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44  Lesson 5

Plug In

LESS

ON 5Finding the Slope of a Line

Slope

Theslopeofalineistheratioofthechangeinthey-valuestothechangeinthecorrespondingx-values .

x

y

1–1

–1 2 3 4 5 6 7 80

1

2

3

4

5

6

3

6

Slope5changeiny-values

______________changeinx-values

5522_____82253__651__2

They-interceptiswherealinecrossesthey-axis .

x

y

1–1

–1 2 3 4 5 6 7 80

1

2

3

4

5

6

(0, 1) y-intercept is 1.

Theequationofalinecanbewrittenintheformy5mx1b,wheretheslopeofthelineism,andthey-interceptisb .

x

y

1–1

–1 2 3 4 5 6 7 80

1

2

3

4

5

6

(0, 1)

3y �12x � 1

6

slope aratioofthechangeiny-coordinates(rise)ofagraphtothechangeincorrespondingx-coordinates(run)

y-interceptthey-coordinateofthepointatwhichalinecrossesthey-axis

Wordsto

Know

I see! I choose two coordinate pairs and then

divide the difference of the y-values by the difference of the corresponding x-values.

The y-intercept is the y-coordinate of the point where the line

intersects the y-axis.

I get it! The slope is m 5 1 _ 2 , and the y-intercept is b 5 1, so the equation of

the line is y 5 1 _ 2 x 1 1.

Willtheslopeofthelinethroughthepoints(1,6)and(7,22)bepositiveornegative?Howdoyouknow?

DISCUSS

A Youcanusetheslopeformulam5y22y1

_____x22x1tofindtheslopeofaline .

Findtheslopeofthelinethroughpoints(5,1)and(24,3) .

1 Choose(x1,y1)and(x2,y2) .

2 Writetheslopeformula .Substituteforx1,x2,y1,andy2 .

3 Subtract .Simplify,ifpossible .

(x1,y1)5(5,1) (x2,y2)5( , )

slope5y22y1

_____x22x1532

_________

25

5 ______

DO

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PRACTICEFind the slope of the line that contains the given points.

1 (2,1)and(4,2)

m522

_________

22

5 ______

2 (1,3)and(5,6)

m5 2

_____________

2

5 ______

Find the slope and y-intercept of each line.

3

x

y

1–1

–2

–3

–4

–5

–1–2–3–4–5 2 3 4 50

1

2

3

4

5

slope5

y-intercept5

4

x

y

1–1

–2

–3

–4

–5

–1–2–3–4–5 2 3 4 50

1

2

3

4

5

slope5

y-intercept5

5 y51__2x15

slope5

y-intercept5

6 y522x13

slope5

y-intercept5

B Youcanusetheequationofalinetofinditsslopeandy-intercept .

Findtheslopeandy-interceptofthelinewiththeequationy53x12 .

1 Comparetheequationofthegivenlinetoy5mx1b .

2 Identifytheslope,m,andthey-intercept,b .

y5 x1

y5m3x1b

m5

b5

Theslopeofthelineis .

They-interceptofthelineis .

DO

Whydoyouthinktheslopeofaverticallineisdescribedas“undefined”?Explainbycomputingtheslopeoftheverticallinethroughpoints(2,6)and(2,23) .

DISCUSS

I have to make sure the equation of a line is in the form

y 5 mx 1 b to determine the slope

and y-intercept.

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46  Lesson 5

POWER uPYoucanwritetheequationofalinefromitsgraphusingtheformy5mx1b .

1 Find the slope.Choosetwopointsontheline,andusetheslopeformulatocalculatetheslope .Usingthepoints(23,21)and(2,4):

m5changeiny-values

______________changeinx-values

5y22y1

_____x22x1

542(21)

_______22(23)

55__5

51

2 Find the y-intercept.Identifywherethelinecrossesthey-axis .

Thelinecrossesthey-axisat(0,2) .

They-interceptis2 .

3 Substitute the slope, m, and the y-intercept, b, in the equation y 5 mx 1 b.

y5mx1b

y5x12isanequationfortheline .

Thexandyintheequationrepresentallthepossiblecoordinatepairsofpointsontheline .

x

y

1–1

–2

–3

–4

–5

–1–2–3–4–5 2 3 4 50

1

2

3

4

5

Doestheslopechange,dependingonwhichpointyouassignas(x1,y1)or(x2,y2)?Explain .

DISCUSS

A Youcanwritetheequationofalineifgiventheslopeandy-intercept .

Whatistheequationofalinethathasaslopeof8anday-interceptof22 .

1 Identifymandb .

2 Substitutetheslope,m,andthey-intercept,b,intheequationy5mx1b .

3 Writetheequation .

m5

b5

y5mx1b

y5 x1

y5

DO

Writing an Equation of a Line

I see! If I know the slope and y-intercept of a line, then I can

write an equation for the line.

I remember! m stands for slope and b represents

the y-intercept.

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LC5 Slope

Andrewsaysthathecanusetheformy5btowritetheequationforanyhorizontallinethatpassesthroughthepoint(0,b) .IsAndrewcorrect?Explain .

DISCUSS

B Youcanwritetheequationofalinethatpassesthroughtheorigin .

Writeanequationforthelineshowninthegraph .

1 Choosetwopointsontheline .

2 Findtheslope .m5y22y1

_____x22x1

3 Identifythey-intercept .

4 Writetheequation .Simplify,ifpossible .

(x1,y1)5(2, )

(x2,y2)5( ,22)

m5 2

_____________

2

5 ______

Thelinecrossesthey-axisattheorigin .

Sob5 .

y5 ______x1 5 ______x

DO

x

y

1–1

–2

–3

–4

–5

–1–2–3–4–5 2 3 4 50

1

2

3

4

5

I remember! The coordinates of the

origin are represented by the point (0,0).

PRACTICEWrite the equation for the line. 

1

x

y

1–1

–2

–3

–4

–5

–1–2–3–4–5 2 3 4 50

1

2

3

4

5

2

x

y

1–1

–2

–3

–4

–5

–1–2–3–4–5 2 3 4 50

1

2

3

4

5

I remember! All horizontal lines have

a slope of 0. That means m 5 0 for all

horizontal lines.

Write an equation of the line with the given slope and y-intercept.

3 slope53;y-intercept57 4 slope522___5 ;y-intercept523

Write an equation of the line that passes through the given points.

5 (0,4)and(25,2) 6 (0,2)and(4,6)

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48  Lesson 5

READY TO gOYoucanuseanytwopointsonalinetofindtheslopeoftheline .Youcancomparetheslopesoftwosegmentsofthelinetoshowthis .

1 Usethepoints(0,23)and(3,21)tofindtheslopeoftheline .

m5212(23)

________320

52__3

2 Usethepoints(3,21)and(9,3)tofindtheslopeoftheline .

m532(21)

_______923

54__6

52__3

3 Comparetheslopes .

2__352__3

Theslopesareequal .

x

y

1–1

–2

–3

–4

–5

–1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

44

22 66

33

Margaritalooksatatableofx-yvaluesandcalculatestheslopeusingonepairofpoints .Sheusesanotherpairofpointsandcalculatesadifferentslope .WhatcanMargaritaconcludeaboutagraphofthosepoints?

DISCUSS

Plug In POWER uP gO!lEssOn lInk

You can find the slope and y-intercept of a line given any two points on the line, its graph, or its equation.

x

y

1–1

–1–2 2 3 4 50

1

2

3

4

5

y �12x � 1

slope:m51__2

y-intercept:b51

You can use the slope and y-intercept of a line to write an equation for the line.

Slope:m55__8

y-intercept:b522

y5mx1b

y55__8x1(22)

55__8x22

Slope

I get it! Every segment of a

line will have the same slope!

I get it! I can investigate slope and y-intercept further

to better understand the characteristics

of lines!

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LC5 Slope

WORk TOgEThERUsetwopairsofpointstoconfirmtheslopeofaline .

• Thislineincludesthepoints(21,6),(1,2),(2,0),and(5,26) .

• Theslopebetween(21,6)and(1,2)is24__2,or22 .

• Theslopebetween(2,0)and(5,26)is26___3 522__1,or22 .

• Theslopeofthelineisthesamewitheitherpairofpoints .

x

y

1–1

–2

–3

–4

–5

–6

–1–2–3–4–5–6 2 3 4 5 60

1

2

3

4

5

6

4

66

2

3

A Youcanuseslopetocheckwhetherthreepointslieonthesameline .

Determineifthepoints(0,6),(3,3),and(6,21)areonthesameline .

1 Findtheslopebetweenanytwoofthethreepoints .

2 Nowfindtheslopebetweentwootherpoints .

3 Comparetheslopestodetermineifthepointsareonthesameline .

Use(0,6)and(3,3) .

m5 2

_____________

2

5 ______5

Use(3,3)and(6,21) .

m5 2

_____________

2

5 ______

Thethreepoints onthesameline

becausetheslopesare .

DO

I could also use any other two pairs of

points on this line to confirm the slope.

B Youcandetermineiftwosegmentsthatshareanendpointareonthesameline .

Determineiftwosegments,onewithendpoints(0,2)and(4,3)andonewithendpoints(0,2)and(8,4),liealongthesameline .

1 Findtheslopeofthefirstsegment .

2 Findtheslopeofthesecondsegment .

3 Comparetheslopestodetermineiftheyareonthesameline .

Use(0,2)and(4,3) .

m5 2

_____________

2

5 ______

Nowuse(0,2)and(8,4) .

m5 2

_____________

2

5 ______5 ______

Theslopesare .The3points onthesameline .

DO

Willalinesegmentthatispartofalargerlinesegmentalwayshavethesameslopeasthelargerlinesegment?Explain .

DISCUSS

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50  Lesson 5

READY TO gO

PRACTICESolve.

1 Alinesegmenthasendpointsat(22,5)and(1,4) .Anotherlinesegmenthasendpointsat(1,4)and(2,7) .Arethelinesegmentspartsofthesameline?

Findtheslopeofthefirstlinesegment .

m542

_________

2(22)

5 ______

Findtheslopeofthesecondlinesegment .

m5 24_______

22

5 ____5

Arethelinesegmentsonthesameline?Explain .

2 Choosetwopairsofpointsonthelinetoshowtheslopeofthelineisthesame .

Usepoints( , )and( , ) .

m5 2

_________

2

5 ______5

Nowusepoints( , )and( , ) .

m5 2

_________

2

5 ____5

Aretheslopesthesame?

3 Usetheendpointsoftwosegmentsalongthislinetoshowthattheslopesofthesegmentsarethesame .

Aretheslopesofthesegmentsthesame?

x

y

1–1

–2

–3

–4

–5

–1–2–3–4–5 2 3 4 50

1

2

3

4

5

x

y

1–1

–2

–3

–4

–5

–1–2–3–4–5 2 3 4 50

1

2

3

4

5

REMEMBER Any two pairs of points on the line will show the same slope.

HINT You can check your answer by graphing the segments on a coordinate grid.

Coordinate Grid can be 

found on p.213 

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LC5 Slope

Solve.

4 Twolinesegmentsarepartoftheliney521__5x110 .Whatistheslopeofeachsegment?

5 Barbdrewalinethroughtheoriginandpoint(2,3) .Jamaldrewalinethroughtheoriginandpoint(4,6) .HowcanyoutellthatBarbandJamaldrewthesameline?

Use the slope formula to determine if the three points lie on a line.

6 (23,22),(21,3),(3,13)

Dothepointslieonthesameline?Explain .

7 (23,11),(2,7),(6,3)

Dothepointslieonthesameline?Explain .

Reasoning with Graphs 

Doanysegmentswithineachofthesegraphsshareanendpoint?Explain .

x

y

1–1

–1 2 3 4 50

1

2

3

4

5

x

y

1–1

–1 2 3 4 50

1

2

3

4

5

x

y

1–1

–1 2 3 4 50

1

2

3

4

5

m � 14

m � 1 4m �

Whatdoyounoticeabouttheslopesasthelinesgetssteeper?

DISCUSS

I see! I can think of any two points as the endpoints

of a segment.

I get it! I can relate the steepness of a line with the value

of its slope.

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52  Lesson 5

READY TO gO

PROblEm sOlvIng

sOlvE

READ

UnderWater PressUreTheunitsofmeasureforpressurearecalledatmospheres .Thepressureatsealevelis1atmosphere .Atadepthof20meters,thepressureis3atmospheres .Writeanequation,andfindthepressureatadepthof100meters .

• Youneedtofindthenumberofatmospheresatadepthof .

• Use(0,1)forthepressureat .

• Use forthepressureatadepthof20meters .

• Usethepointstofindthe .

• Usethemeasureofpressureat asthey-intercept .

• Usey5mx1bandsubstitute forx .

Writeanequation .

Theslope,m5 2

_____________

2

5 ______5 ______

They-interceptis .Theequationisy5 ______x1 .

Findthenumberofatmospheresatadepthof100meters:y51__10( )115

Writeanorderedpairfortheatmosphereatadepthof100meters .(100, )

Checkthatthispointandthegiventwopointslieonthesameline .

Theslopebetween(0,1)and(20,3)is .

Theslopebetween(20,3)and(100,11)is:m5 2

_____________

2

5 ______5 ______

Theslopesare ,sothepoints .

Atadepthof100meters,thepressureis atmospheres .

3 1

PlAn

52  Lesson 5

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  53

5 Slope

PRACTICEUse the problem-solving steps to help you. 

1 Acomputerrepaircompanycharges$50foranappointment .For2hoursofrepairwork,itchargesatotalof$120 .Whatisthetotalchargefor5hoursofrepairwork?

2 Atdawn,thetemperatureis70°F .Thetemperaturefallsatthesameratefortherestoftheday .In4hours,thetemperatureis60°F .Whatisthetemperature6hoursafterdawn?

3 Jennieworksforacellphonecompany .Sheearns$250perweekpluscommissionsshemakesonsales .oneweek,shehadsalesof$1000,andherpaywas$275 .Howmuchwouldherpaybeifshehadsalesof$2,000?

CHECKLIST

READ

PLAN

SoLVE

CHECk

CHECKLIST

READ

PLAN

SoLVE

CHECk

CHECKLIST

READ

PLAN

SoLVE

CHECk

  53

I see! For the company to perform any repair work, a customer is charged for making

an appointment. That amount is the

y-intercept!

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