CHAPTER1:SYSTEMSOFLINEAREQUATIONSANDMATRICES
1.1IntroductiontoSystemsofLinearEquations
2.Equationsinpart(a)donotformalinearsystemsincethesecondoneisnotalinearequation.
Equationsinparts(b),(c),and(d)formlinearsystems.
3.(a)Notlinearbecauseoftheterm
x
2
x
3.(b)Linear
4.(b)Wecansolvethissystembyinspectionsincethesecondequationisamultipleofthefirst.
Letting4solvesbothequations,thereforethesystemisconsistent.
(c)Aneasywaytofindanexampleofasolutionofthissystemwouldbebyletting0first,
thenlookforvaluesofandthatsolvetheresultingsystem
4x+2z=–1
–x–3z=0
Adding4timesthesecondequationtothefirstwillyieldasimplifiedsystem
–10z=–1
–x–3z=0
whichissolvedby
and
.Togetherwith0,thisisasolutionoftheoriginal
system,showingittobeconsistent.
(d)Weadd6timesthefirstequationtothethird,thenproceedtoaddthefirstequationto
thefourth,andthesecondonetothefourth,obtainingasimplifiedsystem
34
51
1627
71
Clearly,thelasttwoequationsarecontradictory,thusthesystemisinconsistent.