the ordered triple in the threeoriginal equations.Az 2 x 1Multiply by 2.Multiply by 1.Pair up equations andtoeliminate y.Pair up equations andtoeliminate y.C BB ASee e-ProfessorSolving Systems of LinearEquations in Three Variablesexample2 Solving a Dependent System of Linear EquationsSolvethesystem. Ifthereisnotauniquesolution, labelthesystemaseitherdependent or inconsistent.Solution:Thefirststepistomakeadecisionregardingthevariabletoeliminate. The y-variable is particularly easy to eliminate because the coefficients of y in equa-tionsandarealreadyopposites. They-variablecanbeeliminatedfrom equationsandby multiplying equationby 2.DEBecause equationsandare equivalent equations, it appears that this isa dependent system. By eliminating variables we obtain the identity Theresult0 0indicatesthatthereareinfinitelymanysolutionsandthatthesystem is dependent.5x z 11 5x z 115x z 115x z 110 0DE0 0.E D2x y 2z 3 4x 2y 4z 6x 2y 3z 5x 2y 3z 55x z 11BC3x y z 82x y 2z 35x z 11ABB C BB A3x y z 82x y 2z 3x 2y 3z 5ABCmiL06198_ch03_127-17012/19/0609:39 AMPage 152 Nishant Nishant:Desktop Folder:prakash 19/12/06:CONFIRMING PAGESexample3 Solving an Inconsistent System of Linear EquationsSolve the system. If there is not a unique solution, identify the system as eitherdependent or inconsistent.Solution:We will eliminate the x-variable.A(contradiction)Theresult0 9isacontradiction, indicatingthatthesystemhasnosolution.The system is inconsistent.4x 6y 14z 84x 6y 14z 1 0 92x 3y 7z 44x 6y 14z 1 5x y 3z 6 2x 3y 7z 44x 6y 14z 1 5x y 3z 6Section 3.5 Systems of Linear Equations in Three Variables and Applications 153Multiply by 2.BC3. ApplicationsofLinearEquationsin Three Variablesexample5 Applying Systems of Linear Equations to NutritionDoctorshavebecomeincreasinglyconcernedaboutthesodiumintakeintheU.S. diet. Recommendations by the American Medical Association indicate thatmost individuals should not exceed 2400 mg of sodium per day.Objective 2: ExercisesFor Exercises 14, solve the system of equations. If there is not a unique solution, label the system as eitherdependent or inconsistent.1. 2.3. 4.Expanding Your SkillsThe systems in Exercises 56 are called homogeneous systems because each system has (0, 0, 0) as a solution.However, if a system is dependent, it will have infinitely many more solutions. For each system determine whether(0, 0, 0) is the only solution or if the system is dependent.5. 6. 5x y 04y z 05x 5y z 02x 4y 8z 0x 3y z 0x 2y 5z 02x y 3z 33x 2y 4z 14x 2y 6z 66x 2y 2z 24x 8y 2z 52x 4y z 23x y z 84x 2y 3z 32x 3y 2z 1x y z 6x y z 22x 3y z 11 miL06198_ch03_127-17012/19/0609:39 AMPage 153 Nishant Nishant:Desktop Folder:prakash 19/12/06:CONFIRMING PAGES154 Chapter 3 Systems of Linear EquationsMultiply by 2.y 150 660 y 220 1030 x y z 1030z 220 660 z 880 x z 880Multiply by 1.DFrom Lizs meal we have:From Davids meal we have:From Melindas meal we have:Equationis missing the y-variable. Eliminating y from equationsand,we haveSolve the system formed by equationsand.From equationwe haveFrom equationwe haveTherefore, 1 slice of pizza has 660 mg of sodium, 1 serving of ice cream has 150 mgof sodium, and 1 glass of soda has 220 mg of sodium.AD3x 2z 24202x 2z 1760 x 6603x 2z 2420 x z 880BDD Bx y z 10302x y 2z 1910 x z 880x y z 10302x y 2z 1910ACC A B2x y 2z 1910 C3x 2z 2420 Bx y z 1030 AObjective 3: Exercises1. A triangle has one angle that measures 5 more than twice the smallest angle, and the largest anglemeasures 11 less than 3 times the measure of the smallest angle. Find the measures of the three angles.2. A movie theater charges $7 for adults, $5 for children under age 17, and $4 for seniors over age 60. For oneshowing of Batman the theater sold 222 tickets and took in $1383. If twice as many adult tickets were sold asthe total of children and senior tickets, how many tickets of each kind were sold?3. Combining peanuts, pecans, and cashews makes a party mixture of nuts. If the amount of peanuts equals theamount of pecans and cashews combined, and if there are twice as many cashews as pecans, how manyounces of each nut is used to make 48 oz of party mixture?Solution:Let x represent the sodium content of 1 slice of pizza.Let y represent the sodium content of 1 serving of ice cream.Let z represent the sodium content of 1 glass of soda.Liz ate 1 slice of pizza, 1 serving of ice cream, and 1 glass of soda for a totalof 1030 mg of sodium. David ate 3 slices of pizza, no ice cream, and 2 glasses ofsoda for a total of 2420 mg of sodium. Melinda ate 2 slices of pizza, 1 serving oficecream, and2glassesofsodaforatotalof1910mgofsodium. Howmuchsodium is in one serving of each item?miL06198_ch03_127-17012/19/0609:39 AMPage 154 Nishant Nishant:Desktop Folder:prakash 19/12/06:CONFIRMING PAGESSection 3.6 Solving Systems of Linear Equations by Using Matrices 155section 3.5 Practice Exercises1. Define the key terms.a. Linear equation in three variables b. Ordered tripleReview ExercisesSolve the system by using two methods: (a) the substitution method and (b) the addition method.2.3. Two cars leave Kansas City at the same time. One travels east and one travels west. After 3 hr the cars are369 mi apart. If one car travels 7 mph slower than the other, find the speed of each car.Mixed ExercisesForExercises46, solvethesystemsofequations. Ifasystemdoesnothaveauniquesolution, indicatewhetheritis dependent or inconsistent.4. 5. 6.7. Three pumps are working to drain a construction site. Working together, the pumps can pump 950 gal/hr ofwater. The slowest pump pumps 150 gal/hr less than the fastest pump. The fastest pump pumps 150 gal/hrless than the sum of the other two pumps. How many gallons can each pump drain per hour?Also see MathZone IQ Chapter 3, Section 5.2x 5y 8 x 2y z 8 10x 6y 2z 42y 3z 2 x 2y z 6 x y z 23x 4z 5 5x 3y z 5 5x 5y 5z 303x y 44x y 5 PracticeProblems e-Professors Self-Tests Videos NetTutorsection 3.6 Solving Systems of Linear Equations by Using Matrices1. IntroductiontoMatricesA matrix is a rectangular array of numbers (the plural of matrix is matrices). Therows of a matrix are read horizontally, and the columns of a matrix are read ver-tically. Every number or entry within a matrix is called an element of the matrix.The order of a matrix is determined by the number of rows and number ofcolumns. Amatrixwithm rowsandn columnsisan(readas m byn)matrix. Notice that with the order of a matrix, the number of rows is given first,followedbythenumberofcolumns.m nObjectives 1. Introduction to Matrices2. Solving Systems of LinearEquations by Using theGauss-Jordan MethodGoonlinetoformiL06198_ch03_127-17012/19/0609:40 AMPage 155 Nishant Nishant:Desktop Folder:prakash 19/12/06:CONFIRMING PAGESexample2 Writing the Augmented Matrix of a System of Linear EquationsWritetheaugmentedmatrixforthelinearsystem.Solution:2 0 30 2 11 1 01424x y 42y z 22x 3z 14Tip: Notice that zeros areinserted to denote thecoefficient of each missingterm.156 Chapter 3 Systems of Linear Equationsexample1 Determining the Order of a MatrixDeterminetheorderofeachmatrix.a. b. c. d.Solution:a. This matrix has two rows and three columns. Therefore, it is a 2 3 matrix.b. This matrix has four rows and one column. Therefore, it is a 4 1 matrix.A matrix with one column is called a column matrix.c. This matrix has three rows and three columns. Therefore, it is a 3 3 matrix. Amatrix with the same number of rows and columns is called a square matrix.d. This matrix has one row and three columns. Therefore, it is a 1 3 matrix.A matrix with one row is called a row matrix.A matrix can be used to represent a system of linear equations written in stan-dard form. To do so, we extract the coefficients of the variable terms and the con-stants within the equation. For example, consider the systemThe matrix A is called the coefficient matrix.Ifweextractboththecoefficientsandtheconstantsfromtheequations, wecanconstruct the augmented matrix of the system:A vertical bar is inserted into an augmented matrix to designate the position of theequal signs.c2 11 2`55dA c2 11 2dx 2y 5 2x y 53 abc4 1 0 00 1 00 0 1 1.907.26.1 c2 4 15 p 27dSee e-ProfessorIntroduction to MatricesmiL06198_ch03_127-17012/19/0609:40 AMPage 156 Nishant Nishant:Desktop Folder:prakash 19/12/06:CONFIRMING PAGESSection 3.6 Solving Systems of Linear Equations by Using Matrices 157example3 Writing a Linear System from an Augmented MatrixWrite a system of linear equations represented by the augmented matrix.Solution: 3x y z 2 x y 2z 5 2x y 3z 142 1 31 1 23 1 11452Objective 1: ExercisesFor Exercises 13, (a) determine the order of each matrix and (b) determine if the matrix is a row matrix, a columnmatrix, a square matrix, or none of these.1. 2. 3.For Exercises 45, set up the augmented matrix.4. 5.For Exercises 67, write a system of linear equations represented by the augmented matrix.6. 7. 1 0 0 0 1 0 0 0 1 0.56.13.9 c2 5 7 15 `1545d5x 17 2z8x 6z 268x 3y 12z 24 yx 3y 32x 5y 4c133462 1 78d 9 4 31 8 45 8 7 512See e-ProfessorIntroduction to MatricesElementary Row OperationsThe following elementary row operations performed on an augmented ma-trixproduceanequivalentaugmentedmatrix:1. Interchangetworows.2. Multiplyeveryelementinarowbyanonzerorealnumber.3. Addamultipleofonerowtoanotherrow.2. SolvingSystemsofLinearEquationsbyUsing theGauss-JordanMethodmiL06198_ch03_127-17012/19/0609:40 AMPage 157 Nishant Nishant:Desktop Folder:prakash 19/12/06:CONFIRMING PAGESexample4 Solving a System of Linear Equations by Using the Gauss-Jordan MethodSolve by using the Gauss-Jordan method.Solution:First write each equation in the system in standard form.3R1 R31R32R1 R21R21 1 0 2 1 2 3 6 7 510143x 6y 7z 14 3x 6y 7z 142x 2z y 10 2x y 2z 10x y 5x y 53x 6y 7z 142x 2z y 10x y 5Set up the augmented matrix.Multiply row 1 by2 and add the resultto row 2. Multiplyrow 1 by 3 andadd the result torow 3.1 1 0 0 1 2 0 3 7 501158 Chapter 3 Systems of Linear Equations3R2 R31R31R2 R11R11 0 0 0 1 0 0 0 1 3211 0 2 0 1 2 0 0 1 501Multiply row 2 by1 and add the re-sult to row 1. Multi-ply row 2 by 3and add the resultto row 3.Multiply row 3 by 2and add the result torow 1. Multiply row3 by 2 and addthe result to row 2.The Gauss-Jordan method uses a series of elementary row operations performedon the augmented matrix to produce a simpler augmented matrix. In particular, wewant to produce an augmented matrix that has 1s along the diagonal of the matrixof coefficients and 0s for the remaining entries in the matrix of coefficients. A matrixwritten in this way is said to be written in reduced row echelon form. For example,the augmented matrix shown is written in reduced row echelon form.Thesolutiontothecorrespondingsystemofequationsiseasilyrecognizedasx 4, y 1, and z 0.1 0 00 1 00 0 1410From the reduced row echelon form of the matrix, we have andThe solution to the system is (3, 2, 1). z 1.x 3, y 2,2R3 R21R22R3 R11R1See e-ProfessorSolving Systems of LinearEquations Using the Gauss-Jordan MethodmiL06198_ch03_127-17012/19/0609:40 AMPage 158 Nishant Nishant:Desktop Folder:prakash 19/12/06:CONFIRMING PAGESSection 3.6 Solving Systems of Linear Equations by Using Matrices 159The order in which we manipulate the elements of an augmented matrix to pro-duce reduced row echelon form was demonstrated in Example 4. In general, theorderisasfollows. First produce a 1 in the first row, first column. Then use the first row to obtain0s in the first column below this element. Next, if possible, produce a 1 in the second row, second column. Use thesecond row to obtain 0s above and below this element. Next, if possible, produce a 1 in the third row, third column. Use the third rowto obtain 0s above and below this element. The process continues until reduced row echelon form is obtained.It is particularly easy to recognize a dependent or inconsistent system of equationsfrom the reduced row echelon form of an augmented matrix. If the last row repre-sentsanidentity, thesystemisdependent. Ifthelastrowrepresentsacontradic-tion, the system is inconsistent. This is demonstrated in Example 5.example5 Solving an Inconsistent System of Equations by Using the Gauss-Jordan MethodSolve by using the Gauss-Jordan method.Solution:Set up the augmented matrix.Multiply row 1 byand add theresult to row 2.Thesecondrowoftheaugmentedmatrixrepresentsthecontradiction 0 5; hence, the system is inconsistent. There is no solution.15c2 50 0 `105d15R1 R21R2c2 5251` 107d 25x y 7 2x 5y 10Objective 2: Exercises1. Given the matrix Fa. What is the element in the second row and second column?b. What is the element in the first row and third column?2. Given the matrix Jwrite the matrix obtained by multiplying the elements in the second row by 13.J c1 1 0 3 `76dF c1 8 12 13 `02dmiL06198_ch03_127-17012/19/0609:40 AMPage 159 Nishant Nishant:Desktop Folder:prakash 19/12/06:CONFIRMING PAGES3. Given the matrix Nwrite the matrix obtained by multiplying the first row by 2 and adding the result to row 2.4. What does the notationmean when one is performing the Gauss-Jordan method?For Exercises 58, solve the systems by using the Gauss-Jordan method.5. 6. 7. 8.3x y 14z 2x 3y 8z 1 6x 2y 3 4x 10y 2 2x 5y 42x 3y 2z 11 3x y 4 2x 5y 1 x 3y 32R31R3N c123 2 `512d160 Chapter 3 Systems of Linear Equations CalculatorConnectionsMany graphing calculators have a matrix editor in which the user defines theorder of the matrix and then enters the elements of the matrix. For example,the 2 3 matrixis entered as shown.D c2 3 3 1 `138dOnceanaugmentedmatrixhasbeenenteredintoagraphing calculator, a rref. function can be used to transformthe matrix into reduced row echelon form (see figure).1. Define the key terms.a. Matrix b. Order of a matrix c. Column matrix d. Square matrixe. Row matrix f. Coefficient matrix g. Augmented matrix h. Reduced row echelon formsection 3.6 Practice Exercises PracticeProblems e-Professors Self-Tests Videos NetTutorGoonlinetoformiL06198_ch03_127-17012/19/0609:40 AMPage 160 Nishant Nishant:Desktop Folder:prakash 19/12/06:CONFIRMING PAGESSection 3.6 Solving Systems of Linear Equations by Using Matrices 161Review ExercisesFor Exercises 23, solve the system by using any method.2. 3.Mixed ExercisesFor Exercises 45, determine the order of each matrix.4. 5.Set up the augmented matrix.6.Write a corresponding system of equations from the augmented matrix.7.For Exercises 89, solve the system by using the Gauss-Jordan method.8. 9.Also see MathZone IQ Chapter 3, Section 6.2x 2y z 92x y 3z 8 12x 5y 6x y z 4 4x 3y 6 1 0 0 0 1 0 0 0 1 528x y z 4 2x y 3z 8 2x 2y z 97124 5 69 20 3x 3y 4z 22x y 3z 7 x 2y 132x y z 4 x 6y 9miL06198_ch03_127-17012/19/0609:40 AMPage 161 Nishant Nishant:Desktop Folder:prakash 19/12/06:CONFIRMING PAGESchapter 3 summarysection 3.1 SolvingSystemsofEquationsbyGraphingKey ConceptsAsystemoflinearequations intwovariablescanbesolved by graphing.A solution to a system of linear equations is an or-deredpairthatsatisfieseachequationinthesystem.Graphically, this represents a point of intersection of thelines.Theremaybeonesolution, infinitelymanysolu-tions, or no solution.One solution Infinitely manyNo solutionConsistent solutionsInconsistentIndependent Consistent IndependentDependentA system of equations is consistent if there is at leastonesolution. Asystemisinconsistent ifthereisnosolution.A linear system in x and y is dependent if two equa-tions represent the same line. The solution set is the setof all points on the line.Iftwolinearequationsrepresentdifferentlines,then the system of equations is independent.ExamplesExample 1Solve by graphing.Write each equation inform to graph.The solution is the point of intersection (1, 2).4 5 4 5 3 1 2 3213454511 2yx322x y 0x y 3(1, 2)y 2xy x 3y mx b 2x y 0 x y 3162 Chapter 3 Systems of Linear EquationsmiL06198_ch03_127-17012/19/0609:40 AMPage 162 Nishant Nishant:Desktop Folder:prakash 19/12/06:CONFIRMING PAGESSummary 163Key ConceptsSubstitution Method1. Isolate one of the variables.2. Substitute the quantity found in step 1 into theother equation.3. Solve the resulting equation.4. Substitute the value from step 3 back into theequation from step 1 to solve for the remainingvariable.5. Check the ordered pair in both equations, andwrite the answer as an ordered pair.ExamplesExample 1Now solve for y.The solution is (2, 1) and checks in both equations.y 1y 2122 5y 2x 5 x 2 2x 4 2x 10 14 4x 10 6x 14 212x 52 6x 14y 2x 52x y 52y 6x 14section 3.2 SolvingSystemsofEquationsbyUsingtheSubstitutionMethodIsolate a variable.SubstitutevSummary 163miL06198_ch03_127-17012/19/0609:40 AMPage 163 Nishant Nishant:Desktop Folder:prakash 19/12/06:CONFIRMING PAGESMult. by 2.164 Chapter 3 Systems of Linear EquationsExample 3Identity. The system is dependent. 0 0 2x 6y 2 2x 6y 2 2x 6y 2 x 3y 1Asystemisindependentifthetwoequationsrep-resent different lines. A system is dependent if the twoequationsrepresentthesameline. Thisproducesinfi-nitelymanysolutions. Adependentsystemisdetectedby an identity (such as 0 02.section 3.3 SolvingSystemsofEquationsbyUsingthe AdditionMethodKey ConceptsAddition Method1. Write both equations in standard form.2. Clear fractions or decimals (optional).3. Multiply one or both equations by nonzeroconstants to create opposite coefficients for oneof the variables.4. Add the equations from step 3 to eliminate onevariable.5. Solve for the remaining variable.6. Substitute the known value from step 5 back intoone of the original equations to solve for theother variable.7. Check the ordered pair in both equations.Asystemisconsistentifthereisatleastonesolution. A system is inconsistent if there is no solution.Aninconsistentsystemisdetectedbyacontradiction(such as 0 52.Ax By CExamplesExample 1The solution is (2, 3) and checks in both equations.Example 2Contradiction. The system is inconsistent. 0 7 4x 2y 1 4x 2y 1 4x 2y 6 2x y 3y 34y 126 4y 183122 4y 18x 229x 5820x 12y 4 5x 3y 19x 12y 54 3x 4y 18Mult. by 3.Mult. by 4.Mult. by 2.miL06198_ch03_127-17012/19/0609:40 AMPage 164 Nishant Nishant:Desktop Folder:prakash 19/12/06:CONFIRMING PAGESSummary 165section 3.4 ApplicationsofSystemsofLinearEquationsin Two VariablesKey ConceptsSolveapplicationproblemsbyusingsystemsoflinearequations in two variables. Cost applications Mixture applications Applications involving principal and interest Applications involving distance, rate, and time Geometry applicationsSteps to Solve Applications:1. Label two variables.2. Construct two equations in words.3. Write two equations.4. Solve the system.5. Write the answer.ExamplesExample 1Mercedes invested $1500 more in a certificate of de-positthatpays6.5%simpleinterestthanshedidina savings account that pays 4% simple interest. If hertotal interest at the end of 1 year is $622.50, find theamountsheinvestedinthe6.5%account.Letx representtheamountofmoneyinvestedat6.5%.Let y represent the amount of money invested at 4%.Using substitution givesMercedes invested $6500 at 6.5% and $5000 at 4%.x 1500 5000 6500 y 5000 0.105y 525 97.5 0.065y 0.04y 622.50 0.06511500 y2 0.04y 622.500.065x 0.04y 622.50x 1500 yInterest earnedfrom 6.5%account interest earnedfrom 4%account $622.50aAmount investedat 6.5%b aamount investedat 4%b $1500miL06198_ch03_127-17012/19/0609:40 AMPage 165 Nishant Nishant:Desktop Folder:prakash 19/12/06:CONFIRMING PAGES166 Chapter 3 Systems of Linear Equationssection 3.5 SystemsofLinearEquationsin Three Variablesand ApplicationsKey ConceptsAlinearequationinthreevariables canbewrittenintheformwhereA, B, andC arenot all zero. The graph of a linear equation in three vari-ables is a plane in space.A solution to a system of linear equations in threevariablesisanorderedtriple thatsatisfieseachequa-tion. Graphically, asolutionisapointofintersectionamong three planes.A system of linear equations in three variables mayhave one unique solution, infinitely many solutions (de-pendent system), or no solution (inconsistent system).Ax By Cz D,ExamplesExample 1and and Substituteinto either equationor.Substituteandintoequation, ,or.The solution is (2, 1, 0).z 0122 2112 z 4 ACB A y 1 x 2x 24x 84x 112 9 DE D y 1y 16y 64x 7y 15 4x 7y 15 E4x y 94x y 9 DE 4x 7y 152x 3y 2z 72x 4y 2z 8 C A 2 D 4x y 9 3x y z 5x 2y z 4 B A2x 3y 2z 7 C3x y z 5 Bx 2y z 4 AmiL06198_ch03_127-17012/19/0609:40 AMPage 166 Nishant Nishant:Desktop Folder:prakash 19/12/06:CONFIRMING PAGESSummary 167section 3.6 SolvingSystemsofLinearEquationsbyUsingMatricesKey ConceptsA matrix is a rectangular array of numbers displayed inrows and columns. Every number or entry within a ma-trix is called an element of the matrix.The order of a matrix is determined by the numberof rows and number of columns. A matrix with m rowsand n columns is anmatrix.A system of equations written in standard form canberepresentedbyanaugmentedmatrix consistingofthecoefficientsofthetermsofeachequationinthesystem.TheGauss-Jordanmethodcanbeusedtosolveasystemofequationsbyusingthefollowingelementaryrow operations on an augmented matrix.1. Interchange two rows.2. Multiply every element in a row by a nonzero realnumber.3. Add a multiple of one row to another row.Theseoperationsareusedtowritethematrixinreduced row echelon form.which represents the solution, and y b. x ac1 0 0 1 `abdm nExamplesExample 1[1 2 5] is amatrix (called a row matrix).is amatrix (called a square matrix).is amatrix (called a column matrix).Example 2The augmented matrix for isExample 3SolvethesystemfromExample2byusingthe Gauss-Jordan method.Solution:x 2andy 4c1 0 0 1 `24d2R2 R11R1c1 2 0 1 `64d19R21R2c1 2 0 9 `636d4R1 R21R2c1 2 4 1 `612d R13R2c4 1 1 2 `126dx 2y 64x y 122 1 c41d2 2 c1185d1 3miL06198_ch03_127-17012/19/0609:40 AMPage 167 Nishant Nishant:Desktop Folder:prakash 19/12/06:CONFIRMING PAGES168 Chapter 3 Systems of Linear Equationschapter 3 test1. Determine if the ordered pairis a solutionto the system.Match each figure with the appropriate description.2. 3. 12x 2y 7 4x 3y 5114, 224.a. The system is consistent and dependent.There are infinitely many solutions.b. The system is consistent and independent.There is one solution.c. The system is inconsistent and independent.There are no solutions.xyxy5. Solve the system by graphing.3x y 74x 2y 44 5 4 5 3 1 2 3213454511 2yx326. Solve the system by using the substitution method.7. Solve the system by using the addition method.For Exercises 813, solve the system of equations.8. 9.10. 11.12.13.14. How many liters of a 20% acid solution shouldbe mixed with a 60% acid solution to produce200 L of a 44% acid solution?15. Two angles are complementary. Two times themeasure of one angle isless than themeasure of the other. Find the measure of eachangle.16. Working together, Joanne, Kent, and Geoff canprocess 504 orders per day for their business.Kent can process 20 more orders per day thanJoanne can process. Geoff can process 104 fewerorders per day than Kent and Joanne combined.Find the number of orders that each person canprocess per day.60x y z 443x y 2z 292x 2y 4z 6 0.4x 2 0.5y 0.03y 0.06x 0.3y 2x 4 141x 22 16y4x 5 2y15x 12y 175 9y 2x 273x 5y 718x 30y 42 7y 5x 21 3x 2y 1 6x 8y 5 y x 9 3x 5y 13xymiL06198_ch03_127-17012/19/0609:41 AMPage 168 Nishant Nishant:Desktop Folder:prakash 19/12/06:CONFIRMING PAGESCumulative review exercises 16917. Write an example of amatrix.18. Given the matrix Aa. Write the matrix obtained by multiplying thefirst row by 4 and adding the result torow 2.A 1 2 1 4 0 1 5 6 3 3203 2ForExercises1920, solvebyusingtheGauss-Jordanmethod.19. 20.2y z 52x y 0x y z 1 5x 4y 34x 2y 8b. Using the matrix obtained in part (a),write the matrix obtained by multiplying the first row by 5 and adding the result to row 3.For Exercises 12, solve the equation.1.2.3. Solve the inequality. Write the answer in intervalnotation.4. Identify the slope and the x- and y-intercepts ofthe line For Exercises 56, graph the lines.5. 6.7. Find the slope of the line passing through thepointsand16, 102. 14, 102x 2 y 13x 45x 2y 15.3y 21y 12 6 5121a 22 3412a 12 16512x 12 213x 12 7 218x 12chapters 13 cumulative review exercises4 5 4 5 3 1 2 3213454511 2yx324 5 4 5 3 1 2 3213454511 2yx328. Find an equation for the line that passes throughthe pointsandWrite the answerin slope-intercept form.9. Solve the system by using the addition method.10. Solve the system by using the substitution method.11. A childs piggy bank contains 19 coins consistingof nickels, dimes, and quarters. The total amountof money in the bank is $3.05. If the number ofquarters is 1 more than twice the number ofnickels, find the number of each type of coin inthe bank.12. Two video clubs rent tapes according to thefollowing fee schedules:Club 1: $25 initiation fee plus $2.50 per tapeClub 2: $10 initiation fee plus $3.00 per tapey 3x 12x y 412x 34y 12x 3y 612, 42. 13, 82miL06198_ch03_127-17012/19/0609:41 AMPage 169 Nishant Nishant:Desktop Folder:prakash 19/12/06:CONFIRMING PAGES170 Chapter 3 Systems of Linear Equationsa. Write a linear equation describing the totalcost, y, of renting x tapes from club 1.b. Write a linear equation describing the totalcost, y, of renting x tapes from club 2.c. How many tapes would have to be rented tomake the cost for club 1 the same as the costfor club 2?13. Solve the system.3x 2y 3z 34x 5y 7z 12x 3y 2z 614. Determine the order of the matrix.15. Write an example of amatrix.16. List at least two different row operations.17. Solve the system by using the Gauss-Jordanmethod.4x y 52x 4y 22 4c4 5 12 6 0dmiL06198_ch03_127-17012/19/0609:41 AMPage 170 Nishant Nishant:Desktop Folder:prakash 19/12/06:CONFIRMING PAGES