Download Reteach Using Graphs and Tables to Solve Linear Systems

Name
LESSON
3-1
Date
Class
Reteach
Using Graphs and Tables to Solve Linear Systems
A linear system of equations is a set of two or more linear equations. To solve a linear
system, find all the ordered pairs (x, y) that make both equations true. Use a table and a
graph to solve a system of equations.
yx2
y x 2
Solve each equation for y.→
y 2x 5
y 2x 5
Make a table of values for each equation.
When x 1, y 3 for
y x 2
y 2x 5
both equations.
x
y
x
y
{
{
2
4
1
3
0
2
1
}
1
2
1
1
3
0
5
1
7
Y
On a graph, the point where the lines intersect is the solution.
Use the table to draw the graph of each equation.
The lines appear to intersect at 1, 3 .
Substitute 1, 3 into the original equations to check.
yx2
y 2x 5
? 2
? 5
3 (1)
3 2(1) 2 2✓
5 5✓
YX
X
YX
Solve the system using a table and a graph. Give the ordered pair
that solves both equations.
1.
{2xx yy15
Solution:
y x 1
y 2x 5
x
y
x
y
0
1
0
1
2
0
5
3
1
1
1
2
3
1
2
3
2, 1 Y
X
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
a207c03-1_rt.indd 6
6
Holt Algebra 2
12/29/05 7:50:18 PM
Process Black
Name
Date
Class
Reteach
LESSON
3-1
Using Graphs and Tables to Solve Linear Systems
(continued)
To classify a linear system:
Step 1 Write each equation in the form y mx b.
Step 2 Compare the slopes and y-intercepts.
Step 3 Classify by the number of solutions of the system.
Exactly One Solution
Independent
The lines have different
slopes and intersect at
one point.
xy3
xy1
Infinitely Many Solutions
Dependent
No Solution
Inconsistent
The lines have the same
slope and y-intercept. Their
graph is the same line.
The lines have the same slope
and different y-intercepts. The
lines are parallel.
{ 2x4y y8x1 4
3
{ yy 2x1 2x
{
Solve each equation for y. Solve each equation for y.
y 2x 1; m 2, b 1
y x 3; m 1
y 2x 1; m 2, b 1
y x 1; m 1
The slopes and the
The slopes are different.
y-intercepts are the same.
The system has
The system has infinitely many
one solution and is
solutions and is dependent.
independent.
{
{
XY
Y
XY
X
Remember: m slope
and b y-intercept.
Solve each equation for y.
2, b 3
{yy 2x2x 1;3; mm 2,
b1
The slopes are the same but the
y-intercepts are different.
The system has no solution and
is inconsistent.
Y
XY
YX
YX
X
Y
X
YX
Classify each system and determine the number of solutions.
yx2
y 1 3x
3.
2.
y 1 x
2y 6x 2
{
{
y
x 2
1 , b 2
y
3x 1 , m 3
,b
1
y
x 1 , m 1 , b 1
y
3x 1 , m 3
,b
1
,m
Number of solutions:
none
Number of solutions:
inconsistent
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
a207c03-1_rt.indd 7
infinitely many
dependent
7
Holt Algebra 2
12/29/05 7:50:21 PM
Process Black
*À>V̈ViÊ
5SING'RAPHSAND4ABLESTO3OLVE,INEAR3YSTEMS
,%33/.
·£
·£
#LASSIFYEACHSYSTEMANDDETERMINETHENUMBEROFSOLUTIONS
$OESTHEGIVENORDEREDPAIRSOLVETHESYSTEMOFEQUATIONS
3UBSTITUTEEACHVALUEFORXANDYINTOTHEEQUATIONS7RITEYESORNO
OÊ
OÊ
XY
SD XY
SDSD
SDSD
.O
OÊ
XY
SD XY
9ES
XY
SD XY
.O
5SEATABLEANDAGRAPHTOSOLVETHESYSTEM
YX
XY
OÊ
A -AKEATABLEOFVALUESFOREACHEQUATION
YX
Y
X
“>˜ÞÊ܏Ṏœ˜Ã
OÊ
YX
SD XY
XY
XY
˜Vœ˜ÃˆÃÌi˜ÌÆʘœÊ
œ˜ÃˆÃÌi˜Ì]ʈ˜`i«i˜‡
܏Ṏœ˜Ã
`i˜ÌÆʜ˜iÊ܏Ṏœ˜
ÌʈÃÊ̅iÊÊ
܏Ṏœ˜°
OÊ
ÊT£]Ê{ÊEÊ
XY
SD XY
OÊ
YX
X
Y??
OÊ
??
??
œ˜ÃˆÃÌi˜Ì]ÊÊ
`i«i˜`i˜ÌÆʈ˜vˆ˜ˆÌiÞÊ
YX
SD XY
Y
Y
OÊ
OÊ
YX
XY
5SESUBSTITUTIONTODETERMINEIFTHEGIVENORDEREDPAIRISANELEMENT
OFTHESOLUTIONSETFORTHESYSTEMOFEQUATIONS)FITISNOTGIVETHE
CORRECTSOLUTION
XY
X
*À>V̈ViÊ
5SING'RAPHSAND4ABLESTO3OLVE,INEAR3YSTEMS
--"
X
ÊT£]ÊÓÊEÊ
OÊ
XY
SDO Ê XY
XY
SDO Ê
XY
YX
SD XY
3OLVEBYGRAPHINGASYSTEMOFEQUATIONS
!PUPPYPENISFOOTLONGERTHANTWICEITSWIDTH
*OHNWANTSTOINCREASETHELENGTHANDWIDTHBY
FEETEACHTOENLARGETHEAREABYSQUAREFEET7HAT
WILLBETHEAREAOFTHENEWPEN
£ÓÈÊõÕ>ÀiÊviiÌ
5SEAGRAPHTOSOLVEEACHSYSTEM
TE
X
·£
+EESHAHASMOREQUARTERSTHANDIMESWHICH
TOGETHERTOTAL(OWMANYCOINSDOESSHE
HAVEINQUARTERSANDDIMES
Y
Y
X
X
(OLT!LGEBRA
*À>V̈ViÊ
5SING'RAPHSAND4ABLESTO3OLVE,INEAR3YSTEMS
-ATCHEACHSYSTEMOFEQUATIONSWITHTHECORRESPONDINGCONDITIONS
THATITSATISFIES
4HESYSTEMISCONSISTENTANDDEPENDENT
!LINEARSYSTEMOFEQUATIONSISASETOFTWOORMORELINEAREQUATIONS4OSOLVEALINEAR
SYSTEMFINDALLTHEORDEREDPAIRSXYTHATMAKEBOTHEQUATIONSTRUE5SEATABLEANDA
GRAPHTOSOLVEASYSTEMOFEQUATIONS
YX
YX
3OLVEEACHEQUATIONFORY→ YX
YX
-AKEATABLEOFVALUESFOREACHEQUATION
7HENXYFOR
YX
YX
BOTHEQUATIONS
X
Y
X
Y
OÊ
OÊ
OÊ
Y??
X
YX
4HESYSTEMISCONSISTENTANDINDEPENDENT
Y??
X
YX
OÊ
3OLVE
!TUBCONTAININGGALLONSOFWATERISDRAININGATARATEOFGALLONPER
HOUR!BASINOFGALLONSOFWATERISDRAININGATARATEOFGALLON
EVERYHOURS
Z
OÊ
YX
4HELINESAPPEARTOINTERSECTATSD
3UBSTITUTESDINTOTHEORIGINALEQUATIONSTOCHECK
YX
YX
B )FBOTHCONTAINERSBEGANDRAININGATTHESAMETIMEHOWSOONWILLTHETUBANDBASIN
HOLDTHESAMEAMOUNTOFWATER
Y
5SETHETABLETODRAWTHEGRAPHOFEACHEQUATION
YÊÊXÊÊ£È
ÊÊÊÊÊÊ
ÊÊ ÊÊ
ÊÊ
YÊÊÊÚÚ
Ê£ÈÊXÊΰx
X
YX
3OLVETHESYSTEMUSINGATABLEANDAGRAPH'IVETHEORDEREDPAIR
THATSOLVESBOTHEQUATIONS
OÊ
XY
XY
£xʅ
YX
X
Y
Ê Ê£
C 7HENTHEAMOUNTSAREEQUALHOWMUCHWATERWILLBEINEACHCONTAINER
£Ê}>œ˜
*ENNAHASINASAVINGSACCOUNT3HEADDSTOHERACCOUNT
EACHMONTH,UISHASINHISSAVINGSACCOUNT(EWITHDRAWS
FROMHISACCOUNTEACHMONTH
B 7HATWILLBETHEBALANCEINEACHACCOUNT
OÊ
/NAGRAPHTHEPOINTWHERETHELINESINTERSECTISTHESOLUTION
A 7RITEASYSTEMOFEQUATIONSTHATREPRESENTSYTHENUMBEROFGALLONSLEFTINTHE
CONTAINERAFTERXHOURS
A )NHOWMANYMONTHSWILLTHEYHAVETHESAME
BALANCEINTHEIRSAVINGSACCOUNTS
œÌʏ}iLÀ>ÊÓ
,iÌi>V…
5SING'RAPHSAND4ABLESTO3OLVE,INEAR3YSTEMS
·£
Y??
X
YX
#OPYRIGHT©BY(OLT2INEHARTAND7INSTON
!LLRIGHTSRESERVED
--"
4HESYSTEMISINCONSISTENT
ÎxʵÕ>ÀÌiÀÃÊÊÓxÊ`ˆ“iÃÊÊÊ
ÈäÊVœˆ˜Ã
#OPYRIGHT©BY(OLT2INEHARTAND7INSTON
!LLRIGHTSRESERVED
--"
TE
X
OÊ
YX
XY
Y
Y
TE
C 7HICHORDEREDPAIRSOLVESBOTHEQUATIONS
OÊ
ÊTx]ÊxÊEÊ
ÌʈÃÊ̅iÊÊ
܏Ṏœ˜°
B 5SETHEVALUESINTHETABLESTOGRAPHEACHEQUATION
XY
XY
ÊTÈ]ÊÓÊEÊ
Ê Êä
£
Ó
3OLUTION
YX
X
Y
x
Î
£
Ê Ê£
ÊTÓ]Ê£ÊEÊ
Y
X
£xʓœ˜Ì…Ã
f£™xä
#OPYRIGHT©BY(OLT2INEHARTAND7INSTON
!LLRIGHTSRESERVED
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
AK4up.indd 51
œÌʏ}iLÀ>ÊÓ
#OPYRIGHT©BY(OLT2INEHARTAND7INSTON
!LLRIGHTSRESERVED
51
œÌʏ}iLÀ>ÊÓ
Holt Algebra 2
4/25/06 3:06:20 PM
,iÌi>V…
5SING'RAPHSAND4ABLESTO3OLVE,INEAR3YSTEMS
CONTINUED
--"
·£
4OCLASSIFYALINEARSYSTEM
3TEP 7RITEEACHEQUATIONINTHEFORMYMXB
3TEP #OMPARETHESLOPESANDYINTERCEPTS
3TEP #LASSIFYBYTHENUMBEROFSOLUTIONSOFTHESYSTEM
%XACTLY/NE3OLUTION
)NDEPENDENT
OÊ
·£
OÊ
XY
XBYC
.O3OLUTION
)NCONSISTENT
4HELINESHAVETHESAME
4HELINESHAVETHESAMESLOPE
SLOPEANDYINTERCEPT4HEIR ANDDIFFERENTYINTERCEPTS4HE
GRAPHISTHESAMELINE
LINESAREPARALLEL
XY
OÊ
YX
…>i˜}i
3YSTEM#LASSIFICATIONS
!SYSTEMOFLINEAREQUATIONSCANBECONSISTENTANDINDEPENDENT
CONSISTENTANDDEPENDENTORINCONSISTENT5SETHEFOLLOWINGSYSTEMOF
EQUATIONSFOR%XERCISESn
2EMEMBERMSLOPE
ANDBYINTERCEPT
)NFINITELY-ANY3OLUTIONS
$EPENDENT
4HELINESHAVEDIFFERENT
SLOPESANDINTERSECTAT
ONEPOINT
XY
XY
--"
OÊ
YX
YX
&INDVALUESOFBANDCSOTHATTHESYSTEM
ISCONSISTENTANDDEPENDENT
LÊʙ]ÊVÊÊÎÈ
&INDVALUESOFBANDCSOTHATTHESYSTEM
ISINCONSISTENT
LÊʙ]ÊVÊqÊÎÈ
&INDVALUESOFBANDCSOTHATTHESYSTEM
ISCONSISTENTANDINDEPENDENT
LÊqʙ
5SETHISSYSTEMOFEQUATIONSFOR%XERCISES
3OLVEEACHEQUATIONFORY 3OLVEEACHEQUATIONFORY
3OLVEEACHEQUATIONFORY
XBYC
XMB
Y
A
XMB
YXM
Y Y X
XB
YC
A
MB
YXMB
YXM
XBYC
A
4HESLOPESANDTHE
4HESLOPESAREDIFFERENT
4HESLOPESARETHESAMEBUTTHE
$ESCRIBEHOWTHEGRAPHOFTHESEEQUATIONSLOOKSWHENTHESYSTEM
YINTERCEPTSARETHESAME
YINTERCEPTSAREDIFFERENT
HASEXACTLYONESOLUTION
4HESYSTEMHAS
4HESYSTEMHASINFINITELYMANY 4HESYSTEMHASNOSOLUTIONAND
ONESOLUTIONANDIS
SOLUTIONSANDISDEPENDENT
Îʏˆ˜iÃʈ˜ÌiÀÃiVÌÊ>ÌÊ>ÊȘ}iÊ«œˆ˜Ì
ISINCONSISTENT
INDEPENDENT
OÊ
OÊ
XY
Y
Y
XY
OÊ
OÊ
XY
YX
X
YX
X
%XPLAINWHYTHESYSTEMOFEQUATIONSMUSTBECONSISTENTIFTHECONSTANT
TERMSCCANDCAREALLZERO
YX
OÊ
Y ÝÊÓ M £ B
Y
Ó
ÝÊ£ M £ B £ ˜œ˜i
.UMBEROFSOLUTIONS
Y
Y
ÎÝÊÊ£ M Î
.UMBEROFSOLUTIONS
--"
·£
OÊ
5NDERWHATCONDITIONSWILLTHISSYSTEMBECONSISTENTANDINDEPENDENT
B £
*œÃÈLiÊ>˜ÃÜiÀ\Ê/œÊLiʈ˜`i«i˜`i˜Ì]Ê̅iÊϜ«iÃÊV>˜˜œÌÊLiÊiµÕ>°Ê/…ˆÃÊV>˜Ê
VÊ Ê]Ê Ê>`ÊqÊLV]ʜÀÊ>`ÊÊLVÊqÊä°
LiÊÃÌ>Ìi`Ê>ÃÊÊÚÚÚ
Ê>Ê ÊÊqÊÊÊÚÚÚ
L
`
ˆ˜vˆ˜ˆÌiÞʓ>˜Þ
`i«i˜`i˜Ì
#OPYRIGHT©BY(OLT2INEHARTAND7INSTON
!LLRIGHTSRESERVED
5SETHISSYSTEMOFEQUATIONSFOR%XERCISE
AXBYE
CXDYF
B £
ÎÝÊÊ£ M Î
ˆ˜Vœ˜ÃˆÃÌi˜Ì
*œÃÈLiÊ>˜ÃÜiÀ\ʘʜÀ`iÀÊ̜ÊLiʈ˜Vœ˜ÃˆÃÌi˜Ì]Ê̅iÊiµÕ>̈œ˜ÃʓÕÃÌÊLiÊ
«>À>iÊ>˜`ʅ>ÛiÊ`ˆvviÀi˜ÌÊއˆ˜ÌiÀVi«ÌðÊvÊ̅iÊVœ˜ÃÌ>˜ÌÊÌiÀ“ÃÊ>ÀiÊiµÕ>Ê̜Ê
ä]Ê̅i˜Ê>Êއˆ˜ÌiÀVi«ÌÃÊ>ÀiÊä]Ê̅iÀivœÀiÊ̅iÀiÊ>ÀiʘœÊ«>À>iÊˆ˜iÃÊ>˜`Ê̅iÊ
ÃÞÃÌi“Ê“ÕÃÌÊLiÊVœ˜ÃˆÃÌi˜Ì°
#LASSIFYEACHSYSTEMANDDETERMINETHENUMBEROFSOLUTIONS
YX
YX
YX
YX
OÊ
œÌʏ}iLÀ>ÊÓ
*ÀœLi“Ê-œÛˆ˜}
5SING'RAPHSAND4ABLESTO3OLVE,INEAR3YSTEMS
--"
·£
{ÝÊÊnÞÊÊÓÈ
X
OÊ
Ê{ÝÊÊnÞÊÓÈ
Ê ÊÊÊÊÊ
Ê
Ê Ê
ÊÝÊ£ÊÊÞ
B 7RITEONEEQUATIONATTHETOPOFEACH
TABLEANDCOMPLETETHETABLE
C 7HATISTHECOSTOFEACHSIZEDRINK
,i>`ˆ˜}Ê-ÌÀ>Ìi}ˆiÃ
5SE'RAPHIC!IDS
#ONSISTENT
ÝÊÊ£ÊÊÞ
Y
X
Ó°Çx
Ó°xä
Ó°Óx
Ó
£°Çx
Ó
Ó°x
Î
ΰx
{
$EPENDENT
OÊ
ÊÞÊʙxÊÊ£xÝ
ÊÊ
Ê ÊÊÊÊÊÊ
Ê ÊÊÊ
Ê
Ê
ÞÊnäÊÊÓäÝ
C &ORWHATNUMBEROFHOURSWOULD3ANDY
AND0AULINEPAYTHESAMETORENTAJETSKI
Îʅ
XY
YX
YX
X
"OTHLINESTHESAME
)NFINITELYMANY
SOLUTIONS
XY
X
XY
,INESINTERSECT
/NESOLUTION
,INESPARALLEL
.OSOLUTIONS
2INGALLTHETERMSTHATDESCRIBETHESYSTEM
YX
YX
OÊ
X
4IMEH
f£{ä
OÊ
OÊ
XY
XY
XY
YX
A #ONSISTENT
A #ONSISTENT
A #ONSISTENT
B )NCONSISTENT
C $EPENDENT
D )NDEPENDENT
B )NCONSISTENT
C $EPENDENT
D )NDEPENDENT
B )NCONSISTENT
C $EPENDENT
D )NDEPENDENT
$RAWALINEFROMEACHSYSTEMOFEQUATIONSTOTHEMATCHINGGRAPH
OÊ
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
œÌʏ}iLÀ>ÊÓ
Y
X
{
X
È
#OPYRIGHT©BY(OLT2INEHARTAND7INSTON
!LLRIGHTSRESERVED
52
XY
XY
Y
x
OÊ
YX
YX
Y
OÊ
OÊ
YX
YX
" O Ê
$ O Ê
YX
YX
OÊ
XY
YX
*UANSTARTEDWITHGALLONSOFWATERINHISPOOLANDHEISFILLINGITATARATE
OFGALLONSPERMINUTE(ISNEXTDOORNEIGHBOR3AMSTARTEDWITHGALLONS
OFWATERINHISPOOLANDHEISFILLINGITATARATEOFGALLONSPERMINUTE
7HICHSYSTEMOFEQUATIONSCOULDYOUUSETOFINDWHENTHEPOOLSWILLCONTAIN
THESAMEAMOUNTOFWATER
YX
YX
! # YX
YX
AK4up.indd 52
#HOOSETHELETTERFORTHEBESTANSWER
#OPYRIGHT©BY(OLT2INEHARTAND7INSTON
!LLRIGHTSRESERVED
D (OWMUCHWOULDITCOSTTORENTTHEJETSKIFOR
THISAMOUNTOFTIME
Y
X
XY
Y
Y
B 'RAPHTHESYSTEM
2ENTAL#OST
)NDEPENDENT
Y
3ANDYRENTEDAJETSKIFORPLUSPERHOUR0AULINERENTEDAJETSKI
FORPLUSPERHOUR
)NCONSISTENT
Y
-“>\Êf£°xäÆʏ>À}i\ÊfÓ°xä
A 7RITEALINEARSYSTEMOFEQUATIONSTOFIND
THENUMBEROFHOURSFORWHICHTHERENTAL
COSTISTHESAME
œÌʏ}iLÀ>ÊÓ
!SYSTEMOFLINEAREQUATIONSCANBECLASSIFIED
,INEAR%QUATIONS
!FTERTHELESSON#ARLTAKESTHEWAKEBOARDINGCLASSTOTHE'LASS#AFE
(EPAYSFORLARGEANDSMALLJUICEDRINKS!LARGEGLASSCOSTS
MORETHANASMALLGLASS
A 7RITEALINEARSYSTEMOFEQUATIONSTO
FINDTHECOSTOFEACHSIZEDRINK
#OPYRIGHT©BY(OLT2INEHARTAND7INSTON
!LLRIGHTSRESERVED
Solve.
ÎÊ«>À>iÊˆ˜iÃ
X
Îʏˆ˜iÃÊ̅>ÌÊVœˆ˜Vˆ`i
$ESCRIBEHOWTHEGRAPHOFTHESEEQUATIONS
LOOKSWHENTHESYSTEMHASNOSOLUTION
$ESCRIBEHOWTHEGRAPHOFTHESEEQUATIONSLOOKSWHEN
THESYSTEMHASANINFINITENUMBEROFSOLUTIONS
Y
X
œÌʏ}iLÀ>ÊÓ
Holt Algebra 2
12/29/05 7:52:46 PM