NAME:___________________________ Math7.2,Period____________

Mr.Rogove Date:__________

SolvingLinearEquationsStudyGuide1

Study Guide: Solving Linear Equations

Writing Equations

Whenreadingwordproblemsandtryingtocreateanequationusingsymbols,itisextremelyimportanttobeginbydefiningyourvariable.Example:YourfamilyistakingaroadtriptoChicago.Theydrive450milesonthefirstday,!

!oftheentiredistanceonthesecondday,!

!onthethirdday,andfinally

drivethefinal325milesonthefourthday.Letdbethedistanceoftheentiretripinmiles.Trytobreakmorecomplicatedwordsentencesdownintomoremanageablepieces.Example:Whenanumberistakenawayfrom57,whatremainsisfourmorethan5timesthenumber.Letxbethenumber.“Whenanumberistakenawayfrom57”à57− 𝑥“Whatremainsis”à=“Fourmorethan5timesthenumberà5𝑥 + 4

Solving Linear Equations LinearExpressionscontainconstants,variablesraisedtothefirstpower,orvariables(raisedtothefirstpower)multipliedbycoefficients.

Examples:12

−3𝑥

14 𝑥 − 2

32+ 21𝑥

Non-Examples:3𝑥!

14𝑥! + 19

𝑥! − 1

(𝑥 + 3)(𝑥 − 2)

WHENSOLVINGLINEAREQUATIONS,YOURULTIMATEGOALISTOISOLATETHEVARIABLE,

ALSOCALLEDSOLVINGFORX.

CHECKINGYOURSOLUTIONISAPARTOFSOLVINGTHELINEAREQUATION!!

NAME:___________________________ Math7.2,Period____________

Mr.Rogove Date:__________

SolvingLinearEquationsStudyGuide2

Inordertokeepyourequationinbalance,usethefollowingpropertiesofequality

Addition Property of Equality If 𝐴 = 𝐵, then 𝐴 + 𝐶 = 𝐵 + 𝐶

Subtraction Property of Equality If 𝐴 = 𝐵, then 𝐴 − 𝐶 = 𝐵 − 𝐶

Multiplication Property of Equality If 𝐴 = 𝐵, then 𝐴 × 𝐶 = 𝐵 × 𝐶

Division Property of Equality If 𝐴 = 𝐵 𝑎𝑛𝑑 𝐶 ≠ 0 , then

𝐴𝐶 =

𝐵𝐶

Different Moves to Make When Solving Equations

Whensolvingequationsyoumighthavetodooneormoreofthefollowingmoves:CombineLikeTermsbeforeyourusepropertiesofequality.Example:3𝑥 + 4− 5𝑥 + 13 = 26à−2𝑥 + 17 = 26Gatherallvariabletermsononesideoftheequationbeforeusingpropertiesofequalitytoisolateyourvariable.Example:3𝑥 − 23+ 4𝑥 = 24+ !

!𝑥 − 21à7𝑥 − !

!𝑥 − 23 = 24− 21

Usethedistributivepropertytoclearparenthesespriortousingpropertiesofequality.Example:43𝑥 − 3 21+ 12𝑥 = 3− 4𝑥 − 5 à43𝑥 − 63− 36𝑥 = 3− 4𝑥 + 5UseRulesofProportionstocreatelinearequationswhenyourequationscontainapairoffractions.Example: !!!!

!"!!!!= !

!à7 3𝑥 − 4 = 5 13𝑥 + !

!

Classifying Solutions to Linear Equations

OneSolution NoSolution InfinitelyMany

SolutionsExample:

7𝑥 − 3 = 5𝑥 + 5

Example:7𝑥 − 3 = 7𝑥 + 5

Example:7𝑥 − 3 = −3+ 7𝑥

-Differentcoefficients-Sameordifferentconstantterms(Ifthesameconstantterm,then𝑥 = 0)

-Samecoefficients-Differentconstantterms

-Samecoefficients-Sameconstantterms

NAME:___________________________ Math7.2,Period____________

Mr.Rogove Date:__________

SolvingLinearEquationsStudyGuide3

The Problem Set

Completethisproblemsetbeforeyoutakethetest!Ifyouarestuckorhavespecificquestionsaboutanyoftheseproblems,comefindmebeforethetestsowecanaddressyourconcerns.Writeanequationforeachwordproblemandanswerthequestion.Shannon’sgoalistorun40mileseachweek.Thisweek,shehasalreadyrundistancesof5.3miles,6.5miles,and6.2miles.Ifshewantstospreadouttheremainingmilesevenlyoverthenextfourdays,howmanymilesperdaydoessheneedtoruneachday?Writeanequationandsolve.

TheYchargesa$44signupfeeand$30/monthforanindividualgymmembership.CitySportsFitnesschargesa$99signupfeeandcharges$25/monthformembership.Writeanequationthatwillhelpyoudeterminehowmanymonthsitwilltakeforthetotalcostofmembershiptobethesame?Howmanymonthswillittake?

Twoverticalanglesaredescribedasfollows:oneoftheanglesisfourtimesthequantityofanumberaddedtothree,andtheotherangleis6lessthanthirteentimesthatsamenumber.Whatarethemeasuresofbothangles?

Oneacuteangleofarighttriangleisninelessthantwotimesthemeasureoftheotheracuteangle.Thethirdangleistherightangle.Drawthetriangleandfindtheanglemeasures.

NAME:___________________________ Math7.2,Period____________

Mr.Rogove Date:__________

SolvingLinearEquationsStudyGuide4

Solveeachequationandcheckyourwork.

−3 4𝑥 − 8 − 1 = 𝑥 + 3 𝑥 + 9

6 𝑥 − 11 − 3 11− 𝑥 = 11 𝑥 + 7

223 𝑥 + 2− 𝑥 = 9 𝑥 + 1 +

53 𝑥

2𝑥 − 3− 9𝑥 = 14+ 𝑥 − 1

NAME:___________________________ Math7.2,Period____________

Mr.Rogove Date:__________

SolvingLinearEquationsStudyGuide5

Solveeachequation.Noneedtocheckyourwork.

− −3+ 13𝑥 − 5𝑥 = 623 𝑥 −

25

165𝑥 − 12 +

233 − 5𝑥 = −4 − (𝑥 + 12)

−5 − 4 − 3𝑥 + 7𝑥 =58

16𝑥 − 20 + 4𝑥

23 12−

92 𝑥 =

3492−

32 𝑥

NAME:___________________________ Math7.2,Period____________

Mr.Rogove Date:__________

SolvingLinearEquationsStudyGuide6

Solveeachequation.Noneedtocheckyourwork.

2𝑥 − 346 =

3− 2𝑥10

5𝑥 − 33− 2𝑥 =

611

Pleaseclassifyeachsolutionashavingoneuniquesolution,nosolution,orinfinitelymanysolutions.Youshouldsimplifytheexpressions,butthereisnoneedtosolvetheequations.

9𝑥 + 12− 3𝑥 ÷ 3 = 2(2+ 𝑥)

3 4𝑥 − 1 − 13𝑥 = −(𝑥 + 3)

4 2𝑥 − 3 = 3 3𝑥 − 3 − 3

14 3+ 72𝑥 = 6 3𝑥 +

14