* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Reteach Using Graphs and Tables to Solve Linear Systems
Survey
Document related concepts
Debye–Hückel equation wikipedia , lookup
Schrödinger equation wikipedia , lookup
Two-body problem in general relativity wikipedia , lookup
BKL singularity wikipedia , lookup
Equation of state wikipedia , lookup
Derivation of the Navier–Stokes equations wikipedia , lookup
Calculus of variations wikipedia , lookup
Maxwell's equations wikipedia , lookup
Euler equations (fluid dynamics) wikipedia , lookup
Navier–Stokes equations wikipedia , lookup
Schwarzschild geodesics wikipedia , lookup
Equations of motion wikipedia , lookup
Differential equation wikipedia , lookup
Transcript
Name LESSON 3-1 Date Class Reteach Using Graphs and Tables to Solve Linear Systems A linear system of equations is a set of two or more linear equations. To solve a linear system, find all the ordered pairs (x, y) that make both equations true. Use a table and a graph to solve a system of equations. yx2 y x 2 Solve each equation for y.→ y 2x 5 y 2x 5 Make a table of values for each equation. When x 1, y 3 for y x 2 y 2x 5 both equations. x y x y { { 2 4 1 3 0 2 1 } 1 2 1 1 3 0 5 1 7 Y On a graph, the point where the lines intersect is the solution. Use the table to draw the graph of each equation. The lines appear to intersect at 1, 3 . Substitute 1, 3 into the original equations to check. yx2 y 2x 5 ? 2 ? 5 3 (1) 3 2(1) 2 2✓ 5 5✓ YX X YX Solve the system using a table and a graph. Give the ordered pair that solves both equations. 1. {2xx yy15 Solution: y x 1 y 2x 5 x y x y 0 1 0 1 2 0 5 3 1 1 1 2 3 1 2 3 2, 1 Y X Copyright © by Holt, Rinehart and Winston. All rights reserved. a207c03-1_rt.indd 6 6 Holt Algebra 2 12/29/05 7:50:18 PM Process Black Name Date Class Reteach LESSON 3-1 Using Graphs and Tables to Solve Linear Systems (continued) To classify a linear system: Step 1 Write each equation in the form y mx b. Step 2 Compare the slopes and y-intercepts. Step 3 Classify by the number of solutions of the system. Exactly One Solution Independent The lines have different slopes and intersect at one point. xy3 xy1 Infinitely Many Solutions Dependent No Solution Inconsistent The lines have the same slope and y-intercept. Their graph is the same line. The lines have the same slope and different y-intercepts. The lines are parallel. { 2x4y y8x1 4 3 { yy 2x1 2x { Solve each equation for y. Solve each equation for y. y 2x 1; m 2, b 1 y x 3; m 1 y 2x 1; m 2, b 1 y x 1; m 1 The slopes and the The slopes are different. y-intercepts are the same. The system has The system has infinitely many one solution and is solutions and is dependent. independent. { { XY Y XY X Remember: m slope and b y-intercept. Solve each equation for y. 2, b 3 {yy 2x2x 1;3; mm 2, b1 The slopes are the same but the y-intercepts are different. The system has no solution and is inconsistent. Y XY YX YX X Y X YX Classify each system and determine the number of solutions. yx2 y 1 3x 3. 2. y 1 x 2y 6x 2 { { y x 2 1 , b 2 y 3x 1 , m 3 ,b 1 y x 1 , m 1 , b 1 y 3x 1 , m 3 ,b 1 ,m Number of solutions: none Number of solutions: inconsistent Copyright © by Holt, Rinehart and Winston. All rights reserved. a207c03-1_rt.indd 7 infinitely many dependent 7 Holt Algebra 2 12/29/05 7:50:21 PM Process Black *À>VÌViÊ 5SING'RAPHSAND4ABLESTO3OLVE,INEAR3YSTEMS ,%33/. Σ Σ #LASSIFYEACHSYSTEMANDDETERMINETHENUMBEROFSOLUTIONS $OESTHEGIVENORDEREDPAIRSOLVETHESYSTEMOFEQUATIONS 3UBSTITUTEEACHVALUEFORXANDYINTOTHEEQUATIONS7RITEYESORNO OÊ OÊ XY SD XY SDSD SDSD .O OÊ XY SD XY 9ES XY SD XY .O 5SEATABLEANDAGRAPHTOSOLVETHESYSTEM YX XY OÊ A -AKEATABLEOFVALUESFOREACHEQUATION YX Y X >ÞÊÃÕÌà OÊ YX SD XY XY XY VÃÃÌiÌÆÊÊ ÃÃÌiÌ]Ê`i«i ÃÕÌà `iÌÆÊiÊÃÕÌ ÌÊÃÊÌ iÊÊ ÃÕÌ° OÊ ÊT£]Ê{ÊEÊ XY SD XY OÊ YX X Y?? OÊ ?? ?? ÃÃÌiÌ]ÊÊ `i«i`iÌÆÊvÌiÞÊ YX SD XY Y Y OÊ OÊ YX XY 5SESUBSTITUTIONTODETERMINEIFTHEGIVENORDEREDPAIRISANELEMENT OFTHESOLUTIONSETFORTHESYSTEMOFEQUATIONS)FITISNOTGIVETHE CORRECTSOLUTION XY X *À>VÌViÊ 5SING'RAPHSAND4ABLESTO3OLVE,INEAR3YSTEMS --" X ÊT£]ÊÓÊEÊ OÊ XY SDO Ê XY XY SDO Ê XY YX SD XY 3OLVEBYGRAPHINGASYSTEMOFEQUATIONS !PUPPYPENISFOOTLONGERTHANTWICEITSWIDTH *OHNWANTSTOINCREASETHELENGTHANDWIDTHBY FEETEACHTOENLARGETHEAREABYSQUAREFEET7HAT WILLBETHEAREAOFTHENEWPEN £ÓÈÊõÕ>ÀiÊviiÌ 5SEAGRAPHTOSOLVEEACHSYSTEM TE X Σ +EESHAHASMOREQUARTERSTHANDIMESWHICH TOGETHERTOTAL(OWMANYCOINSDOESSHE HAVEINQUARTERSANDDIMES Y Y X X (OLT!LGEBRA *À>VÌViÊ 5SING'RAPHSAND4ABLESTO3OLVE,INEAR3YSTEMS -ATCHEACHSYSTEMOFEQUATIONSWITHTHECORRESPONDINGCONDITIONS THATITSATISFIES 4HESYSTEMISCONSISTENTANDDEPENDENT !LINEARSYSTEMOFEQUATIONSISASETOFTWOORMORELINEAREQUATIONS4OSOLVEALINEAR SYSTEMFINDALLTHEORDEREDPAIRSXYTHATMAKEBOTHEQUATIONSTRUE5SEATABLEANDA GRAPHTOSOLVEASYSTEMOFEQUATIONS YX YX 3OLVEEACHEQUATIONFORY→ YX YX -AKEATABLEOFVALUESFOREACHEQUATION 7HENXYFOR YX YX BOTHEQUATIONS X Y X Y OÊ OÊ OÊ Y?? X YX 4HESYSTEMISCONSISTENTANDINDEPENDENT Y?? X YX OÊ 3OLVE !TUBCONTAININGGALLONSOFWATERISDRAININGATARATEOFGALLONPER HOUR!BASINOFGALLONSOFWATERISDRAININGATARATEOFGALLON EVERYHOURS Z OÊ YX 4HELINESAPPEARTOINTERSECTATSD 3UBSTITUTESDINTOTHEORIGINALEQUATIONSTOCHECK YX YX B )FBOTHCONTAINERSBEGANDRAININGATTHESAMETIMEHOWSOONWILLTHETUBANDBASIN HOLDTHESAMEAMOUNTOFWATER Y 5SETHETABLETODRAWTHEGRAPHOFEACHEQUATION YÊÊXÊÊ£È ÊÊÊÊÊÊ ÊÊ ÊÊ ÊÊ YÊÊÊÚÚ Ê£ÈÊXÊΰx X YX 3OLVETHESYSTEMUSINGATABLEANDAGRAPH'IVETHEORDEREDPAIR THATSOLVESBOTHEQUATIONS OÊ XY XY £xÊ YX X Y Ê Ê£ C 7HENTHEAMOUNTSAREEQUALHOWMUCHWATERWILLBEINEACHCONTAINER £Ê}> *ENNAHASINASAVINGSACCOUNT3HEADDSTOHERACCOUNT EACHMONTH,UISHASINHISSAVINGSACCOUNT(EWITHDRAWS FROMHISACCOUNTEACHMONTH B 7HATWILLBETHEBALANCEINEACHACCOUNT OÊ /NAGRAPHTHEPOINTWHERETHELINESINTERSECTISTHESOLUTION A 7RITEASYSTEMOFEQUATIONSTHATREPRESENTSYTHENUMBEROFGALLONSLEFTINTHE CONTAINERAFTERXHOURS A )NHOWMANYMONTHSWILLTHEYHAVETHESAME BALANCEINTHEIRSAVINGSACCOUNTS ÌÊ}iLÀ>ÊÓ ,iÌi>V 5SING'RAPHSAND4ABLESTO3OLVE,INEAR3YSTEMS Σ Y?? X YX #OPYRIGHT©BY(OLT2INEHARTAND7INSTON !LLRIGHTSRESERVED --" 4HESYSTEMISINCONSISTENT ÎxʵÕ>ÀÌiÀÃÊÊÓxÊ`iÃÊÊÊ ÈäÊVà #OPYRIGHT©BY(OLT2INEHARTAND7INSTON !LLRIGHTSRESERVED --" TE X OÊ YX XY Y Y TE C 7HICHORDEREDPAIRSOLVESBOTHEQUATIONS OÊ ÊTx]ÊxÊEÊ ÌÊÃÊÌ iÊÊ ÃÕÌ° B 5SETHEVALUESINTHETABLESTOGRAPHEACHEQUATION XY XY ÊTÈ]ÊÓÊEÊ Ê Êä £ Ó 3OLUTION YX X Y x Î £ Ê Ê£ ÊTÓ]Ê£ÊEÊ Y X £xÊÌ Ã f£xä #OPYRIGHT©BY(OLT2INEHARTAND7INSTON !LLRIGHTSRESERVED Copyright © by Holt, Rinehart and Winston. All rights reserved. AK4up.indd 51 ÌÊ}iLÀ>ÊÓ #OPYRIGHT©BY(OLT2INEHARTAND7INSTON !LLRIGHTSRESERVED 51 ÌÊ}iLÀ>ÊÓ Holt Algebra 2 4/25/06 3:06:20 PM ,iÌi>V 5SING'RAPHSAND4ABLESTO3OLVE,INEAR3YSTEMS CONTINUED --" Σ 4OCLASSIFYALINEARSYSTEM 3TEP 7RITEEACHEQUATIONINTHEFORMYMXB 3TEP #OMPARETHESLOPESANDYINTERCEPTS 3TEP #LASSIFYBYTHENUMBEROFSOLUTIONSOFTHESYSTEM %XACTLY/NE3OLUTION )NDEPENDENT OÊ Î£ OÊ XY XBYC .O3OLUTION )NCONSISTENT 4HELINESHAVETHESAME 4HELINESHAVETHESAMESLOPE SLOPEANDYINTERCEPT4HEIR ANDDIFFERENTYINTERCEPTS4HE GRAPHISTHESAMELINE LINESAREPARALLEL XY OÊ YX >i}i 3YSTEM#LASSIFICATIONS !SYSTEMOFLINEAREQUATIONSCANBECONSISTENTANDINDEPENDENT CONSISTENTANDDEPENDENTORINCONSISTENT5SETHEFOLLOWINGSYSTEMOF EQUATIONSFOR%XERCISESn 2EMEMBERMSLOPE ANDBYINTERCEPT )NFINITELY-ANY3OLUTIONS $EPENDENT 4HELINESHAVEDIFFERENT SLOPESANDINTERSECTAT ONEPOINT XY XY --" OÊ YX YX &INDVALUESOFBANDCSOTHATTHESYSTEM ISCONSISTENTANDDEPENDENT LÊÊ]ÊVÊÊÎÈ &INDVALUESOFBANDCSOTHATTHESYSTEM ISINCONSISTENT LÊÊ]ÊVÊqÊÎÈ &INDVALUESOFBANDCSOTHATTHESYSTEM ISCONSISTENTANDINDEPENDENT LÊqÊ 5SETHISSYSTEMOFEQUATIONSFOR%XERCISES 3OLVEEACHEQUATIONFORY 3OLVEEACHEQUATIONFORY 3OLVEEACHEQUATIONFORY XBYC XMB Y A XMB YXM Y Y X XB YC A MB YXMB YXM XBYC A 4HESLOPESANDTHE 4HESLOPESAREDIFFERENT 4HESLOPESARETHESAMEBUTTHE $ESCRIBEHOWTHEGRAPHOFTHESEEQUATIONSLOOKSWHENTHESYSTEM YINTERCEPTSARETHESAME YINTERCEPTSAREDIFFERENT HASEXACTLYONESOLUTION 4HESYSTEMHAS 4HESYSTEMHASINFINITELYMANY 4HESYSTEMHASNOSOLUTIONAND ONESOLUTIONANDIS SOLUTIONSANDISDEPENDENT ÎÊiÃÊÌiÀÃiVÌÊ>ÌÊ>ÊÃ}iÊ«Ì ISINCONSISTENT INDEPENDENT OÊ OÊ XY Y Y XY OÊ OÊ XY YX X YX X %XPLAINWHYTHESYSTEMOFEQUATIONSMUSTBECONSISTENTIFTHECONSTANT TERMSCCANDCAREALLZERO YX OÊ Y ÝÊÓ M £ B Y Ó ÝÊ£ M £ B £ i .UMBEROFSOLUTIONS Y Y ÎÝÊÊ£ M Î .UMBEROFSOLUTIONS --" Σ OÊ 5NDERWHATCONDITIONSWILLTHISSYSTEMBECONSISTENTANDINDEPENDENT B £ *ÃÃLiÊ>ÃÜiÀ\Ê/ÊLiÊ`i«i`iÌ]ÊÌ iÊëiÃÊV>ÌÊLiÊiµÕ>°Ê/ ÃÊV>Ê VÊ Ê]Ê Ê>`ÊqÊLV]ÊÀÊ>`ÊÊLVÊqÊä° LiÊÃÌ>Ìi`Ê>ÃÊÊÚÚÚ Ê>Ê ÊÊqÊÊÊÚÚÚ L ` vÌiÞÊ>Þ `i«i`iÌ #OPYRIGHT©BY(OLT2INEHARTAND7INSTON !LLRIGHTSRESERVED 5SETHISSYSTEMOFEQUATIONSFOR%XERCISE AXBYE CXDYF B £ ÎÝÊÊ£ M Î VÃÃÌiÌ *ÃÃLiÊ>ÃÜiÀ\ÊÊÀ`iÀÊÌÊLiÊVÃÃÌiÌ]ÊÌ iÊiµÕ>ÌÃÊÕÃÌÊLiÊ «>À>iÊ>`Ê >ÛiÊ`vviÀiÌÊÞÌiÀVi«ÌðÊvÊÌ iÊVÃÌ>ÌÊÌiÀÃÊ>ÀiÊiµÕ>ÊÌÊ ä]ÊÌ iÊ>ÊÞÌiÀVi«ÌÃÊ>ÀiÊä]ÊÌ iÀivÀiÊÌ iÀiÊ>ÀiÊÊ«>À>iÊiÃÊ>`ÊÌ iÊ ÃÞÃÌiÊÕÃÌÊLiÊVÃÃÌiÌ° #LASSIFYEACHSYSTEMANDDETERMINETHENUMBEROFSOLUTIONS YX YX YX YX OÊ ÌÊ}iLÀ>ÊÓ *ÀLiÊ-Û} 5SING'RAPHSAND4ABLESTO3OLVE,INEAR3YSTEMS --" Σ {ÝÊÊnÞÊÊÓÈ X OÊ Ê{ÝÊÊnÞÊÓÈ Ê ÊÊÊÊÊ Ê Ê Ê ÊÝÊ£ÊÊÞ B 7RITEONEEQUATIONATTHETOPOFEACH TABLEANDCOMPLETETHETABLE C 7HATISTHECOSTOFEACHSIZEDRINK ,i>`}Ê-ÌÀ>Ìi}ià 5SE'RAPHIC!IDS #ONSISTENT ÝÊÊ£ÊÊÞ Y X Ó°Çx Ó°xä Ó°Óx Ó £°Çx Ó Ó°x Πΰx { $EPENDENT OÊ ÊÞÊÊxÊÊ£xÝ ÊÊ Ê ÊÊÊÊÊÊ Ê ÊÊÊ Ê Ê ÞÊnäÊÊÓäÝ C &ORWHATNUMBEROFHOURSWOULD3ANDY AND0AULINEPAYTHESAMETORENTAJETSKI ÎÊ XY YX YX X "OTHLINESTHESAME )NFINITELYMANY SOLUTIONS XY X XY ,INESINTERSECT /NESOLUTION ,INESPARALLEL .OSOLUTIONS 2INGALLTHETERMSTHATDESCRIBETHESYSTEM YX YX OÊ X 4IMEH f£{ä OÊ OÊ XY XY XY YX A #ONSISTENT A #ONSISTENT A #ONSISTENT B )NCONSISTENT C $EPENDENT D )NDEPENDENT B )NCONSISTENT C $EPENDENT D )NDEPENDENT B )NCONSISTENT C $EPENDENT D )NDEPENDENT $RAWALINEFROMEACHSYSTEMOFEQUATIONSTOTHEMATCHINGGRAPH OÊ Copyright © by Holt, Rinehart and Winston. All rights reserved. ÌÊ}iLÀ>ÊÓ Y X { X È #OPYRIGHT©BY(OLT2INEHARTAND7INSTON !LLRIGHTSRESERVED 52 XY XY Y x OÊ YX YX Y OÊ OÊ YX YX " O Ê $ O Ê YX YX OÊ XY YX *UANSTARTEDWITHGALLONSOFWATERINHISPOOLANDHEISFILLINGITATARATE OFGALLONSPERMINUTE(ISNEXTDOORNEIGHBOR3AMSTARTEDWITHGALLONS OFWATERINHISPOOLANDHEISFILLINGITATARATEOFGALLONSPERMINUTE 7HICHSYSTEMOFEQUATIONSCOULDYOUUSETOFINDWHENTHEPOOLSWILLCONTAIN THESAMEAMOUNTOFWATER YX YX ! # YX YX AK4up.indd 52 #HOOSETHELETTERFORTHEBESTANSWER #OPYRIGHT©BY(OLT2INEHARTAND7INSTON !LLRIGHTSRESERVED D (OWMUCHWOULDITCOSTTORENTTHEJETSKIFOR THISAMOUNTOFTIME Y X XY Y Y B 'RAPHTHESYSTEM 2ENTAL#OST )NDEPENDENT Y 3ANDYRENTEDAJETSKIFORPLUSPERHOUR0AULINERENTEDAJETSKI FORPLUSPERHOUR )NCONSISTENT Y ->\Êf£°xäÆÊ>À}i\ÊfÓ°xä A 7RITEALINEARSYSTEMOFEQUATIONSTOFIND THENUMBEROFHOURSFORWHICHTHERENTAL COSTISTHESAME ÌÊ}iLÀ>ÊÓ !SYSTEMOFLINEAREQUATIONSCANBECLASSIFIED ,INEAR%QUATIONS !FTERTHELESSON#ARLTAKESTHEWAKEBOARDINGCLASSTOTHE'LASS#AFE (EPAYSFORLARGEANDSMALLJUICEDRINKS!LARGEGLASSCOSTS MORETHANASMALLGLASS A 7RITEALINEARSYSTEMOFEQUATIONSTO FINDTHECOSTOFEACHSIZEDRINK #OPYRIGHT©BY(OLT2INEHARTAND7INSTON !LLRIGHTSRESERVED Solve. ÎÊ«>À>iÊià X ÎÊiÃÊÌ >ÌÊVV`i $ESCRIBEHOWTHEGRAPHOFTHESEEQUATIONS LOOKSWHENTHESYSTEMHASNOSOLUTION $ESCRIBEHOWTHEGRAPHOFTHESEEQUATIONSLOOKSWHEN THESYSTEMHASANINFINITENUMBEROFSOLUTIONS Y X ÌÊ}iLÀ>ÊÓ Holt Algebra 2 12/29/05 7:52:46 PM