Download Section 1.1 - Gordon State College

6/16/2016
TERMINOLOGY
• Anequationinonevariable isa
(mathematical)statementinwhichtwo
expressions,atleastoneofwhichcontainsa
variableareequal.
Section 1.1
• Tosolve anequationmeanstofindallthe
valuesofthevariablethatsatisfythe
equation.Thesevaluesarecalledsolutions
orroots.
Linear Equations
EXAMPLE:3
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TERMINOLOGY (CONTINUED)
SOLVING AN EQUATION
• Oftenwewritethesolutionofanequationas
asolutionset.Forexample,thesolutionset
of
16 0 is 4, 4 .
Onemethodforsolvinganequationisto
replacetheoriginalequationbyasuccessionof
equivalentequationsuntilanequationwithan
obvioussolutionisobtained.
• Anequationthatissatisfiedforeveryvalueof
thevariableforwhichbothsidesaredefined
iscalledanidentity.Forexample.
3
2
4
3
Equivalentequations areequationsthathave
exactly thesamesolutionset.
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isanidentitybecausethisstatementistrue
foranyrealnumber .
PROCEDURES THAT RESULT IN
EQUIVALENT EQUATIONS
1. Interchangethetwosidesofanequation.
2. Simplifythesidesofanequationbycombining
liketerms,eliminatingparentheses,etc.
3. Addorsubtractthesameexpressionfromboth
sidesoftheequation.
4. Multiplyordividebothsidesoftheequationby
thesamenonzeroexpression.
5. Ifonesideoftheequationis0andtheotherside
canbefactored,thenwemayusetheZero‐
ProductPropertyandseteachfactorequalto0.
ZERO-PRODUCT PROPERTY
Let and bealgebraicexpressions.If
0,then
0 or
0 orbothequal0.
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STEPS FOR SOLVING
EQUATIONS
1. Listanyrestrictionsonthedomainofthe
variable.
2. Simplifytheequationbyreplacingthe
originalequationbyasuccessionof
equivalentequationsfollowingthe
procedureslistedearlier.
3. IftheresultofStep2isaproductoffactors
equalto0,usetheZero‐ProductProperty
andseteachfactorequalto0.
4. Checkyoursolution(s).
LINEAR EQUATIONS
Alinearequationinonevariable is
equivalenttoanequationoftheform
0
where and arerealnumbersand
0.
Alinearequationisalsocalledafirst‐degree
equation.
STEPS FOR SOLVING A
LINEAR EQUATON
1. Listanyrestrictionsonthedomainofthe
variable.
2. Ifnecessary,cleartheequationoffractionsby
multiplyingbothsidesbytheleastcommon
multiple(LCM)ofallthedenominatorsofallthe
fractions.
3. Removeallparenthesesandsimplify.
4. Collectalltermscontainingthevariableonone
sideoftheequationandallremainingtermson
theotherside.
5. Simplifyandsolve.
6. Checkyoursolution(s).
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