Linear in Three Variables LinearLinearLinearLinearEquationsEquations Equations Equationsinin in inThreeThree Three ThreeVariablesVariables Variables Variables R2 is the space of 2 dimensions. There is an -coordinate that can be any 22 JR2JR2realRRisis theis number,theis the thespacespace space space andofof of22 of theredimensions.dimensions. 2 2 dimensions. dimensions. is a y-coordinateThereThere There Thereisis an isanthat is anx-coordiuatu anx-coordiuatux canx-coordinate-coordinate be anyIJIHIIJIHI real that that number. can can be be any any realrealrealrealnumber,number, number,3 number,is theandand space and andtherethere therethere ofis 3is dimensions.a isa isy-coordinatey-coordinate a ayy-coordinate-coordinate Therethatthat that that iscancan an can canbebex, beanyy beany,and any anyrealreal realz realnumber.coordinate.number. number. number. Each R3R3coordinateRRisis33theistheis the thespacespace space can spaceofof be of33 of anydimensions.dimensions. 3 3 dimensions. dimensions. .ThereThere There Thereisis an isan is an an.x,.x, xy,y,x,,yandandy,and,andzz coordinate.zcoordinate.zcoordinate.coordinate.EachEach Each Each coordinatecoordinatecoordinatecoordinatecancan can canbebe beany beany any anyrealreal real realnumber.number. number. number.

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LinearLinear equations equations in in three three variables variables LinearIfIfaa,,bb,,cc equationsandandrrareare real real in numbers numbers three (and (and variables if ifaa,,bb,and,andccareare not not all all equal equal to to 0) 0) LinearLinearthenthenIf aaxax,equationsequationsb+,+cbybyand++czrczare==ininr realr isthreeisthree callednumbers calledvariables. avariables. alinear (andlinear if equationa, ,and in inc threeare three not variables variables all equal.. (The (The to 0) IfIf“three“threethena,a, b,b, axc variables” variables”andand+ byrr +arearecz are arerealreal= the thernumbersnumbersisxx,the,the calledyy(and,andthe(and,andthe a linearifif a,a, b, equationb,z.).)andand cc areare innot threenot allall variablesequalequal toto.0)0) (The The numbers a, b,andc are called the coecients of the equation. The thenthen“threeaxTheax++ by numbersby variables”++ ezez ==arr, areisisb,andcalledcalled the xca,theaarelinearlinear calledy,andtheequationequation the coezin.)intwotwocientsvariables.variables.of the equation.(TJir’(TJir’ “tliv’~“tliv’~ The variables”variables”numbernumberarerarerisisthethe called calledx,x, thethe the they,y,constantconstantandand thetheofofz.)z.) the the equation. equation. Examples. 3x +4y 7z =2, 2x + y z = 6, x 17z =4,4y =0,and Examples.Examples.Examples.Examples.x + y + z3x+4y—7z=2,3x+4y—7z=2,=2arealllinearequationsinthreevariables.33xx+4+4yy 77zz=2,=2,—2x+y—z=—6,x—17z=4,4y=0,and—2x+y—z=—6,x—17z=4,4y=0,and22xx++yy zz== 6,6,xx 1717zz=4,4=4,4yy=0,and=0,and x + y + z =2arealllinearequationsinthreevariables. xx ++xyy+++yzz+==z22=2arealllinearequationsinthreevariables.areare allall linearlinear equationsequations inin threethree variables.variables. Solutions to equations SolutionsSolutionsSolutionsSolutionsA solutiontoto to toequationsequations a equations linear equations equation in three variables ax+by+cz = r is a specific AAsolutionsolution3 toof a a linear linear equation equation in in three three variables variablesax+axby++bycz =+ rczis= a specificr is a AApointsolutionsolution in Rtoto asucha linearlinear thatequationequation when wheninin threethree thevariablesvariablesx-coordinateax+by+czax+by+cz of the point== rr isis isaaspecific multipliedspecific 3 3 pointpointspecificpointbyinina,theJR3JR3 in pointsuchRsuchy-coordinatesuch inthatthatR thatsuchwhenwhen when of thatwhenwhen the when when pointthethe the thex-coordinatex-coordinate is multipliedx-coordinateof byof thethe of ofb,the the thepointpoint pointz point-coordinateisis multiplied ismultiplied is multiplied multiplied of the bybybya,bya,pointathethea,the,the isy-coordinatey-coordinate multipliedyy-coordinate-coordinate byofof ofthe ofcthe, the the andpointpoint point point thenisis is is thosemultipliedmultiplied multiplied multiplied threebyby productsby byb,b,bb,theand,theand zthe arethez-coordinate-coordinatez-coordinate addedz-coordinate together, of of the the ofof thepointthepointthepointpoint answer is is multiplied multipliedisis multiplied equalsmultiplied byr by. (Therecbybyc,, and andc,c, andand then are thenthosethose always those thosethreethree three infinitely threenumbersnumbers products products manyareare are solutionsareaddedadded added addedtogether,together, to together, together, a linear thethethetheequationansweranswer answer answerequalsequals in equals equals threer.r. r variables.)(There(Therer.. (There (Thereareare are arealwaysalways always alwaysinfinitelyinfinitely infinitely infinitelymanymany many manysolutionssolutions solutions solutionstoto toa toa linear alinear a linear linear equationequationequationequationinin three inthree in three threevariables.)variables.) variables.) variables.) 1 255192 11 Example.Example.Example.Example.The pointTheThex pointThe= point1, pointxy=1,x= =1,x2, =1,yand=2,andy =2,andyz =2,and= —1z is=z a=zsolution1isasolutionoftheequation=1isasolutiontotheequation1isasolutiontotheequationto the equation —Zx + 5y + z = 7 since —2(1) + 5(2) 2+x(—1)2+5x2+5xy =+5+y—2z+y=7+z+=7z10=7— 1 = 7. The point x = 3, y = —2, and z = 4 is a not a solution to the equation sincesincesince —2x + 5y + z = 7 since —2(3) + 5(—2) + (4) = —6 — 10 + 4 = —12, and 2(1)2(1) +2(1) 5(2) + +5(2) + 5(2) ( + +1)( ( =1)1) =2+10 =2+102+101=71=71=7 —12 $ 7. TheThe pointThe point pointx =3,x x=3,=3,y =y =y 2,= 2, and2, andz and=4isaz =4isaz =4isanotnotasolutionoftheequationnotasolutiontotheequationasolutiontotheequation Linear2xequations2+5x2+5xy+5+yz+y=7since+z =7sincezand=7sinceplanes. The set of solutions in2(3)R22(3) +2(3)to 5( +a +5(2)linear 5( +2) (4)2) +equation +(4) = (4) =6 =610in6 10 +two10 4 + =variab1r’~ +4 = 412 =1212 1 1 - dimensional . andandand The set of solutions in to a linear equation in three variables is a 2- 12 12=712=7=7 dimensional . 6 ⇥ 6 A solutionLinearLinearLinearto equationsa linear equations equationsequation and andin and planesthree planes planesvariables — ax + by + cz = r — is a point in R3 that lies on the plane corresponding to ax + by + cz = r. So TheTheThe set set of set of solutions of solutions solutions in inR2 inof2 to2 ato linear a lineara linear equation equation equation in in two in two variables two variables variables is ais 1- is a 1-a 1- we see that there are more solutionsR Rto a linear eqnation in three variables dimensionaldimensionaldimensional line. line. line. (2-dimensions worth) then there are to a linear eqnation in two variables TheThe set set of ofsolutions solutions in inR3 of3 to a3 linear a linear equation equation in inthree three variables variables is a is 2- a 2- (1-dimensionsTheworth) set of solutions inR R to a linear equation in three variables is a 2- dimensionaldimensionaldimensional plane. plane. plane.

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SolutionsSolutionsSolutions of to systems to systems systems of of linear of linear linear equations equations equations SolutionsAsAs inAsto thein in thesystems previous the previous previous chapter,of chapter, chapter,linear we we can weequations. can have can have have a system a system a system of linearof of linear linear equations, equations, equations, and and and As inwethewe canweprevious can try can try to try tofindchapter, to find solutions find solutions solutionswe can that thathave arethat are commona aresystem common common toof toeachlinear to each eachofequations, the of of the equations the equations equationsand in thein in the the ask forsystem.a system.commonsystem.solution to each equation in the system. If thereWeisWe callaWesolution call a call solution a solutionato solutiona tosystem to a to system aof systema systemequations, of ofequations of equations equationswe calluniquethatuniqueuniquesolutionif thereif thereif there areunique are no are no other no other other solutions.solutions.solutions. if there are no other solutions. 256 9 193 2 Example.Example.Example.TheTheThe point point pointxxx=3,=3,=3,yyy=0,and=0,and=0,andzzz=1isasolutiontothefollowing=1isasolutiontothefollowing= 1 is a solution of the following systemsystemsystem of of of three three three linear linear linear equations equations equations in in in three three three variables variables variables 333xxx+2+2+2yyy 555zzz=== 1414 222xxx 333yyy+4+4+4zzz=10=10=10 xxx+++ yyy +++ zzz=4=4=4 That’sThat’sThat’ because because because we we we can can can substitute substitute substitute 3, 3, 3, 0, 0, 0, and and and 1 1 1 for for forxx,,,yy,and,and,andzzrespectivelyrespectively in in thethethe equations equations equations above above above and and and check check check that that that 3(3)3(3)3(3) + + + 2(0) 2(0) 2(0) 5(1)5(1)5(1) = = = 999 5=5= 1414 2(3)2(3)2(3) 3(0)3(0)3(0) + + + 4(1) 4(1) 4(1) = = = 6 6 6 + + + 4 4 4 = = 10 10 (3)(3)(3) + + + (0) (0) (0) + + + (1) (1) (1) = = = 3 3 3 + + + 1 1 1 = = 4 4

GeometryGeometryGeometry of of of solutions solutions solutions SupposeSupposeSuppose you you you have have have a a a system system system of of of three three three linear linear linear equations equations in in three three variables. variables. 33 EachEachEach of of of the the the three three three equations equations equations has has has a a a set set set of of of solutions solutions solutions that’s that’s a a plane plane in inRR3.A.A.A solutionsolutionsolution to to of the the system system of of equations equations is is a a point point that that lies lies on on all all three three of of those those planes.planes.planes. If If If there there there is is is only only only one one one point point point that that that lies lies lies on on on all all three three planes, planes, then then that that solutionsolutionsolution is is is unique. unique. unique. IfIfIf you you you randomly randomly randomly write write write down down down three three three di di di↵erenterenterent linear linear linear equations equations in in three three vari- vari- ables,ables,ables, the the the odds odds odds are are are that that that the the the three three three corresponding corresponding corresponding planes planes will will intersect intersect in in one, one, andandand only only only one, one, one, point. point. point. That That That means means means that that that for for for most most most systems systems of of three three linear linear equa- equa- tionstionstions in in in three three three variables, variables, variables, there there there will will will be be be a a a unique unique unique solution. solution.

1942573 ItIt might might be be that that the the three three planes planes from from the the system system of of three three equations equations will will It might be that the three planes from the system of three equations will beItbeIt parallel. might mightparallel. be be Then that Then that the the the the three three three three planes planes planes planes from wouldn’ from wouldn’t the the system intersect. system intersect. of of three There’ three There’d equations equations be be no no point will point will be parallel. Then the three planes wouldn’t intersect. There’d be no point bebecommoncommon parallel. parallel. to to allThen Then all three three the the planes, planes, three three planesand planes and hence hence wouldn’t wouldn’t the the system system intersect. intersect. will will not notThere’d There’d have have any be any be solutions.no no solutions. point point common to all three planes, and hence the system will not have any solutions. commoncommon to to all all three three planes, planes, and and hence hence the the system system will will not not have have any any solutions. solutions.

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There might not be a point that lies on all three planes even if the planes ThereThere might might not not be be a a point point that that lies lies on on all all three three planes planes even even if if the the planes planes Therearen’tThere might parallel. might not not In be be this a a point casepoint again, that that lies lies there’d on on all all be three three no solution planes planes even at even all. if if the the planes planes aren’taren’t parallel. parallel. In In this this case case again, again, there’d there’d be be no no solution solution at at all. all. aren’taren’t parallel. parallel. In In this this case case again, again, there’d there’d be be no no solution solution at at all. all.

1954 258 1954 Sometimes, the three planes will intersect in a way that allows for more Sometimes,Sometimes, the the three three planes planes will will intersect intersect in in a a way way that that allows allows for for more more than one point to be on all three planes at once. In this case, there are thanthan one one point point to to be be on on all all three three planes planes at at once. once. In In this this case, case, there there are are multiple solutions. Because there’s more than one solution, there’s not a multiplemultiple solutions. solutions. Because Because there’s there’s more more than than one one solution, solution, there’s there’s not not a a unique solution. uniqueunique solution. solution.

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Visualize di↵erent arrangements of three planes in 3 3 and try to convince VisualizeVisualize di dierenterent arrangements arrangements of of three three planes planes in inRR3andand try try to to convince convince yourself that either there is exactly one point containedR in all three planes, yourselfyourself that that either either there there is is exactly exactly one one point point contained contained in in all all three three planes, planes, or no points contained in all three planes, or that there are infinitely many oror no no points points contained contained in in all all three three planes, planes, or or that that there there are are infinitely infinitely many many points that are contained in all three planes. That means that a system pointspoints that that are are contained contained in in all all three three planes. planes. That That means means that that a a system system of three linear equations in three variables will always have either a unique ofof three three linear linear equations equations in in three three variables variables will will always always have have either either a a unique unique solution, no solution at all, or infinitely many solutions. solution,solution, no no solution solution at at all, all, or or infinitely infinitely many many solutions. solutions.

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259 1965 Exercises

1.) What are the coecients of the equation 3x 2y +5z =13

2.) What is the constant of the equation 3x 2y +5z =13

3.) Is x =4,y = 3, and z = 1asolutionoftheequation 3x 2y +5z =13

4.) Is x = 2, y =0,andz =4asolutionoftheequation x +7y 8z =10

5.) Is x =5,y =7,andz = 2asolutionoftheequation 3y 5z =31

Questions #6-10 refer to the following system of three linear equations in three variables. 3x 4y +2z = 9 4x +4y +10z =32 x +2y 7z = 7

6.) Is x =1,y =4,andz =2asolutionofthesystem?

7.) Is x = 1, y =1,andz = 1asolutionofthesystem? 8.) Is x =25,y =23,andz =4asolutionofthesystem?

9.) Does the system have a unique solution?

10.) Does the system have infinitely many solutions? 260 For #11-13, solve the logarithmic equations for x.

11.) log (2x 3) = 4

12.) log2(5x +1)=3 13.) log (x2 + x) log (x)=log(2) e e e

261