Section 5.1 - Different Forms of Linear Equations ♦ 191
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Standard Form of a Linear Equation
IfA,BandCarerealnumbers,theequationAx By C+ = iscalledthestandard formoftheequationofaline. Wheneverpossible,itisbesttowritetheequationwithA,BandCasintegers,andA 0$ .
Forexample: x y3 4- + = canbeexpressedas x y3 4- =- ←multiply each term by ( − 1)
x y32 2 3+ = canbeexpressedas x y2 6 9+ = ←multiply each term by 3
Slope - Intercept Form of a Linear Equation
Theequationy mx b= + istheslope-intercept formoftheequationofaline.They-interceptofthelineis (0,b),andtheslopeofthelineism.
Thestandardformofanequationofalinecanbere-writteninslope-interceptformasfollows:
Ax By C By Ax C yBA x
BC
" "+ = =- + =- +
TheslopeofAx By C+ = isBA-
They-interceptofAx By C+ = isBC ,andthepointatwhichthegraphcrossesthey-axisis ,
BC0` j.
Forexample,considerthelinearequation x y2 3 12- = .Theslopeinterceptformofthelinecanbefoundin twoways:
x y2 3- =12 mBA
32
32=- =-
-=
y3- = x2 12- + or y-intercept=BC
312 4=-=-
y = x32 4- y x
32 4= -
Theslopeofthelineis32 ,andthey-interceptis(0,−4).
runrise
x
y
5.1
Lambrick Park Secondary
192 ♦ Chapter 5 - Linear Equations
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Graphing a Line Using the Slope and y-Intercept
Step1: Writetheequationinslope-interceptformbysolvingfory. Step2: Identifythey-intercept(0,b)andgraphthispoint. Step3: Graphanotherpointusingtheslope,countingfromthey-intercept. Step4: Drawthelineconnectingthetwopointstoobtainthegraph.
Graph x y3 2 12+ = byusingtheslopeandy-intercept.
►Solution: Step1: x y3 2+ =12 y2 = x3 12- +
y = x23 6- +
Step2: They-interceptis(0,6):markthispoint.
Step3: Theslopeis: runrisem
23= =- .
From(0,6),gototheright2units, andgodown3units,toobtainthe point(3,2).
Step4: Drawthelinethroughthepoints (0,6)and(3,2).
Graphing a Line Using the Slope and a Point
Step1: Locateandgraphthegivenpoint. Step2: Graphanotherpointusingtheslope,countingfromthefirstpoint. Step3: Drawalineconnectingthetwopointstoobtainthegraph.
Graphthelinethrough(−2,−4)withslope3.
►Solution: Theslopeis3,therefore,fromthepoint(−2,−4),goup3units,andtotheright1unitto obtainthepoint(−1,−1).
Example 1
Example 2
(0,6)
(3,2)–3
+2
x
y
+3
+1 x
y
Lambrick Park Secondary
Section 5.1 - Different Forms of Linear Equations ♦ 193
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Writing an Equation of a Line Using a Slope and a Point
Bysubstitutinggivenvaluesforaslopeandpointofalineintoy mx b= + ,theline’sequationcanbefound.
Writetheequationofthelinewithslope2thatrunsthrough(−4,1)inslopeintercept-form.
►Solution: Thepoint(−4,1)givesax-valueof−4anday-valueof1.
y mx b= + " b1 2 4= - +^ h b1 8=- +
b 9=
Therefore,theequationofthelineisy x2 9= + .
Point - Slope Form of a Linear Equation
Theequationy y m x x1 1- = -^ histhepoint-slopeequationofaline.Thegivenpointis ,x y1 1^ handthe slopeofthelineism.Thisformulacomesfromre-arrangingthedefinitionofslope,m
x xy y
1
1=-- .
Writetheequationofalinewithslope2thatpassesthrough(−4,1)inslopeinterceptform.
►Solution: Substitutingthegivenpointandslopeintothepoint-slopeequationgives:
y y m x x1 1- = -^ h " y x1 2 4- = - -^^ hh y x1 2 4- = +^ h y x1 2 8- = +
y x2 9= +
Writetheequationofalinewithslope54 thatpassesthrough(3,−2)instandardform.
►Solution: Substitutingthegivenpointandslopeintothepoint-slopeequationgives:
y y m x x1 1- = -^ h " y x254 3- - = -^ ^h h
y x254 3+ = -^ h
y x5 2 4 3+ = -^ ^h h
y x5 10 4 12+ = -
x y4 5 22- =
Example 3
Example 4
Example 5
Lambrick Park Secondary
194 ♦ Chapter 5 - Linear Equations
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5.1 Exercise Set
1. Completeeachstatement.
a) Theformulaforthepoint-slopeformofalineis.
b) Intheequationy mx b= + ,(0,b)iscalledthe.
c) Theequationy mx b= + iscalledtheformoftheequationofaline.
d) Thestandardformoftheequationofalineis.
e) TheslopeofAx By C+ = is.
f) They-interceptofAx By C+ = is.
2. Findtheslopeandy-intercept.
a) x y3 2 6- = slope b) x y4 3 12+ = slope
y-intercept y-intercept
c) x y2 5 7- =- slope d) x y5 2 0+ = slope
y-intercept y-intercept
e) x y4 4- =- slope f) x y6 3- =- slope
y-intercept y-intercept
3. Rewritethestandardformequationinslope-interceptform.
a) x y2 6+ = b) x y3 4- =
c) x y4 3 12+ = d) x y2 3 6- =
e) x y5 4 3+ = f) x y6 3 4- =
Lambrick Park Secondary
Section 5.1 - Different Forms of Linear Equations ♦ 195
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4. Rewritetheslope-interceptequationinstandardform.
a) y x2 1=- + b) y x3 1= -
c) y x3= d) y x32 1=- +
e) y x43 5= + f) y x
52
21=- +
5. Rewritethepoint-slopeequationinslope-interceptform.
a) y x2 3 1- = +^ h b) y x4 2 1+ =- -^ h
c) y x131 2- = +^ h d) y x4
52 3+ =- -^ h
e) y x32
41 8- = -^ h f) y x
41
21
32- = +` j
6. Rewritethepoint-slopeequationinstandardform.
a) y x2 3 1- = +^ h b) y x4 2 1+ =- -^ h
c) y x131 2- = +^ h d) y x4
52 3+ =- -^ h
e) y x32
41 8- = -^ h f) y x
41
21
32- = +` j
Lambrick Park Secondary
196 ♦ Chapter 5 - Linear Equations
Copyright © 2009 by Crescent Beach Publishing. No part of this publication may be reproduced without written permission from the publisher.
7. Matcheachdescriptionwithanequation.
a) Slope=−3,passingthrough(−1,2) i) y x3=
b) Slope=3,y-intercept(0,−6) ii) y x31=-
c) Passingthrough(0,0)and(3,−1) iii) y x3=-
d) Passingthrough(0,0)and(−1,3) iv) x y3 6- =
e) Passingthrough(2,0)and(0,−6) v) x y3 6- =
vi) y x2 3 1- =- +^ h
vii)y x2 3 1+ =- -^ h
8. Matcheachequationwiththegraphitmostcloselyresembles.
a) y x 2= - i) ii)
b) y x 2=- -
c) y x 2=- +
d) y x 2= +
iii) iv)
9. Writetheequationofeachlineinslope-interceptform.
a) (0,2),m=2 b) (0,−3),m=21
c) (0,3),m=0 d) (0,−2),m=32-
e) ,021-` j,m=
43- f) (0,2.3),m=0.4
x
y
x
y
x
y
x
y
Lambrick Park Secondary
Section 5.1 - Different Forms of Linear Equations ♦ 197
Copyright © 2009 by Crescent Beach Publishing. No part of this publication may be reproduced without written permission from the publisher.
10. Graphthelinearequation.
a) x y4 3 12- = b) y x32 4=- +
c) y x321 4- = +^ h d) x y2 3 10+ =
e) y x232 5+ =- +^ h f) x y5 2 0- =
g) y x25
21
23- =- +` j h) y x
35
27= -
x
y
x
y
x
y
x
y
x
y
x
y
x
y
x
y
Lambrick Park Secondary
198 ♦ Chapter 5 - Linear Equations
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11. Writetheequationinstandardform,slope-interceptform,andpoint-slopeform.
a) b)
standardform standardform
slope-interceptform slope-interceptform
point-slopeform point-slopeform
c) d)
standardform standardform
slope-interceptform slope-interceptform
point-slopeform point-slopeform
e) f)
standardform standardform
slope-interceptform slope-interceptform
point-slopeform point-slopeform
x
y
x
y
x
y
x
y
x
y
x
y
Lambrick Park Secondary
Section 5.2 - Special Cases of Linear Equations ♦ 199
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Horizontal Lines
Ahorizontallinecanbethoughtofasallthepointsonagraphwhereyhasthesamevalue.Fromsection5.1, itwasshownthattheslopeofahorizontallineis0.
Usingaslopeof0intheslope-interceptequationofaline,y mx b y x b y b0" "$= + = + =
Equation of a Horizontal Line with y-Intercept k
y k=
Forexample:y 3=
Vertical Lines
Averticallinecanbethoughtofasallthepointsonagraphwherexhasthesamevalue.Fromsection5.1it wasshownthattheslopeofaverticallineisundefined.
Theequationofaverticallineisx=kbydefinition,sincetheslopeisundefined.
Equation of a Vertical Line with x-Intercept k
x k=
Forexample:x 3=
x
y
x
y
Special Cases of Linear Equations5.2
Lambrick Park Secondary
200 ♦ Chapter 5 - Linear Equations
Copyright © 2009 by Crescent Beach Publishing. No part of this publication may be reproduced without written permission from the publisher.
Writing the Equation of a Line Through Two Points
Withourknowledgefromsection6.1itispossibletowritetheequationofalinewhenthecoordinatesoftwo pointsonthelineareknown.
WritetheequationofthelinepassingthroughA(5,2)andB(1,−4)inslope-interceptform.
►Solution: First,findtheslopeoftheline.
mx xy y
5 1
2 4
46
23
2 1
2 1=
--
=-
- -= =
^ h
Pickeitherpoint,andsubstituteitintothepoint-slopeformequation.Forthisexample,A(5,2) isused.
y y m x x1 1- = -^ h → y x223 5- = -^ h
y x223
215- = -
y x23
215 2= - +
y x23
211= -
Parallel and Perpendicular Lines
Inchapter5itwasshownthatparallellineshavethesameslopebutdifferenty-interceptsandperpendicular lineshaveslopesthatarenegativereciprocalsofeachother.Wecannowusetheseconceptstodetermineif equationsareparallel,perpendicularorneither.
Inthesystemofequationsx yx y2 32 6
- + =+ =' 1,determineifthelinesareparallel,perpendicular,
orneither.
►Solution: Theslopeofthestandardformoftheequationofaline,Ax By C+ = ,isBA- .
x y2 6+ = hasslope21-
x y2 3- + = hasslope2
Theslopesarenegativereciprocalsofeachother,thereforethelinesareperpendicular.
Example 1
Example 2
Lambrick Park Secondary
Section 5.2 - Special Cases of Linear Equations ♦ 201
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Inthesystemofequationsx yx y6 2 123 5- + =
- =' 1,determineifthelinesareparallel,perpendicular, orneither.
►Solution: Thisproblemcanbesolvedbychangingbothequationstoslope-interceptform.
x y3 5- = x y6 2 12- + =
y x3 5- =- + y x2 6 12= +
,y x m3 5 3= - = ,y x m3 6 3= + =
Theslopesareequal,thereforethelinesareparallel.
Inthesystemofequationsx yx y2 44 3 7- =+ =' 1,determineifthelinesareparallel,perpendicular,
orneither.
►Solution: Leavingthesystemofequationsinstandardform:
x y4 3 7+ = hasslopemBA
34=- =-
x y2 4- = hasslopemBA
12 2=- =--=
Changingthesystemofequationstoslope-interceptform:
x y4 3 7+ = x y2 4- =
y x3 4 7=- + y x2 4- =- +
,y x m34
37
34=- + =- ,y x m2 4 2= - =
Bothmethodsproducethesameresult:theslopesareneitherthesame,nornegativereciprocals, thereforethelinesareneitherparallelnorperpendicular.
Example 3
Example 4
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202 ♦ Chapter 5 - Linear Equations
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5.2 Exercise Set
1. Matchthegraphy mx b= + withitsclosestdescription.
a) ,m b0 01 1 i) ii) iii)
b) ,m b0 02 1
c) ,m b0 01 2
d) ,m b0 02 2
e) m 0= iv) v) vi)
f) b 0=
2. Matchthegraphwiththelinearrelation.
a) x y4 8- =- i) ii)
b) x y4 8+ =
c) x y4 2- =-
d) x y4 2+ =
iii) iv)
3. Determinetheequationofthegraph.
a) b)
c) d)
x
y
x
y
x
y
x
y
x
y
x
y
x
y
x
y
x
y
x
y
x
y
x
y
x
y
x
y
Lambrick Park Secondary
Section 5.2 - Special Cases of Linear Equations ♦ 203
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4. Determinetheequationofalinethroughthegivenpairofpoints.
a) (−4,1)and(6,1) b) (1,−4)and(1,6)
c) (−2,0)and(5,0) d) (0,−2)and(0,5)
e) (a,b)and(c,b) f) (b,a)and(b,c)
5. Writetheequationofthelinewiththegivencharacteristics.
a) vertical,passesthrough(3,6) b) vertical,passesthrough(−2,−4)
c) horizontal,passesthrough(3,6) d) horizontal,passesthrough(−2,−4)
6. Foreachpairofequations,determinewhetherthelinesareparallel,perpendicular,orneitherparallelnor perpendicular.
a) x y2 5 7+ = b) x y4 3 7- + =
x y4 10 2+ = x y8 6 0- + =
c) x y4 3 6- = d) x y3 5 4- = x y4 6 3+ =- x y5 3 4- =
e) x y4 3 5- = f) x y2 5 3- =-
x y3 4 2+ = x y10 4 1+ =
g) x y4 3- = h) x y5 2 7- =
x y4 2- =- x y2 5 7+ =
Lambrick Park Secondary
204 ♦ Chapter 5 - Linear Equations
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7. Writetheequationofalinepassingthroughthegivensetofpointsinslope-interceptform.
a) (3,5)and(2,4) b) (5,−2)and(−3,1)
c) (−4,1)and(−2,−3) d) (−1,−2)and(−6,−4)
e) (6,−2)and(−3,2) f) (0,0)and(−3,2)
g) (0,−6)and(−4,0) h) (5.2,−6.8)and(−1.6,−3.8)
i) (2,5)and(−2,5) j) (3,7)and(3,−1)
Lambrick Park Secondary
Section 5.2 - Special Cases of Linear Equations ♦ 205
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8. Reasoning.
a) Ifalineishorizontal,whatistheslopeofany b) Ifthegraphofalinearequationhasonepointthat linethatisperpendiculartoit? isboththex-interceptandy-intercept,whatpoint thatbe?
c) Whatistheequationofthex-axis? d) Findthex-interceptof x y3 2 8- = .
e) Findthevalueofcsothatthegraphof f) Findthey-interceptof x y4 3 2=- + . x c y4 3+ = hasanx-interceptof(−2,0).
g) Findthevalueofcsothatthegraphof h) IfAisnotzero,whatwillthegraphofAx C D+ =
x c y3 2- = hasay-interceptof(0,5). looklike?
i) IfBisnotzero,whatwillthegraphof j) Whatistheequationofalinewithxandycoordinates By E F+ = looklike? thatareequal,andpassesthroughtheorigin?
k) Whatistheequationofalinewithxandy l) Whatistheequationofalinepassingthroughthe coordinatesthatareoppositeinvalueand point ,a b^ hwithslope0? passesthroughtheorigin?
m) Whatistheequationofalinepassingthrough n) Whatisthey-interceptofax by c+ = ? thepoint ,a b^ hwithanundefinedslope?
o) Whatisthex-interceptofax by c+ = ? p) Whatistheslopeofthelineax by c+ = ?
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206 ♦ Chapter 5 - Linear Equations
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9. Findthexandyinterceptsoftheline 10. Showthattheequationofalinewithx-intercept ax by ab+ = . (a,0)andy-intercept(0,b)canbewritteninthe
formaxby1+ = .
11. Showthattheequationofalinewithpoints 12. Determinetherelationshipbetweenthegraphsof ,x y1 1^ hand ,x y2 2^ hcanbewrittenintheform theequationsAx By C+ = andBx Ay C- = . y y x x y y x x1 2 1 2 1 1- - = - -^ ^ ^ ^h h h h.
13. Ifthetwopointsthatalinepassesthroughare 14. Thinkofdifferentpointsonthegraphofthe known,itsequationcanbefound.Explainhow horizontalliney 2= .Whatdothepointshave thisisdone. incommon?Howdotheydiffer?
15. Whatistheslopeofallorderedpairsoftheform 16. Giventhat0cCisthesametemperatureas32cF, ,x x3-^ h? and100cCisequivalentto212cF,determinethe equivalentof68cFincC.
17. Intheequation,ax by x y2 3 6+ = - + ,find 18. Intheequation,ax by x y2 3 6+ = - + ,find a andb ifthegraphisahorizontallinepassing a andb ifthegraphisaverticallinepassing through ,0 3^ h. through ,3 0^ h.
Lambrick Park Secondary
Section 5.3 - Equations of Parallel and Perpendicular Lines ♦ 207
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Towritetheequationofaline,apointandslopeisneeded.However,insomeproblemsthisinformationisnot directlygiven,andfurtherstepsmustbetakentofindapointorslope.
Whendeterminingtheequationofalinethatisparalleltoagivenslope,theconcepttorememberisthat parallellineshaveequalslopes.Whendeterminingtheequationofalinethatisperpendiculartoagivenslope, theconcepttorememberisthatperpendicularlineshaveslopesthatarenegativereciprocalsofeachother.
Writetheequationofalineparallelto x y3 2 6- = ,andwhichgoesthroughthepointA(4,−2).
►Solution: x y3 2 6- = hasslope m=BA- =
23--
=23
Therefore,theslopeofalineparallelto x y3 2 6- = hasslopem=23
Substitutingthegivenpointandslopeintothepoint-slopeequationofalinegives:
y y m x x1 1- = -^ h → y x223 4- - = -^ ^h h
y x223 6+ = -
y x23 8= - (slope-intercept form)
x y3 2 16- = (standard form)
Writetheequationofalineperpendicularto x y4 2 7+ = goingthroughthepointB(−2,5).
►Solution: x y4 2 7+ = hasslopem=BA- =
24- =−2
Thereforetheslopeofalineperpendicularto x y4 2 7+ = hasslopem=21 .
Substitutingthegivenpointandslopeintothepoint-slopeequationofalinegives:
y y m x x1 1- = -^ h → y x521 2- = - -^^ hh
y x521 2- = +^ h
y x521 1- = +
y x21 6= + (slope-intercept form)
x y2 12- = (standard form)
Example 1
Example 2
Equations of Parallel and Perpendicular Lines5.3
Lambrick Park Secondary
208 ♦ Chapter 5 - Linear Equations
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5.3 Exercise Set
1. Findtheslopesoflinesparallelandperpendiculartotheequation.
a) y x3= m ; ; b) y x2=- m ; ;
m= m=
c) y x32 2=- + m ; ; d) y x
53 1= - m ; ;
m= m=
e) x y2 3 4- = m ; ; f) x y3 2+ = m ; ;
m= m=
g) x y5 0- = m ; ; h) x 2= m ; ;
m= m=
i) y 2=- m ; ; j) x y2 1= - m ; ;
m= m=
k) x y43
31
21= + m ; ; l) . . .x y0 2 2 3 1 4+ = m ; ;
m= m=
Lambrick Park Secondary
Section 5.3 - Equations of Parallel and Perpendicular Lines ♦ 209
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2. Findtheequationoftheline,instandardform,thatpassesthroughthegivenpointandisparalleltothegiven line.
a) P(0,0);y x2 5= - b) P(0,0);x y2 5= +
c) P(1,3); x y3 6- = d) P(−2,0); x y2 5 3+ =
e) P(−6,3);y x4 8+ =- f) P(5,−2); y x3 1 4+ =-
g) P(−4,−3);x y43 2= - h) P(0,−5);x y
32 1=- +
i) P(−5,2);x 3= j) P(−5,2);y 4=-
k) P(−4,1); x y32
43 12+ = l) ,P
21
32-` j; .x y
31 0 4 2- =
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210 ♦ Chapter 5 - Linear Equations
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3. Findtheequationoftheline,instandardform,thatpassesthroughthegivenpointandisperpendiculartothe givenline.
a) P(0,0);y x2 5= - b) P(0,0);x y2 5= +
c) P(1,3); x y3 6- = d) P(−2,0); x y2 5 3+ =
e) P(−6,3);y x4 8+ =- f) P(5,−2); y x3 1 4+ =-
g) P(−4,−3);x y43 2= - h) P(0,−5);x y
32 1=- +
i) P(−5,2);x 3= j) P(−5,2);y 4=-
k) P(−4,1); x y32
43 12+ = l) ,P
21
32-` j; .x y
31 0 4 2- =
Lambrick Park Secondary
Section 5.3 - Equations of Parallel and Perpendicular Lines ♦ 211
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4. Determinetheequationofalineparalleltothegraphgoingthroughthegivenpoint,instandardform.
a) (4,−2) b) (−5,−4)
c) (2,1) d) (5,−3)
5. Determinetheequationofalineperpendiculartothegraphgoingthroughthegivenpoint,instandardform.
a) (4,−2) b) (−5,−4)
c) (2,1) d) (5,−3)
x
y
x
y
x
y
x
y
x
y
x
y
x
y
x
y
Lambrick Park Secondary
212 ♦ Chapter 5 - Linear Equations
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6. Findtheequationofalineparallelto x y3 4 8+ = 7. Findtheequationofalineparalleltox y3 8- =
withthesamey-interceptas x y5 3 10- = . withthesamey-interceptas x y3 2 6+ = .
8. Findtheequationofalineparallelto x y2 7 10+ = 9. Findtheequationofalineperpendicularto withthesamex-interceptas x y3 4 5- = . x y2 3 7- = withthesamey-interceptas x y5 2 10- = .
10. Findtheequationofalineperpendicularto 11. Findtheequationofalineperpendicularto x y3 2 9+ = withthesamex-interceptas 1x y2
321= + withthesamex-interceptas
x y2 5 0- = . x y2 3 9+ = .
12. Acirclecentredattheoriginpassesthroughthepoint 13.Arhombushascoordinates ,0 0^ h, ,3 4^ h, ,8 4^ h, ,3 4-^ h.Whatistheequationofalineperpendicular and ,5 0^ h.Whataretheequationsofthediagonals totheradiusatthispoint? oftherhombus?Whatrelationshipistherebetween thediagonals?
Lambrick Park Secondary
Section 5.4 - Linear Applications and Modelling ♦ 213
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Graphsareeffectivevisualtoolsbecausetheypresentinformationquicklyandeasily.Sometimes,datacanbe betterunderstoodwhenpresentedbyagraphthanbyatablebecausethegraphcanrevealatrendor comparison.
Waterfreezesat32°F,or0°C.Waterboilsat212°F,or100°C.Graphthelinearrelation between°Cand°F,andfindaformulathatconvertsCelsiustoFahrenheit.
►Solution: Thefreezingpointonthegraphis(0,32) Theboilingpointonthegraphis(100,212)
m100 0212 32
100180
59=
-- = =
Byslope-intercept,F C59 32= +
Itcostsapopcornvendor$490tomake150bagsofpopcornand$610tomake350bags.
a) Graphthelinearrelationbetweencostandnumberofbags. b) Findthecostequation. c) Findthefixedcost. d) Findthecostof250bagsofpopcorn. e) Howmanybagsofpopcorncanbeboughtfor$724?
►Solution: b) .m350 150610 490
200120 0 60=
-- = =
.C B490 0 60 150- = -^ h .C B490 0 60 90- = -
.C B0 60 400= +
c) Thefixedcostis$400
d) . $C 0 60 250 400 550= + =^ h
e) . B0 6724 0 400= +
. B0 6724 400 0- =
. B0 6 3240 =
B 540=
Example 1
Example 2
212
32
100°C
°F
Cost($)
BagsofPopcorn
490
610
150 350
Linear Applications and Modelling5.4
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214 ♦ Chapter 5 - Linear Equations
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Afamilyhasamedicalplanthatpays70%ofallprescriptioncosts,lessa$200deductibleeach year.
a) Writeafunctionthatmodelsthefamily’sresponsibilityforprescriptioncosts. b) Determinetheamountthemedicalplanwillpayon$1250inprescriptioncosts. c) Determinetheamountspentonprescriptionpurchasesiftheamounttheplanpaidwas $1250. d) Graphthisfunctionandlabeltheanswersfromb)andc).
►Solution: a) LetRbetherefund,andCbetheprescriptioncost.
m=0.70,y-intercept=−200, .R C0 70 200= -
b) R = . C0 70 200-
= .0 70 1250 200-^ h = 675
Theplanwillpay$675on$1250inprescriptioncosts.
c) R = . C0 70 200-
1250 = . C0 70 200-
1450 = . C0 70
C = 2071.43
$2071.43isspentonprescriptionpurchases,togeta$1250refund.
d)
Example 3
PrescriptionCost($)
c)
b)
Refund($)
01250
675
2071.73–200
1250
Lambrick Park Secondary
Section 5.4 - Linear Applications and Modelling ♦ 215
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5.4 Exercise Set
Assumelinearappreciationorlineardepreciationforallproblems.
1. Aninsurancecompanypurchasedcomputersforits 2. Inherfirstyearofpractice,apsychologisthas160 office.Thevalueofthecomputersaftertwoyears patients.Bythethirdyear,thenumberofpatients was$80000,and$56000afterfouryears. grewto246.Ifthistrendcontinues,howmany Determinethepurchasepriceofthecomputers. patientswouldshehaveinthefourthyear?
3. Thepercentof18-25yearoldswhosmoke 4. Ataxicabispurchasedfor$36000.Attheendof worldwidehaschangedfrom46.8%in1987to 10yearsitissoldforscrapfor$1800.Findthe 37.2%in2000.Predictthepercentageof18-25 depreciationequation. yearoldsthatwillsmokein2012.
5. Ahomewaspurchasedfor$410000.Theowner 6. Aprintercosts$960newandisexpectedtobe expectsthehometodoubleinvalueinthenext worth$140aftersixyears.Whatwillitbeworth 10years.Findtheappreciationequation. afterfouryears?
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216 ♦ Chapter 5 - Linear Equations
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7. Apaintingisexpectedtoappreciate$75eachyear. 8. Agrandfatherclockisexpectedtobeworth$2700 Ifthepaintingwillbeworth$620intwoyears, inthreeyearsand$3200infiveyears.Whatwill whatwillitbeworthin14years? itbeworthineightyears?
9. Atimesharecottagepurchasedfouryearsagois 10. Aprintingcompanychargesafixedratetosetup nowworth$36200.Ifthecottagehasappreciated theprintingpress,plusacostof$3.50forevery $2150peryear,finditsoriginalpurchaseprice. 100copies.If800copiescost$64.00,howmuch willitcosttoprint1500copies?
11. Acitywithapopulationof62000had480police 12. Anelectricalsubstationisworth$246000when investigationsinayear.Whenthepopulationof itisinstallednew,butisworthnothingafterits thecityroseto74000,thenumberofinvestigations 15yearlifecycle.Findthedepreciationequation. was640inayear.Ifthistrendcontinues,howmany investigationswillthecityhavewhenitspopulation reaches100000?
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Section 5.4 - Linear Applications and Modelling ♦ 217
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13. Thetotalcostofacomputeristhesumoftheselling 14. Itcosts$1200tostartupabusinesssellinghot price,plusasalestaxof12%,plusa$20disposalfee. dogsonthebeach.Eachhotdogcosts40¢to produce.
a) Expressthetotalcostofthecomputerasalinear a) Findthecostequation. functionofthesellingprice.
b) Whatisthetotalcostofacomputerthatsells b) Howmanyhotdogsareproducedifthetotal for$1540? costis$1560?
c) Whatisthesellingpriceofacomputerwhose c) Howmanyhotdogsmustbesoldattwo totalcostwas$1061.60? for$1.00tobreakeven?
15. Itcostsacompany$2140toproduce500widgets 16. ToshipapackagefromVancouvertoWinnipeg and$3660toproduce900widgets. overnightcosts$27.30foraonepoundpackage, and$38.80forathreepoundpackage.
a) Whatisthefixedcostforproducingwidgets? a) Findthecostequation.
b) Findanequationrelatingthecostofproducing widgets. b) Findthecostofshippinga6.5poundpackage.
c) Whatisthetotalcostofproducing200widgets? c) Ifapackagecost$45.70toship,howmuch doesitweigh?
d) Howmanywidgetscanbeproducedfor$7308?
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218 ♦ Chapter 5 - Linear Equations
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Thenotation f x^ hisanotherwayofwritingyasafunction.Forexample,thefunctiony x2 4= - maybe writtenas f x x2 4= -^ h ,where f x^ hisread“fofx”.
Withoutfunctionnotation,aproblemcouldbestated:Giveny x2 4= - ,findywhenx=5.Usingfunction notation,thesameproblemwouldbestated:Given f x x2 4= -^ h ,find f 5^ h.The notation f 5^ h implies the value of y when x is 5.Thestatement f 5 6=^ h saysthevalueofyis6whenxis5.Thisisthepoint(5,6).
Given f x x3 5= +^ h ,determinethecoordinatesofonepointonthelinefor f 2^ h.
►Solution: f 2 3 2 5 11= + =^ ^h h
Thereforethepointis(2,11).
Given f x x3 5= +^ h ,determinethecoordinatesofthepointwhere f x 7=-^ h .
►Solution: f x x3 5= +^ h x7 3 5- = +
x7 5 3- - =
x12 3- =
x4- =
Thereforethepointis(−4,−7).
Completethetablefor f x x3 5= +^ h .
x x3 5+ f x^ h ,x y^ h3
►Solution:
Example 1
Example 2
Example 3
x x3 5+ f x^ h ,x y^ h3 14 f(3) (3,14)
Function Notation5.5
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Section 5.5 - Function Notation ♦ 219
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Determinetheslope-interceptfunction f x mx b= +^ h if f 1 4=^ h and f 3 2=-^ h .
►Solution: f 1 4=^ h meansthepoint(1,4)
f 3 2=-^ h meansthepoint(3,−2)
m1 3
4 2
26 3=
-
- -=-=-
^ h
f x mx b= +^ h f b1 3 1=- +^ ^h h b4 3=- +
b 7=
Therefore f x x3 7=- +^ h
If f x x2 1= +^ h ,
a)Whatis f x3^ h?b)Whatis f x 3+^ h?
►Solution: a) f x3^ h= x2 3 1+^ h = x6 1+
b) f x 3+^ h= x2 3 1+ +^ h = x2 7+
If f x x2 1= +^ h ,determineh
f x h f x+ -^ ^h h ,h 0! .
►Solution: f x x2 1= +^ h , f x h x h2 1+ = + +^ ^h h
Thereforeh
f x h f x+ -^ ^h h =h
x h x2 1 2 1+ + - +^ h6 6@ @
=h
x h x2 2 1 2 1+ + - -
=hh2
= 2
Example 4
Example 5
Example 6
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220 ♦ Chapter 5 - Linear Equations
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5.5 Exercise Set
1. Completethetableforthelinearfunctiondefinedbyg x x2 3=- +^ h .
x x2 3- + g x^ h ,x y^ h2−42c
c−2−c+1
2. For f x x3 2= -^ h ,find:
a) f 3^ h b) f 4-^ h
c) f k^ h d) f x2 1-^ h
e) f x h+^ h f) f x f h+^ ^h h
3. For f x x4 5= +^ h ,find:
a) f 3^ h b) f 4-^ h
c) f k^ h d) f x2 1-^ h
e) f x h+^ h f) f x f h+^ ^h h
4. For f x x5 2=- +^ h ,find:
a) f x 3=-^ h b) f x 7=^ h
c) f x 12=-^ h d) f x 5=-^ h
e) f x a=^ h f) f x a5 7=- +^ h
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Section 5.5 - Function Notation ♦ 221
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5. Grapheachfunctionovertherealnumbers.
a) f x x2 1= +^ h b) f x x21 3=- +^ h
c) f x x43 2= -^ h d) f x x
32 4=- -^ h
e) f x 3=^ h f) f x x41 42=- +^ h
f(x)
x
f(x)
x
f(x)
x
f(x)
x
f(x)
x
f(x)
x
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222 ♦ Chapter 5 - Linear Equations
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6. Grapheachfunctionifthedomainis{−3,−2,−1,0,1,2}
a) f x x2 1= +^ h b) f x x21 3=- +^ h
c) f x x43 2= -^ h d) f x x
32 4=- -^ h
e) f x 3=-^ h f) f x x41 3=^ h
f(x)
x
f(x)
x
f(x)
x
f(x)
x
f(x)
x
f(x)
x
Lambrick Park Secondary
Section 5.5 - Function Notation ♦ 223
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7. Forg r r2r=^ h ,find:
a) .g 0 5^ h b) g38` j
c) g h^ h d) g h 2+^ h
8. Forg r rh2r=^ h ,find:
a) .g 0 5^ h b) g38` j
c) g h^ h d) g h 2+^ h
9. Forg r r2r=^ h ,find:
a) g21` j b) g
38` j
c) g h^ h d) g h 2+^ h
10. Forg r r h2r=^ h ,find:
a) g21` j b) g
38` j
c) g h^ h d) g h 2+^ h
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224 ♦ Chapter 5 - Linear Equations
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11. Usethegraphofeachfunctiontostatethedomain,statetherange,determine f 2^ h,andsolve f x 2=^ h forx.
a) domain
range
f 2^ h
f x 2=^ h
b) domain
range
f 2^ h
f x 2=^ h
c) domain
range
f 2^ h
f x 2=^ h
d) domain
range
f 2^ h
f x 2=^ h
f(x)
x
f(x)
x
f(x)
x
f(x)
x
Lambrick Park Secondary
Section 5.5 - Function Notation ♦ 225
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12. Completethetable,andgraphthefunction.Also,givethedomainandrangeofthefunction.
a) x xg 2 1=- -^ h domain
range
b) g x x32
= -^ h domain
range
c) x xg 4= -^ h domain
range
d) x xg21= -^ h domain
range
x g x^ h−4−204
x g x^ h1−1−3−5
x g x^ h10−1−2
x g x^ h01−1−2
g(x)
x
g(x)
x
g(x)
x
g(x)
x
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226 ♦ Chapter 5 - Linear Equations
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13. Determine f x mx b= +^ h .
a) f 0 3=-^ h b) f 2 4=^ h f 2 5- =^ h f 1 4- =-^ h
c) f 2 5=^ h d) f 3 6- =^ h f 3 3- =^ h f 1 2=-^ h
e) f 3 2=^ h f) f21
32=-` j
f 3 2- =^ h f25
38- =` j
14. Determineh
f x h f x+ -^ ^h h ,h 0! .
a) f x x3=^ h b) f x x3 4= -^ h
c) f x x5 2= -^ h d) f x x2=^ h
Lambrick Park Secondary
Section 5.5 - Function Notation ♦ 227
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15. Thefunction 32f c c59= +^ h determinesthe 16. Aballisdroppedfromahighrisebuilding.The
FahrenheitequivalentofdegreesCelsius.Findthe heightoftheballinmetres,tsecondsafteritis Fahrenheitequivalentof: dropped,isgivenbythefunction .h t t9 8 1002=- +^ h .
a) 30°C a) Findh 0^ h.
b) 0°C b) Findtheheightoftheballafter2seconds.
c) −40°C c) Findthetimeittakesfortheballtohitthe ground.
17. Thefunction 1P d d32= +^ h givesthepressurein 18. ThetemperaturebelowthesurfaceoftheEarthis
atmospheresatadepthofdfeetintheocean. givenbyT d d10 20= +^ h ,whereTisincelsius anddisinkilometres.
a) Findthepressureat160feet. a) Findthetemperature12kmbelowthesurface oftheearth.
b) Atwhatdepthisthepressure9.6atmospheres? b) Whatdepthhasatemperatureof166°C?
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228 ♦ Chapter 5 - Linear Equations
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Section 5.1
1. Findtheslopeandy-intercept.
a) x y2 5 7- = slope b) x y5 2+ =- slope
y-intercept y-intercept
2. Writethestandardformequationinslope-interceptform.
a) x y6 3- = b) x y2 5 7+ =
3. Writetheslope-interceptequationinstandardform.
a) y x32 4=- + b) y x3
52=- +
4. Writethepoint-slopeequationinslope-interceptform.
a) y x132 4+ =- -^ h b) y x
32 4
21- =- +` j
5. Writethepoint-slopeequationinstandardform.
a) y x132 4+ =- -^ h b) y x
32 4
21- =- +` j
6. Writetheequationofeachlineinstandardform.
a) (0,−3),m=−4 b) (2,0),m=31-
Chapter Review5.6
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Section 5.6 - Chapter Review ♦ 229
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7. Determinetheequationin:standardform,slope-interceptformandpoint-slopeform.
a) b)
standardform standardform
slope-interceptform slope-interceptform
point-slopeform point-slopeform
Section 5.2
8. Determinetheequationofthegraph.
a) b)
9. Writetheequationofthelinewiththegivencharacteristics.
a) vertical,passesthrough(−2,5) b) horizontal,passesthrough(−2,5)
c) vertical,passesthrough ,a b^ h d) horizontal,passesthrough ,a b^ h
x
y
x
y
x
y
x
y
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230 ♦ Chapter 5 - Linear Equations
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10. Foreachpairofequations,determinewhetherthelinesareparallel,perpendicularorneitherparallelnor perpendicular.
a) x y3 2 7+ = b) x y5 2 4- =
x y4 6 2+ = x y4 10 3+ =
c) y x2 3= - d) x y3 2- =
x y2 3+ =- x y6 2 2- =
11. Writetheequationofthelinepassingthroughthegivensetofpointsinstandardform.
a) (−3,1)and(−4,−6) b) (−2,−3)and(−5,−1)
Section 5.3
12. Findtheslopesoflinesparallelandperpendiculartothefollowingequations.
a) x y3 4 6- =- m ; ; b) x y3 2= + m ; ;
m= m=
13. Findtheequationofthelinethatpassesthroughthegivenpointandisparalleltothegivenline.
a) P(−2,4); x y2 3 5- = b) P(4,−1); x y4 7 2+ =-
14. Findtheequationofthelinethatpassesthroughthegivenpointandisperpendiculartothegivenline.
a) P(−2,4); x y2 3 5- = b) P(4,−1); x y4 7 2+ =-
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Section 5.6 - Chapter Review ♦ 231
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15. Determinetheequationofaline,instandardformwhichisparalleltothelineandwhichgoesthroughthe givenpoint.
a) (5,2) b) (−3,4)
16. Determinetheequationofaline,instandardformwhichisperpendiculartothelineandwhichgoesthrough thegivenpoint.
a) (5,2) b) (−3,4)
Section 5.4
17. Thecosttoprint1200booksis$11140,andthecosttoprint2000booksis$17940.Assumingthereisalinear relationbetweenthecostsandthenumberofbooksprinted.
a) Findthecostequation. b) Findthe“setup”costofprintingthebooks.
c) Findthecostof3000books. d) Howmanybookscanbepurchasedfor$24740.
x
y
x
y
x
y
x
y
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232 ♦ Chapter 5 - Linear Equations
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Section 5.5
18. For f x x3 2=- -^ h ,find:
a) f 3^ h b) f 4-^ h
c) f x 3=^ h d) f x 4=-^ h
e) f a^ h f) f x a=^ h
g) f x h+^ h h) f x f h+^ ^h h
19. Determine f x mx b= +^ h .
a) f 3 4=^ h b) f 1 4- =-^ h f 2 6- =^ h f 3 7=^ h
c) f 4 2- =-^ h d) f 4 2=^ h f 1 5=^ h f 2 4=^ h
e) f a a2=^ h f) f a b=^ h f b b2=^ h f b a=^ h
Lambrick Park Secondary