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Linear Equations in Three Variables

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Linear Equations in Three Variables Linear Equations in Three Variables LinearLinearLinearLinearEquationsEquations Equations Equationsinin in inThreeThree Three ThreeVariablesVariables Variables Variables R2 is the space of 2 dimensions. There is an x-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,R 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. real number.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. :2.:2. II 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 equation equationa, b,and in inc threeare three not variables variables all equal.. (The (The to 0) IfIf“three“threethena,a, b,b, axc variables”c variables”andand+ byrr +arearecz are arerealreal= the thernumbersnumbersisxx,the,the calledyy(and,andthe(and,andthe a linearifif a,a, b,z equationb,z.).)andand cc areare innot threenot allall variablesequalequal toto.0)0) (The The numbers a, b,andc are called the coefficients of the equation. The thenthen“threeaxTheax++ by numbersby variables”++ ezez ==arr, areisisb,andcalledcalled the xca,theaarelinearlinear calledy,andtheequationequation the coezin.)inffitwotwocientsvariables.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 isaaspecificspecific multiplied 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 line. − − − − − − − −−−− − − − − andandand The set of solutions in F to a linear equation in three variables is a 2- 12 12=712=7=7 dimensional plane. − −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. ax + C I ‘C 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’s 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
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