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Page 1: Study Guide: Slope and Linear Equationsmrrogove2016.weebly.com/uploads/4/3/0/0/43009773/g... · 4 Study Guide: Slope and Linear Equations Problem set Complete the problems below before

NAME:___________________________ Math________,Period____________

Mr.Rogove Date:__________

StudyGuide:SlopeandLinearEquations1

Study Guide: Slope and Linear Equations

Rates and Unit Rates Arateisaproportionalrelationshipbetweentwoquantities.Unitrateisaratewherethesecondquantityis1.Example:Paulinecanmow35squarefeetoflawnis2.5minutes.(thisisarate).Shecanmow14squarefeetperminute(thisisaunitrate).Wesetupproportionstofindunitratesandwritelinearequationsin2variables.Wesolvefory.Example:

𝑦𝑥 =

352.5

Setupproportion

2.5𝑦 = 35𝑥

Multiplythenumeratorsbythedenominatoroftheotherfraction.

𝑦 = 14𝑥 Solvefory.RatesaredirectlyproportionaliftheirgraphliesonastraightlineANDpassesthroughtheorigin.

Slope Slopeisameasureofsteepnessorslantofaline.Itisalsothemeasureoftherateofthechangeofaline.Slopecanbeexpressedasaunitrate:Howmanyverticalunitsdoesalinemoveforeveryonehorizontalunit?Theslopeformulastatesthat:

𝒎 =𝒑𝟐 − 𝒓𝟐𝒑𝟏 − 𝒓𝟏

= 𝒓𝒊𝒔𝒆𝒓𝒖𝒏 =

𝒅𝒊𝒇𝒇𝒆𝒓𝒆𝒏𝒄𝒆 𝒊𝒏 𝒚− 𝒗𝒂𝒍𝒖𝒆𝒔𝒅𝒊𝒇𝒇𝒆𝒓𝒆𝒏𝒄𝒆 𝒊𝒏 𝒙− 𝒗𝒂𝒍𝒖𝒆𝒔

WherethecoordinatesofpointPonalinearerepresentedby𝑃 = (𝑝!,𝑝!)andthecoordinatesofpointRonthelinearerepresentedby𝑅 = (𝑟!, 𝑟!).

Page 2: Study Guide: Slope and Linear Equationsmrrogove2016.weebly.com/uploads/4/3/0/0/43009773/g... · 4 Study Guide: Slope and Linear Equations Problem set Complete the problems below before

NAME:___________________________ Math________,Period____________

Mr.Rogove Date:__________

StudyGuide:SlopeandLinearEquations2

Slope Triangles Wecanusetheconceptofslopetrianglestoprovetheslopeofalineisthesamebetweenanytwopointsontheline:

• ShowAAsimilarityoftwotriangles.• Knowthatcorrespondingsidesofsimilartrianglesare

proportional…andthismeansthattheslopeofalineisthesamefortheentiredistanceoftheline.

Ifyoucreateseveraldifferentrighttrianglesusingthepointsabove,youcanseethatseethatallofthetrianglesaresimilartrianglesusingtheconceptofAAsimilarity.Example:∆𝑃𝐿𝐾 ~∆𝑅𝑀𝐾becausetheyshareacommonangleANDtheyeachhaverightangles…hence,theymeettheAAcriterionforsimilarity.Thismeanstheproportionsofcorrespondingsidesofeachtriangleareequal.Example:Theproportionsofthecorrespondingsidesofthetrianglesmentionedaboveareequal…

𝑅𝑀𝑀𝐾

=𝑃𝐿𝐿𝐾

126 =

105

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NAME:___________________________ Math________,Period____________

Mr.Rogove Date:__________

StudyGuide:SlopeandLinearEquations3

Slope (cont’d)

PositiveSlopeLeft-to-rightinclining

lines

NegativeslopeLeft-to-rightdeclining

lines

ZeroslopeHorizontalLines

Thegreatertheabsolutevalueoftheslope,thesteepertheline!!Horizontallineshavezeroslope,andarewrittenas𝑦 = 𝑐.Verticallineshaveanundefinedslopeandarewrittenas𝑥 = 𝑐.

Equations of Lines StandardFormofanequation𝒂𝒙 + 𝒃𝒚 = 𝒄where𝑎, 𝑏, and 𝑐areallintegersand𝑎 > 0Wecangraphalineknowingthestandardformofalinearequationbyidentifyingthex-interceptandy-intercept.SlopeInterceptformofanequation𝒚 = 𝒎𝒙 + 𝒃wheremistheslopeandbisthey-intercept.Wecangraphalineknowingtheslope-interceptformofalinearequationbyidentifyingthey-interceptandusingtheslopetographanotherpoint.Minimumrequirementsforfindingtheequationofaline:1.Slopeandonepointontheline(oftentheonepointwillbethey-intercept)2.Twopointsontheline.(thiscanbethex-interceptandy-intercept)Convertfromslope-interceptformtostandardform:

• Gatheraxandbytermsonleftsideoftheequation.• Makesurethata,b,andcareallintegersandaispositive.

Convertfromstandardformtoslope-interceptform:

• Solvefory.

Page 4: Study Guide: Slope and Linear Equationsmrrogove2016.weebly.com/uploads/4/3/0/0/43009773/g... · 4 Study Guide: Slope and Linear Equations Problem set Complete the problems below before

NAME:___________________________ Math________,Period____________

Mr.Rogove Date:__________

StudyGuide:SlopeandLinearEquations4

Problem set Completetheproblemsbelowbeforetakingourtest.Ifyouhaveanyquestions,besuretoaskeitheratareviewsession.Aphonecompanyhastwoplansfordatathatareasfollows:PlanAincludesanupfrontservicechargeof$45permonth,andcharges$2.50foreachGBofdataused.PlanBincludesnoservicecharge,andcharges$7.50foreachGBofdata.1.Writeequationsthatrepresenteachcellphoneplan,whereyisthecostofthedata,andxistheamountofdata(inGB).Identifytheslopeandexplainwhatitmeans.2.Graphtheequationsonthecoordinateplane.3.WhenisitbetterforyoutogowithplanA?WhendoesitmakesensetogowithplanB?SophiaandColbyweregiventhesetwopoints(2,−5)and(−3, 5),andtoldtowritetheequationoftheline.Sophiathoughttheequationwas𝑦 = −2𝑥 − 1.Colbythoughtshewaswrongandthattheequationwas𝑦 = − !

!𝑥 − 4.Who’sright?Why?

Whatdidtheotherpersondowrong?

Page 5: Study Guide: Slope and Linear Equationsmrrogove2016.weebly.com/uploads/4/3/0/0/43009773/g... · 4 Study Guide: Slope and Linear Equations Problem set Complete the problems below before

NAME:___________________________ Math________,Period____________

Mr.Rogove Date:__________

StudyGuide:SlopeandLinearEquations5

AgallonofmilkatWholeFoodscost$6.ThegraphbelowcomparesthecostofmilkatWholeFoodsandSafeway.Estimatethecostofahalf-gallonofmilkatSafeway.Justifyyourreasoning.12candybarsatTargetcost$15.00.ThegraphbelowcomparesthecostofcandybarsatTargetandCostco.Estimatethecostof1candybaratCostco.Justifyyourreasoning.Inadditiontothesefourproblems,thefollowingexerciseshavebeenassignedinKhanAcademy.Pleasechecktheboxtoindicatethatyouhaveeithercompleted5inarowor8outof10.

¨ Graphfromslope-interceptequation¨ Slope-interceptequationfromgraph¨ ConvertingBetweenSlope-InterceptandStandardForm

Cost($)

Milk(gallons)

WholeFoods

Safeway

Costco

Target

Candybars

Cost($)

Page 6: Study Guide: Slope and Linear Equationsmrrogove2016.weebly.com/uploads/4/3/0/0/43009773/g... · 4 Study Guide: Slope and Linear Equations Problem set Complete the problems below before

NAME:___________________________ Math________,Period____________

Mr.Rogove Date:__________

StudyGuide:SlopeandLinearEquations6

Usethegraphbelowtocompletethequestions.Onpage2,weidentifiedtwosimilartriangles(∆𝑃𝐿𝐾 𝑎𝑛𝑑 ∆𝑅𝑀𝐾).Identify3othersimilartriangles,andshowthattheyaresimilarbylookingattheproportionsofthecorrespondingsides.3triangles:Proportionsofcorrespondingsides(seepage2foranexample):Whatistheproportionofthecorrespondingsidesalsocalled?(ifyouconsiderthecorrespondingverticalsidesinthenumeratorandthecorrespondinghorizontalsidesinthedenominator).

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