Fundamentals in solving equations in one or more steps

When we want to solve an equation including one unknown variable, as x in the example above, we always aim at isolating the unknown variable. Algebra1/ Howtosolvelinearequations/ Fundamentalsinsolvingequationsinoneormoresteps Formulasareverycommonwithinphysicsandchemistry,forexample,velocityequalsdistancedividedbytime.Thusweusethecommonsymbolsforvelocity(v),distance(d)andtime(t)andexpressitthus: $$v=\frac{d}{t}$$ Wemaysimplydescribeaformulaasbeingavariableandanexpressionseparatedbyanequalsignbetweenthem.Inotherwordsaformulaisthesameasanequation. Example Abookclubrequiresamembershipfeeof$10inadditiontothe$2leviedforeachbookordered.Ifweweretolistthecostoforderinganumberofbooks,itwouldlooklike: Numberofbooks Cost 1 10+2∙1=12 2 10+2∙2=14 3 10+2∙3=16 4 10+2∙4=18 5 10+2∙5=20 x 10+2x IfwedesignatethetotalbookclubcostasC,wemayderivethefollowingformulafortheexpression: $$C=10+2x$$ Ifwethenwanttoknowhowmanybookswemaygetfromthebookclubfor$30wecaneithercontinuefillinginthetableaboveorusethepropertiesofequationsthatwehandledinthelastsection. $$30=10+2x$$ Cwasthecost,i.e.itisnow$30 $$30\:{\color{green}{-\,10}}=10+2x\,{\color{green}{-\,10}}$$ wesubtract$10fromeachside $$20=2x$$ simplify $$\frac{20}{{\color{green}2}}=\frac{2x}{{\color{green}2}}$$ dividebothsidesby2toisolatex $$10=x$$ xequals10 Wemaypurchase10booksfor$30. Whenwewanttosolveanequationincludingoneunknownvariable,asxintheexampleabove,wealwaysaimatisolatingtheunknownvariable.Youcansaythatweputeverythingelseontheothersideoftheequalsign.Itisalwaysagoodideatofirstisolatethetermsincludingthevariablefromtheconstantstobeginwithaswedidabovebysubtractingoraddingbeforedividingormultiplyingawaythecoefficientinfrontofthevariable.Aslongasyoudothesamethingonbothsidesoftheequalsignyoucandowhateveryouwantandinwhichorderyouwant. Abovewebeganbysubtractingtheconstantonbothsides.Wecouldhavebegunbydividingby2instead.Itwouldhavelookedlike $$\frac{30}{{\color{blue}2}}=\frac{10+2x}{{\color{blue}2}}$$ $$\frac{30}{{\color{blue}2}}=\frac{10}{{\color{blue}2}}+\frac{2x}{{\color{blue}2}}$$ $$15=5+x$$ $$15\,{\color{blue}{-\,5}}=5+x\,{\color{blue}{-\,5}}$$ $$10=x$$ Againthesameanswerjustprovingthepoint. Ifyourequationcontainsliketermsitispreferabletobeginbycombiningtheliketermsbeforecontinuingsolvingtheequation. Example $$5x+14+2x+2=30$$ Beginbycombiningtheliketerms(alltermsincludingthesamevariablexandallconstants) $$\left(5x+2x\right)+\left(14+2\right)=30$$ $$7x+16=30$$ Nowit'stimetoisolatethevariablefromtheconstantpart.Thisisdonebysubtracting16frombothsides $$7x+16\,{\color{green}{-\,16}}=30\,{\color{green}{-\,16}}$$ $$7x=14$$ Dividebothsidesby7toisolatethevariable $$\frac{7x}{{\color{green}7}}=\frac{14}{{\color{green}7}}$$ $$x=2$$ Ifyouhaveanequationwhereyouhavevariablesonbothsidesyoudobasicallythesamethingasbefore.Youcollectallliketerms.Beforeyouhaveworkedbyfirstcollectingallconstanttermsononesideandkeepthevariabletermsontheotherside.Thesameapplieshere.Youcollectallconstanttermsononesideandthevariabletermsontheotherside.It'susuallyagoodideatocollectallvariablesonthesidethathasthevariablewiththehighestcoefficienti.e.intheexamplebelowtherearemorex:esontheleftside(4x)comparedtotherightside(2x)andhencewecollectallx:esontheleftside. Example $$4x+3=2x+11$$ subtract2xfrombothsides $$4x+3\,{\color{blue}{-\,2x}}=2x+11\,{\color{blue}{-\,2x}}$$ Nowitlookslikeanyotherequation $$2x+3=11$$ subtract3frombothsides $$2x+3\,{\color{blue}{-\,3}}=11\,{\color{blue}{-\,3}}$$ $$2x=8$$ Divideby2onbothsides $$\frac{2x}{{\color{blue}2}}=\frac{8}{{\color{blue}2}}$$ $$x=4$$ Inthebeginningofthissectionweshowedtheformulaforcalculatingthevelocitywherevelocity(v)equalsthedistance(d)dividedbytime(t)or $$v=\frac{d}{t}$$ Ifwebysomechancewanttoknowhowfaratruckdrivesin3hoursat60milesperhourwecanusetheformulaaboveandrewriteittosolvethedistance,d. $$\frac{d}{t}\,{\color{green}{\cdot\,t}}=v\,{\color{green}{\cdot\,t}}$$ $$d=v\cdott$$ Whenthat'sdonewecanjustputournumbersintheformulaandcalculatetheanswer $$d=60\cdot3=180$$ Thetrucktravels180milesin3hours. Thisholdstrueforallformulasandequations. Videolesson Solvetheequation $$3\left(x+2\right)-3+x+17=40$$ Moreclassesonthissubject Algebra1 Howtosolvelinearequations:Propertiesofequalities Algebra1 Howtosolvelinearequations:Similarfigures Algebra1 Howtosolvelinearequations:Calculatingwithpercents NextChapter: HOWTOSOLVELINEAREQUATIONS–Ratiosandproportionsandhowtosolvethem Search MathPlayground Allcourses MathPlayground Pre-Algebra Allcourses Pre-Algebra Algebra1 Allcourses Algebra1 Discoveringexpressions,equationsandfunctions Algebra1 Discoveringexpressions,equationsandfunctions Overview Expressionsandvariables Operationsintherightorder Composingexpressions Composingequationsandinequalities Representingfunctionsasrulesandgraphs Exploringrealnumbers Algebra1 Exploringrealnumbers Overview Integersandrationalnumbers Calculatingwithrealnumbers TheDistributiveproperty Squareroots Howtosolvelinearequations Algebra1 Howtosolvelinearequations Overview Propertiesofequalities Fundamentalsinsolvingequationsinoneormoresteps Ratiosandproportionsandhowtosolvethem Similarfigures Calculatingwithpercents AboutMathplanet Cookiesettings Mattecentrum Matteboken Formelsamlingen Pluggakuten Visualizinglinearfunctions Algebra1 Visualizinglinearfunctions Overview Thecoordinateplane Linearequationsinthecoordinateplane Theslopeofalinearfunction Theslope-interceptformofalinearequation Formulatinglinearequations Algebra1 Formulatinglinearequations Overview Writinglinearequationsusingtheslope-interceptform Writinglinearequationsusingthepoint-slopeformandthestandardform Parallelandperpendicularlines Scatterplotsandlinearmodels Linearinequalities Algebra1 Linearinequalities Overview Solvinglinearinequalities Solvingcompoundinequalities Solvingabsolutevalueequationsandinequalities Linearinequalitiesintwovariables Systemsoflinearequationsandinequalities Algebra1 Systemsoflinearequationsandinequalities Overview Graphinglinearsystems Thesubstitutionmethodforsolvinglinearsystems Theeliminationmethodforsolvinglinearsystems Systemsoflinearinequalities Exponentsandexponentialfunctions Algebra1 Exponentsandexponentialfunctions Overview Propertiesofexponents Scientificnotation Exponentialgrowthfunctions Factoringandpolynomials Algebra1 Factoringandpolynomials Overview Monomialsandpolynomials Specialproductsofpolynomials Polynomialequationsinfactoredform Quadraticequations Algebra1 Quadraticequations Overview Usegraphingtosolvequadraticequations Completingthesquare Thequadraticformula Radicalexpressions Algebra1 Radicalexpressions Overview Thegraphofaradicalfunction Simplifyradicalexpressions Radicalequations ThePythagoreanTheorem Thedistanceandmidpointformulas Rationalexpressions Algebra1 Rationalexpressions Overview Simplifyrationalexpression Multiplyrationalexpressions Divisionofpolynomials Addandsubtractrationalexpressions Solvingrationalexpressions Algebra2 Allcourses Algebra2 Geometry Allcourses Geometry SAT Allcourses SAT ACT Allcourses ACT