answers (lesson 3-1) · a system of equations is a set of two or more equations containing the same...
TRANSCRIPT
Chapter 3 A1 Glencoe Algebra 2
An
swer
s
STEP
1
Cha
pter
33
Gle
ncoe
Alg
ebra
2
NA
ME
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__D
ATE
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ER
IOD
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3
Copyright ©Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Ant
icip
atio
n G
uide
Sys
tem
s o
f E
qu
atio
ns
and
Ineq
ual
itie
s
Chapter Resources
Bef
ore
you
beg
in C
ha
pte
r 3
•R
ead
each
sta
tem
ent.
•D
ecid
e w
het
her
you
Agr
ee (
A)
or D
isag
ree
(D)
wit
h t
he
stat
emen
t.
•W
rite
A o
r D
in
th
e fi
rst
colu
mn
OR
if
you
are
not
su
re w
het
her
you
agr
ee o
r di
sagr
ee,
wri
te N
S (
Not
Su
re).
Aft
er y
ou c
omp
lete
Ch
ap
ter
3
•R
erea
d ea
ch s
tate
men
t an
d co
mpl
ete
the
last
col
um
n b
y en
teri
ng
an A
or
a D
.
•D
id a
ny
of y
our
opin
ion
s ab
out
the
stat
emen
ts c
han
ge f
rom
th
e fi
rst
colu
mn
?
•F
or t
hos
e st
atem
ents
th
at y
ou m
ark
wit
h a
D,u
se a
pie
ce o
f pa
per
to w
rite
an
exa
mpl
e of
wh
y yo
u d
isag
ree.
STEP
2
ST
EP
1S
TE
P 2
A,D
,or
NS
Sta
tem
ent
A o
r D
1.A
sys
tem
of
equ
atio
ns
con
sist
s of
tw
o or
mor
e eq
uat
ion
s w
ith
di
ffer
ent
vari
able
s.D
2.T
he
solu
tion
of
a sy
stem
of
equ
atio
ns
can
be
fou
nd
by f
indi
ng
the
inte
rsec
tion
of
the
grap
hs
of t
he
equ
atio
ns.
A
3.A
sys
tem
of
equ
atio
ns
that
is
inco
nsi
sten
t h
as a
n i
nfi
nit
e n
um
ber
of s
olu
tion
s.D
4.G
iven
th
e tw
o eq
uat
ion
s y
�x
�7
and
3x�
4y�
10,a
so
luti
on c
an b
e fo
un
d by
su
bsti
tuti
ng
x�
7 fo
r y
in t
he
Ase
con
d eq
uat
ion
.
5.T
he
prod
uct
of
the
equ
atio
n 6
m�
4n�
22 a
nd
�2
is �
12m
�8n
�22
.D
6.W
hen
sol
vin
g a
syst
em o
f in
equ
alit
ies
by g
raph
ing,
if t
he
grap
hs
do n
ot i
nte
rsec
t th
en t
her
e is
no
solu
tion
.A
7.A
ll t
he
orde
red
pair
s in
th
e in
ters
ecti
on o
f th
e gr
aph
s of
a
syst
em o
f in
equ
alit
ies
are
call
ed c
onst
rain
ts.
D
8.If
th
e in
ters
ecti
on o
f th
e gr
aph
s of
a s
yste
m o
f in
equ
alit
ies
is
a po
lygo
nal
reg
ion
,th
at r
egio
n i
s ca
lled
bou
nde
d.A
9.Li
nea
r pr
ogra
mm
ing
is t
he
proc
ess
of f
indi
ng
all
solu
tion
s to
a
syst
em o
f li
nea
r in
equ
alit
ies.
D
10.
Th
e so
luti
on t
o a
syst
em o
f eq
uat
ion
s w
ith
th
ree
vari
able
s is
w
ritt
en a
s (x
,y,z
) an
d is
cal
led
an o
rder
ed t
ripl
e.A
Lesson 3-1
Cha
pter
35
Gle
ncoe
Alg
ebra
2
NA
ME
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____
____
__D
ATE
____
____
____
__P
ER
IOD
____
_
3-1
Copyright ©Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Less
on R
eadi
ng G
uide
So
lvin
g S
yste
ms
of
Eq
uat
ion
s by
Gra
ph
ing
Get
Rea
dy
for
the
Less
on
Rea
d t
he
intr
odu
ctio
n t
o L
esso
n 3
-1 i
n y
our
text
boo
k.
•W
hic
h a
re g
row
ing
fast
er,i
n-s
tore
sal
es o
r on
lin
e sa
les?
on
line
sale
s
•In
wh
at y
ear
is t
he
in-s
tore
an
d on
lin
e sa
les
the
sam
e?20
05
Rea
d t
he
Less
on
1.T
he
Stu
dy T
ip o
n p
age
117
of y
our
text
book
say
s th
at w
hen
you
sol
ve a
sys
tem
of
equ
atio
ns
by g
raph
ing
and
fin
d a
poin
t of
in
ters
ecti
on o
f th
e tw
o li
nes
,you
mu
st a
lway
sch
eck
the
orde
red
pair
in
bot
hof
th
e or
igin
al e
quat
ion
s.W
hy
is i
t n
ot g
ood
enou
gh t
och
eck
the
orde
red
pair
in
just
on
e of
th
e eq
uat
ion
s?S
amp
le a
nsw
er:T
o b
e a
solu
tio
n o
f th
e sy
stem
,th
e o
rder
ed p
air
mu
stm
ake
bo
th o
f th
e eq
uat
ion
s tr
ue.
2.U
nde
r ea
ch s
yste
m g
raph
ed b
elow
,wri
te a
ll o
f th
e fo
llow
ing
wor
ds t
hat
app
ly:c
onsi
sten
t,in
con
sist
ent,
dep
end
ent,
and
ind
epen
den
t.
inco
nsi
sten
tco
nsi
sten
t;co
nsi
sten
t;d
epen
den
tin
dep
end
ent
Rem
emb
er W
hat
Yo
u L
earn
ed
3.L
ook
up
the
wor
ds c
onsi
sten
tan
d in
con
sist
ent
in a
dic
tion
ary.
How
can
th
e m
ean
ing
ofth
ese
wor
ds h
elp
you
dis
tin
guis
h b
etw
een
con
sist
ent
and
inco
nsi
sten
t sy
stem
s of
equ
atio
ns?
Sam
ple
an
swer
:O
ne
mea
nin
g o
f co
nsi
sten
tis
“b
ein
g in
ag
reem
ent,”
or
“co
mp
atib
le,”
wh
ile o
ne
mea
nin
g o
f in
con
sist
ent
is “
no
t b
ein
g in
agre
emen
t”o
r “i
nco
mp
atib
le.”
Wh
en a
sys
tem
is c
on
sist
ent,
the
equ
atio
ns
are
com
pat
ible
bec
ause
bo
th c
an b
e tr
ue
at t
he
sam
e ti
me
(fo
rth
e sa
me
valu
es o
f x
and
y).
Wh
en a
sys
tem
is in
con
sist
ent,
the
equ
atio
ns
are
inco
mp
atib
le b
ecau
se t
hey
can
nev
er b
e tr
ue
at t
he
sam
e ti
me.
x
y
Ox
y
Ox
y
O
Answers (Lesson 3-1)
A1-A19 A2-03-873973 5/22/06 8:25 AM Page A1
Chapter 3 A2 Glencoe Algebra 2
Cha
pter
36
Gle
ncoe
Alg
ebra
2
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
__P
ER
IOD
____
_
3-1
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Stud
y G
uide
and
Inte
rven
tion
So
lvin
g S
yste
ms
of
Eq
uat
ion
s by
Gra
ph
ing
Gra
ph
Sys
tem
s o
f Eq
uat
ion
sA
sys
tem
of
equ
atio
ns
is a
set
of
two
or m
ore
equ
atio
ns
con
tain
ing
the
sam
e va
riab
les.
You
can
sol
ve a
sys
tem
of
lin
ear
equ
atio
ns
bygr
aph
ing
the
equ
atio
ns
on t
he
sam
e co
ordi
nat
e pl
ane.
If t
he
lin
es i
nte
rsec
t,th
e so
luti
on i
sth
at i
nte
rsec
tion
poi
nt.
Sol
ve t
he
syst
em o
f eq
uat
ion
s b
y gr
aph
ing.
x �
2y�
4x
�y
��
2W
rite
eac
h e
quat
ion
in
slo
pe-i
nte
rcep
t fo
rm.
x �
2y�
4→
y�
�2
x�
y�
�2
→y
��
x�
2
Th
e gr
aph
s ap
pear
to
inte
rsec
t at
(0,
�2)
.
CH
ECK
Su
bsti
tute
th
e co
ordi
nat
es i
nto
eac
h e
quat
ion
.x
�2y
�4
x�
y�
�2
0 �
2(�
2) �
40
�(�
2)�
�2
4 �
4✓
�2
��
2✓
Th
e so
luti
on o
f th
e sy
stem
is
(0,�
2).
Sol
ve e
ach
sys
tem
of
equ
atio
ns
by
grap
hin
g.
1.y
��
�1
2.y
�2x
�2
3.y
��
�3
y�
�4
(6,�
1)y
��
x�
4(2
,2)
y�
(4,1
)
4.3x
�y
�0
5.2x
��
�7
6.�
y�
2
x�
y�
�2
(1,3
)�
y�
1(�
4,3)
2x�
y�
�1
(�2,
�3)
( –2,
–3)
x
y
O
( –4,
3)
x
y
O
( 1, 3
)
x
y
O
x � 2
x � 2y � 3
( 4, 1
)
x
y
O
( 2, 2
)
x
y
O
( 6, –
1)
x
y
O
x � 4x � 2
x � 2x � 3
x
y
O
( 0, –
2)
x � 2
Exam
ple
Exer
cise
s
Lesson 3-1
Cha
pter
37
Gle
ncoe
Alg
ebra
2
NA
ME
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____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
__P
ER
IOD
____
_
3-1
Copyright ©Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Cla
ssif
y Sy
stem
s o
f Eq
uat
ion
sT
he
foll
owin
g ch
art
sum
mar
izes
th
e po
ssib
ilit
ies
for
grap
hs
of t
wo
lin
ear
equ
atio
ns
in t
wo
vari
able
s.
Gra
ph
s o
f E
qu
atio
ns
Slo
pes
of
Lin
esC
lass
ific
atio
n o
f S
yste
mN
um
ber
of
So
luti
on
s
Line
s in
ters
ect
Diff
eren
t sl
opes
Con
sist
ent
and
inde
pend
ent
One
Line
s co
inci
de (
sam
e lin
e)S
ame
slop
e, s
ame
Con
sist
ent
and
depe
nden
tIn
finite
ly m
any
y-in
terc
ept
Line
s ar
e pa
ralle
lS
ame
slop
e, d
iffer
ent
Inco
nsis
tent
Non
e y-
inte
rcep
ts
Gra
ph
th
e sy
stem
of
equ
atio
ns
x�
3y�
6an
d d
escr
ibe
it a
s co
nsi
sten
t a
nd
in
dep
end
ent,
2x�
y�
�3
con
sist
ent
an
d d
epen
den
t,or
in
con
sist
ent.
Wri
te e
ach
equ
atio
n i
n s
lope
-in
terc
ept
form
.
x�
3y�
6→
y�
x�
2
2x�
y�
�3
→y
�2x
�3
Th
e gr
aph
s in
ters
ect
at (
�3,
�3)
.Sin
ce t
her
e is
on
e so
luti
on,t
he
syst
em i
s co
nsi
sten
t an
d in
depe
nde
nt.
Gra
ph
th
e sy
stem
of
equ
atio
ns
and
des
crib
e it
as
con
sist
ent
an
d i
nd
epen
den
t,co
nsi
sten
t a
nd
dep
end
ent,
or i
nco
nsi
sten
t.
1.3x
�y
��
2 2.
x�
2y�
5 3.
2x�
3y�
06x
�2y
�10
3x�
15 �
�6y
co
nsi
sten
t4x
�6y
�3
inco
nsi
sten
tan
d d
epen
den
tin
con
sist
ent
4.2x
�y
�3
5.4x
�y
��
26.
3x�
y�
2
x�
2y�
4co
nsi
sten
t 2x
��
�1
con
sist
ent
x�
y�
6 co
nsi
sten
t
and
ind
epen
den
t an
d d
epen
den
t an
d in
dep
end
ent
x
y
O
x
y
Ox
y
O
y � 2
x
y
Ox
y
Ox
y
O
1 � 3
x
y O
( –3,
–3)
Stud
y G
uide
and
Inte
rven
tion
(co
ntin
ued)
So
lvin
g S
yste
ms
of
Eq
uat
ion
s by
Gra
ph
ing
Exam
ple
Exer
cise
s
Answers (Lesson 3-1)
A1-A19 A2-03-873973 5/22/06 8:25 AM Page A2
Chapter 3 A3 Glencoe Algebra 2
An
swer
s
Cha
pter
38
Gle
ncoe
Alg
ebra
2
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
__P
ER
IOD
____
_
3-1
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Skill
s Pr
acti
ceS
olv
ing
Sys
tem
s o
f E
qu
atio
ns
By
Gra
ph
ing
Sol
ve e
ach
sys
tem
of
equ
atio
ns
by
grap
hin
g.
1.x
�2
2.y
��
3x�
63.
y�
4 �
3x
y�
0(2
,0)
y�
2x�
4(2
,0)
y�
�x
�1
(2,�
2)
4.y
�4
�x
5.y
��
2x�
26.
y�
x
y�
x�
2(3
,1)
y�
x�
5(3
,�4)
y�
�3x
�4
(1,1
)
7.x
�y
�3
8.x
�y
�4
9.3x
�2y
�4
x�
y�
1(2
,1)
2x�
5y�
8(4
,0)
2x�
y�
1(�
2,�
5)
Gra
ph
eac
h s
yste
m o
f eq
uat
ion
s an
d d
escr
ibe
it a
s co
nsi
sten
t a
nd
ind
epen
den
t,co
nsi
sten
t a
nd
dep
end
ent,
or i
nco
nsi
sten
t.
10.y
��
3x11
.y�
x�
512
.2x
�5y
�10
y�
�3x
�2
�2x
�2y
��
103x
�y
�15
inco
nsi
sten
tco
nsi
sten
t an
d
con
sist
ent
and
d
epen
den
tin
dep
end
ent
( 5, 0
)
2
2x
y
O
x
y
O
x
y
O
( –2,
–5)
x
y
O( 4
, 0)
x
y
O( 2
, 1)
x
y
O
( 1, 1
)
x
y
O( 3
, –4)
x
y
O
( 3, 1
)
x
y
O
1 � 3
( 2, –
2)x
y
O( 2
, 0)
x
y
O( 2
, 0)
x
y
O
1 � 2
3-1
Lesson 3-1
Cha
pter
39
Gle
ncoe
Alg
ebra
2
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
__P
ER
IOD
____
_
3-1
Copyright ©Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Sol
ve e
ach
sys
tem
of
equ
atio
ns
by
grap
hin
g.
1.x
�2y
�0
2.x
�2y
�4
3.2x
�y
�3
y�
2x�
3(2
,1)
2x�
3y�
1(2
,1)
y�
x�
(3,�
3)
4.y
�x
�3
5.2x
�y
�6
6.5x
�y
�4
y�
1(�
2,1)
x�
2y�
�2
(2,�
2)�
2x�
6y�
4(1
,1)
Gra
ph
eac
h s
yste
m o
f eq
uat
ion
s an
d d
escr
ibe
it a
s co
nsi
sten
t a
nd
ind
epen
den
t,co
nsi
sten
ta
nd
dep
end
ent,
or i
nco
nsi
sten
t.
7.2x
�y
�4
8.y
��
x�
29.
2y�
8 �
x
x�
y�
2x
�y
��
4y
�x
�4
con
sist
ent
and
ind
ep.
inco
nsi
sten
tco
nsi
sten
t an
d d
ep.
SOFT
WA
RE
For
Exe
rcis
es 1
0–12
,use
th
e fo
llow
ing
info
rmat
ion
.L
ocat
ion
Map
pin
g n
eeds
new
sof
twar
e.S
oftw
are
A c
osts
$13
,000
plu
s $5
00 p
er a
ddit
ion
alsi
te l
icen
se.S
oftw
are
B c
osts
$25
00 p
lus
$120
0 pe
r ad
diti
onal
sit
e li
cen
se.
10.W
rite
tw
o eq
uat
ion
s th
at r
epre
sen
t th
e co
st o
f ea
ch s
oftw
are.
A:
y�
13,0
00 �
500x
,B
:y
�25
00 �
1200
x
11.G
raph
th
e eq
uat
ion
s.E
stim
ate
the
brea
k-ev
en
poin
t of
th
e so
ftw
are
cost
s.15
ad
dit
ion
al li
cen
ses
12.I
f L
ocat
ion
Map
pin
g pl
ans
to b
uy
10 a
ddit
ion
al
site
lic
ense
s,w
hic
h s
oftw
are
wil
l co
st l
ess?
B
( 15,
20,
500)
A
B
So
ftw
are
Co
sts
Ad
dit
ion
al L
icen
ses
Total Cost ($)
40
812
1416
1820
26
10
24,0
00
20,0
00
16,0
00
12,0
00
8,00
0
4,00
0
x
y
O
x
y
O( 2
, 0)
x
y
O
1 � 2
y
xO
( 1, 1
)x
y O( 2
, –2)
x
y
O
( –2,
1)
( 3, –
3)
x
y
O( 2
, 1) x
y
O
( 2, 1
)
x
y
O
9 � 21 � 2
Prac
tice
So
lvin
g S
yste
ms
of
Eq
uat
ion
s B
y G
rap
hin
g
Answers (Lesson 3-1)
A1-A19 A2-03-873973 5/22/06 8:25 AM Page A3
Chapter 3 A4 Glencoe Algebra 2
Cha
pter
310
Gle
ncoe
Alg
ebra
2
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
__P
ER
IOD
____
_
3-1
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Wor
d Pr
oble
m P
ract
ice
So
lvin
g S
yste
ms
of
Eq
uat
ion
s B
y G
rap
hin
g1.
STR
EETS
An
drew
is
stu
dyin
g a
map
an
dn
otic
es t
wo
stre
ets
that
ru
n p
aral
lel
toea
ch o
ther
.He
com
pute
s th
e eq
uat
ion
sof
th
e li
nes
th
at r
epre
sen
t th
e tw
o ro
ads.
Are
th
ese
two
equ
atio
ns
con
sist
ent
orin
con
sist
ent?
If
they
are
con
sist
ent,
are
they
in
dep
end
ent
or d
epen
den
t? E
xpla
in.
Th
ey a
re in
con
sist
ent.
Par
alle
llin
es n
ever
inte
rsec
t so
th
ere
isn
o s
olu
tio
n.
2.SP
OTL
IGH
TSS
hip
A h
as c
oord
inat
es(�
1,�
2) a
nd
Sh
ip B
has
coo
rdin
ates
(�
4,1)
.Bot
h s
hip
s h
ave
thei
r sp
otli
ghts
fixa
ted
on t
he
sam
e li
febo
at.T
he
ligh
tbe
am f
rom
Sh
ip A
tra
vels
alo
ng
the
lin
ey
�2x
.Th
e li
ght
beam
fro
m S
hip
Btr
avel
s al
ong
the
lin
e y
�x
�5.
Wh
atar
e th
e co
ordi
nat
es o
f th
e li
febo
at?
(5,1
0)
3.LA
SER
SA
mac
hin
e h
eats
up
a si
ngl
epo
int
by s
hin
ing
seve
ral
lase
rs a
t it
.Th
eeq
uat
ion
s y
�x
�1
and
y�
�x
�7
desc
ribe
tw
o of
th
e la
ser
beam
s.G
raph
both
of
thes
e li
nes
to
fin
d th
eco
ordi
nat
es o
f th
e h
eate
d po
int
(3,4
)
4.PA
THS
Th
e gr
aph
sh
ows
the
path
s of
two
peop
le w
ho
took
a w
alk
in a
par
k.W
her
e di
d th
eir
path
s in
ters
ect?
(6,6
)
PHO
NE
SER
VIC
EF
or E
xerc
ises
5–7
,use
the
foll
owin
g in
form
atio
n.
Bet
h i
s de
cidi
ng
betw
een
tw
o te
leph
one
plan
s.P
lan
A c
har
ges
$15
per
mon
th p
lus
10 c
ents
per
min
ute
.Pla
n B
ch
arge
s $2
0 pe
rm
onth
plu
s 5
cen
ts p
er m
inu
te.
5.W
rite
a s
yste
m o
f eq
uat
ion
s th
atre
pres
ent
the
mon
thly
cos
t of
eac
h p
lan
.y
�0.
1x+
15y
�0.
05x
+ 20
6.G
raph
th
e eq
uat
ion
s.
7.F
or h
ow m
any
min
ute
s pe
r m
onth
do
the
two
phon
e pl
ans
cost
th
e sa
me
amou
nt?
100
min
ute
s
y
xO5
10
y
xO
5
5
y
xO5
5
y
xO
5
5�
5
Ship
B
Ship
A
Exer
cise
s
Exam
ple
s
3-1
Lesson 3-1
Cha
pter
311
Gle
ncoe
Alg
ebra
2
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
__P
ER
IOD
____
_
3-1
Copyright ©Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Enri
chm
ent
So
luti
on
s to
No
nlin
ear
Eq
uat
ion
s R
eal-
life
sit
uat
ion
s ar
e of
ten
not
cap
able
of
bein
g re
pres
ente
d by
a l
inea
r eq
uat
ion
.S
yste
ms
of n
onli
nea
r eq
uat
ion
s ar
e of
ten
use
d in
th
e st
udy
of
popu
lati
on d
ynam
ics,
mod
elin
g ca
rbon
mon
oxid
e ex
posu
re,a
nd
dete
rmin
ing
the
hei
ght
of a
n o
bjec
t in
fre
e fa
ll.
Non
lin
ear
equ
atio
ns
hav
e on
e va
riab
le r
aise
d to
a p
ower
oth
er t
han
on
e or
mu
ltip
lica
tion
of t
wo
or m
ore
vari
able
s.
a.xy
�1
or y
��1 x�
The
firs
t eq
uatio
n ha
s a
prod
uct
of t
wo
varia
bles
.
The
sec
ond
is t
he s
ame
equa
tion
solv
ed fo
r y.
b.y
�x3
� 2
x�
1T
he v
aria
ble
xis
rai
sed
to t
he t
hird
pow
er.
Sys
tem
s of
non
lin
ear
equ
atio
ns
con
sist
of
two
or m
ore
equ
atio
ns,
wh
ere
at l
east
on
e is
non
lin
ear.
Sol
uti
ons
to t
hes
e sy
stem
s ar
e ty
pica
lly
diff
icu
lt t
o fi
nd.
On
e u
sefu
l m
eth
od f
or f
indi
ng
solu
tion
s to
sys
tem
s of
non
lin
ear
equ
atio
ns
is t
he
sam
e as
th
e m
eth
od f
or f
indi
ng
solu
tion
s to
syst
ems
of l
inea
r eq
uat
ion
s—u
se t
ech
nol
ogy
to g
raph
th
esy
stem
an
d fi
nd
the
poin
t(s)
of
inte
rsec
tion
.Th
e gr
aph
of
the
syst
em i
s sh
own
at
the
righ
t.
y�
�1 x�
y �
x3�
2x
� 1
Usi
ng
the
Inte
rsec
tion
fu
nct
ion
of
a gr
aph
ing
calc
ula
tor
you
fin
d th
at o
ne
poin
t of
inte
rsec
tion
is
appr
oxim
atel
y (1
.345
09,0
.743
445)
.
1.U
se a
gra
phin
g ca
lcu
lato
r to
fin
d th
e ot
her
poi
nt
of i
nte
rsec
tion
.(�
1.71
064,
�0.
5845
75)
2.U
se t
he
Zoo
mfe
atu
re o
n t
he
calc
ula
tor
to z
oom
in
aro
un
d th
e po
int
of i
nte
rsec
tion
.W
hat
do
the
two
non
lin
ear
equ
atio
ns
rem
ind
you
of
at t
his
lev
el o
f zo
om?
Th
ey s
ho
uld
ap
pea
r to
be
linea
r.
3.G
OLF
Th
e h
eigh
t of
a g
olf
ball
dro
pped
fro
m t
he
top
of a
100
-foo
t to
wer
aft
er
tse
con
ds i
s gi
ven
by
h�
�16
t3�
100.
Use
a g
raph
ing
calc
ula
tor
to d
eter
min
e w
hen
(in
sec
onds
) th
e go
lf b
all
is 1
0 fe
et f
rom
th
e gr
oun
d.2.
3717
1 se
con
ds
⎧ ⎨ ⎩
Answers (Lesson 3-1)
A1-A19 A2-03-873973 5/22/06 8:25 AM Page A4
Chapter 3 A5 Glencoe Algebra 2
An
swer
s
Exer
cise
s
Cha
pter
312
Gle
ncoe
Alg
ebra
2
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
__P
ER
IOD
____
_
3-1
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Spre
adsh
eet
Act
ivit
yB
reak
-Eve
n P
oin
t
1.If
Car
ly c
ould
dec
reas
e h
er a
nn
ual
ove
rhea
d to
$14
,000
,wh
at w
ould
th
ebr
eak-
even
poi
nt
be?
bet
wee
n 1
400
and
150
0 ca
nd
les
2.S
upp
ose
Car
ly d
ecre
ases
her
an
nu
al o
verh
ead
to $
14,0
00 a
nd
incr
ease
sth
e pr
ice
of a
can
dle
to $
14.0
0.W
hat
is
the
new
bre
ak-e
ven
poi
nt?
bet
wee
n 1
200
and
130
0 ca
nd
les
Car
ly E
rics
on i
s co
nsi
der
ing
open
ing
a ca
nd
le
bu
sin
ess.
Sh
e es
tim
ates
th
at s
he
wil
l h
ave
an a
nn
ual
ove
rhea
d o
f$1
5,00
0.It
cos
ts C
arly
$3.
00 t
o m
ake
a ja
r ca
nd
le,w
hic
h s
he
sell
s fo
r$1
2.50
.Wh
at i
s C
arly
’s b
reak
-eve
n p
oin
t?
Use
Col
um
n A
for
th
e n
um
ber
of c
andl
es.C
olu
mn
s B
and
C a
re t
he
cost
an
d th
e in
com
e,re
spec
tive
ly.
Ext
end
the
row
s of
th
e sp
read
shee
t to
fin
d th
e po
int
atw
hich
the
inco
me
firs
t ex
ceed
s th
e co
st.T
he b
reak
-eve
npo
int
occu
rs b
etw
een
th
is p
oin
t an
d th
e pr
evio
us
poin
t.In
th
is c
ase,
the
brea
k ev
en p
oin
t oc
curs
bet
wee
n 1
500
and
1600
can
dles
.
Th
e ch
art
tool
of
the
spre
adsh
eet
allo
ws
you
to
grap
hth
e da
ta.T
he
grap
h v
erif
ies
the
solu
tion
.
You
hav
e le
arn
ed t
hat
th
e br
eak-
even
poi
nt
is t
he
poin
t at
wh
ich
th
e in
com
eeq
ual
s th
e co
st.Y
ou c
an u
se t
he
form
ula
s an
d ch
arts
in
a s
prea
dsh
eet
to f
ind
a br
eak-
even
poi
nt.
A1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
C
$0$1
,250
$2,5
00$3
,750
$5,0
00$6
,250
$7,5
00$8
,750
$10,
000
$11,
250
$12,
500
$13,
750
$15,
000
$16,
250
$17,
500
$18,
750
$20,
000
Inco
me
B
$15,
000
$15,
300
$15,
600
$15,
900
$16,
200
$16,
500
$16,
800
$17,
100
$17,
400
$17,
700
$18,
000
$18,
300
$18,
600
$18,
900
$19,
200
$19,
500
$19,
800
Cos
tC
andl
es0
100
200
300
400
500
600
700
800
900
1000
1100
1200
1300
1400
1500
1600
Sh
eet
1S
hee
t 2
Sh
eet
3
Exam
ple
Lesson 3-1
Cha
pter
313
Gle
ncoe
Alg
ebra
2
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
__P
ER
IOD
____
_
3-2
Copyright ©Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Less
on R
eadi
ng G
uide
So
lvin
g S
yste
ms
of
Eq
uat
ion
s A
lgeb
raic
ally
Get
Rea
dy
for
the
Less
on
Rea
d t
he
intr
odu
ctio
n t
o L
esso
n 3
-2 i
n y
our
text
boo
k.
•H
ow m
any
mor
e m
inu
tes
of l
ong
dist
ance
tim
e di
d Yo
lan
da u
se i
n F
ebru
ary
than
in
Jan
uar
y?13
min
ute
s
•H
ow m
uch
mor
e w
ere
the
Feb
ruar
y ch
arge
s th
an t
he
Jan
uar
y ch
arge
s?$1
.04
•U
sin
g yo
ur
answ
ers
for
the
ques
tion
s ab
ove,
how
can
you
fin
d th
e ra
te p
er m
inu
te?
Fin
d $
1.04
�13
.
Rea
d t
he
Less
on
1.S
upp
ose
that
you
are
ask
ed t
o so
lve
the
syst
em o
f eq
uat
ion
s 4x
�5y
�7
at t
he
righ
t by
th
e su
bsti
tuti
on m
eth
od.
3x�
y�
�9
Th
e fi
rst
step
is
to s
olve
on
e of
th
e eq
uat
ion
s fo
r on
e va
riab
le
in t
erm
s of
th
e ot
her
.To
mak
e yo
ur
wor
k as
eas
y as
pos
sibl
e,w
hic
h e
quat
ion
wou
ld y
ou s
olve
for
wh
ich
var
iabl
e? E
xpla
in.
Sam
ple
an
swer
:S
olv
e th
e se
con
d e
qu
atio
n f
or
yb
ecau
se in
th
ateq
uat
ion
th
e va
riab
le y
has
a c
oef
fici
ent
of
1.
2.S
upp
ose
that
you
are
ask
ed t
o so
lve
the
syst
em o
f eq
uat
ion
s 2x
�3y
��
2at
th
e ri
ght
by t
he
elim
inat
ion
met
hod
.7x
�y
�39
To
mak
e yo
ur
wor
k as
eas
y as
pos
sibl
e,w
hic
h v
aria
ble
wou
ld
you
eli
min
ate?
Des
crib
e h
ow y
ou w
ould
do
this
.S
amp
le a
nsw
er:
Elim
inat
e th
e va
riab
le y
;m
ult
iply
th
e se
con
d e
qu
atio
n b
y3
and
th
en a
dd
th
e re
sult
to
th
e fi
rst
equ
atio
n.
Rem
emb
er W
hat
Yo
u L
earn
ed
3.T
he
subs
titu
tion
met
hod
an
d el
imin
atio
n m
eth
od f
or s
olvi
ng
syst
ems
both
hav
e se
vera
lst
eps,
and
it m
ay b
e di
ffic
ult
to
rem
embe
r th
em.Y
ou m
ay b
e ab
le t
o re
mem
ber
them
mor
e ea
sily
if
you
not
ice
wh
at t
he
met
hod
s h
ave
in c
omm
on.W
hat
ste
p is
th
e sa
me
inbo
th m
eth
ods?
Sam
ple
an
swer
:A
fter
fin
din
g t
he
valu
e o
f o
ne
of
the
vari
able
s,yo
u f
ind
the
valu
e o
f th
e o
ther
var
iab
le b
y su
bst
itu
tin
g t
he
valu
e yo
u h
ave
fou
nd
in o
ne
of
the
ori
gin
al e
qu
atio
ns.
Answers (Lessons 3-1 and 3-2)
A1-A19 A2-03-873973 5/22/06 8:25 AM Page A5
Chapter 3 A6 Glencoe Algebra 2
Cha
pter
314
Gle
ncoe
Alg
ebra
2
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
__P
ER
IOD
____
_
3-2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Sub
stit
uti
on
To
solv
e a
syst
em o
f li
nea
r eq
uat
ion
s by
su
bst
itu
tion
,fir
st s
olve
for
on
eva
riab
le i
n t
erm
s of
th
e ot
her
in
on
e of
th
e eq
uat
ion
s.T
hen
su
bsti
tute
th
is e
xpre
ssio
n i
nto
the
oth
er e
quat
ion
an
d si
mpl
ify.
Use
su
bst
itu
tion
to
solv
e th
e sy
stem
of
equ
atio
ns.
2x�
y�
9x
�3y
��
6S
olve
th
e fi
rst
equ
atio
n f
or y
in t
erm
s of
x.
2x�
y�
9F
irst
equa
tion
�y
��
2x�
9S
ubtr
act
2xfr
om b
oth
side
s.
y�
2x�
9M
ultip
ly b
oth
side
s by
�1.
Su
bsti
tute
th
e ex
pres
sion
2x
�9
for
yin
to t
he
seco
nd
equ
atio
n a
nd
solv
e fo
r x.
x�
3y�
�6
Sec
ond
equa
tion
x�
3(2x
�9)
��
6S
ubst
itute
2x
�9
for
y.
x�
6x�
27 �
�6
Dis
trib
utiv
e P
rope
rty
7x�
27 �
�6
Sim
plify
.
7x�
21A
dd 2
7 to
eac
h si
de.
x�
3D
ivid
e ea
ch s
ide
by 7
.
Now
,su
bsti
tute
th
e va
lue
3 fo
r x
in e
ith
er o
rigi
nal
equ
atio
n a
nd
solv
e fo
r y.
2x�
y�
9F
irst
equa
tion
2(3)
�y
�9
Rep
lace
xw
ith 3
.
6 �
y�
9S
impl
ify.
�y
�3
Sub
trac
t 6
from
eac
h si
de.
y�
�3
Mul
tiply
eac
h si
de b
y �
1.
Th
e so
luti
on o
f th
e sy
stem
is
(3,�
3).
Sol
ve e
ach
sys
tem
of
lin
ear
equ
atio
ns
by
usi
ng
sub
stit
uti
on.
1.3x
�y
�7
2.2x
�y
�5
3.2x
�3y
��
34x
�2y
�16
3x�
3y�
3x
�2y
�2
(�1,
10)
(2,1
)(�
12,7
)
4.2x
�y
�7
5.4x
�3y
�4
6.5x
�y
�6
6x�
3y�
142x
�y
��
83
�x
�0
no
so
luti
on
(�2,
�4)
(3,�
9)
7.x
�8y
��
28.
2x�
y�
�4
9.x
�y
��
2x
�3y
�20
4x�
y�
12x
�3y
�2
(14,
�2)
��,3
�(�
8,�
6)
10.x
�4y
�4
11.x
�3y
�2
12.2
x�
2y�
42x
�12
y�
134x
�12
y�
8x
�2y
�0
�5,�
infi
nit
ely
man
y�
,�
2 � 34 � 3
1 � 4
1 � 2
Stud
y G
uide
and
Inte
rven
tion
So
lvin
g S
yste
ms
of
Eq
uat
ion
s A
lgeb
raic
ally
Exam
ple
Exer
cise
s
3-2
Lesson 3-2
Cha
pter
315
Gle
ncoe
Alg
ebra
2
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
__P
ER
IOD
____
_
3-2
Copyright ©Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Elim
inat
ion
To
solv
e a
syst
em o
f li
nea
r eq
uat
ion
s by
eli
min
atio
n,a
dd o
r su
btra
ct t
he
equ
atio
ns
to e
lim
inat
e on
e of
th
e va
riab
les.
You
may
fir
st n
eed
to m
ult
iply
on
e or
bot
h o
f th
eeq
uat
ion
s by
a c
onst
ant
so t
hat
on
e of
th
e va
riab
les
has
th
e sa
me
(or
oppo
site
) co
effi
cien
t in
one
equ
atio
n a
s it
has
in
th
e ot
her
.
Use
th
e el
imin
atio
n m
eth
od t
o so
lve
the
syst
em o
f eq
uat
ion
s.2x
�4y
��
263x
�y
��
24
Stud
y G
uide
and
Inte
rven
tion
(co
ntin
ued)
So
lvin
g S
yste
ms
of
Eq
uat
ion
s A
lgeb
raic
ally
Mu
ltip
ly t
he
seco
nd
equ
atio
n b
y 4.
Th
en s
ubt
ract
the
equ
atio
ns
to e
lim
inat
e th
e y
vari
able
.2x
�4y
��
262x
�4y
��
263x
�y
��
24M
ultip
ly b
y 4.
12x
�4y
��
96�
10x
�70
x�
�7
Rep
lace
xw
ith
�7
and
solv
e fo
r y.
2x�
4y�
�26
2(�
7) �
4y�
�26
�14
�4y
��
26�
4y�
�12
y�
3T
he
solu
tion
is
(�7,
3).
Mu
ltip
ly t
he
firs
t eq
uat
ion
by
3 an
d th
e se
con
deq
uat
ion
by
2.T
hen
add
th
e eq
uat
ion
s to
elim
inat
e th
e y
vari
able
.3x
�2y
�4
Mul
tiply
by
3.9x
�6y
�12
5x�
3y�
�25
Mul
tiply
by
2.10
x�
6y�
�50
19x
��
38x
��
2
Rep
lace
xw
ith
�2
and
solv
e fo
r y.
3x�
2y�
43(
�2)
�2y
�4
�6
�2y
�4
�2y
�10
y�
�5
Th
e so
luti
on i
s (�
2,�
5).
Use
th
e el
imin
atio
n m
eth
od t
o so
lve
the
syst
em o
f eq
uat
ion
s.3x
�2y
�4
5x�
3y�
�25
Sol
ve e
ach
sys
tem
of
equ
atio
ns
by
usi
ng
elim
inat
ion
.
1.2x
�y
�7
2.x
�2y
�4
3.3x
�4y
��
104.
3x�
y�
123x
�y
�8
�x
�6y
�12
x�
4y�
25x
�2y
�20
(3,�
1)(1
2,4)
(�2,
�1)
(4,0
)
5.4x
�y
�6
6.5x
�2y
�12
7.2x
�y
�8
8.7x
�2y
��
1
2x�
�4
�6x
�2y
��
143x
�y
�12
4x�
3y�
�13
no
so
luti
on
(2,1
)in
fin
itel
y m
any
(�1,
3)
9.3x
�8y
��
610
.5x
�4y
�12
11.�
4x�
y�
�12
12.5
m�
2n�
�8
x�
y�
97x
�6y
�40
4x�
2y�
64m
�3n
�2
(6,�
3)(4
,�2)
�,�
2 �(�
4,6)
5 � 2
3 � 2y � 2
Exam
ple
1
Exam
ple
2
Exer
cise
s
Answers (Lesson 3-2)
A1-A19 A2-03-873973 5/22/06 8:25 AM Page A6
Chapter 3 A7 Glencoe Algebra 2
An
swer
s
Cha
pter
316
Gle
ncoe
Alg
ebra
2
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
__P
ER
IOD
____
_
3-2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Skill
s Pr
acti
ceS
olv
ing
Sys
tem
s o
f E
qu
atio
ns
Alg
ebra
ical
lyS
olve
eac
h s
yste
m o
f eq
uat
ion
s b
y u
sin
g su
bst
itu
tion
.
1.m
�n
�20
2.x
�3y
��
33.
w�
z�
1m
�n
��
4(8
,12)
4x�
3y�
6(3
,�2)
2w�
3z�
12(3
,2)
4.3r
�s
�5
5.2b
�3c
��
46.
x�
y�
�5
2r�
s�
5(2
,�1)
b�
c�
3(1
3,�
10)
3x�
4y�
13(�
1,4)
Sol
ve e
ach
sys
tem
of
equ
atio
ns
by
usi
ng
elim
inat
ion
.
7.2p
�q
�17
8.2j
�k
�3
9.3c
�2d
�2
3p�
q�
8(5
,�7)
3j�
k�
2(1
,�1)
3c�
4d�
50(6
,8)
10.2
f�
3g�
911
.�2x
�y
��
112
.2x
�y
�12
f�
g�
2(3
,1)
x�
2y�
3(1
,1)
2x�
y�
6n
o s
olu
tio
n
Sol
ve e
ach
sys
tem
of
equ
atio
ns
by
usi
ng
eith
er s
ub
stit
uti
on o
r el
imin
atio
n.
13.�
r�
t�
514
.2x
�y
��
515
.x�
3y�
�12
�2r
�t
�4
(1,6
)4x
�y
�2
��,4
�2x
�y
�11
(3,5
)
16.2
p�
3q�
617
.6w
�8z
�16
18.
c�
d�
6�
2p�
3q�
�6
3w�
4z�
8c
�d
�0
(3,3
)in
fin
itel
y m
any
infi
nit
ely
man
y
19.2
u�
4v�
�6
20.3
a�
b�
�1
21.2
x�
y�
6u
�2v
�3
no
so
luti
on
�3a
�b
�5
(�1,
2)3x
�2y
�16
(4,�
2)
22.3
y�
z�
�6
23.c
�2d
��
224
.3r
�2s
�1
�3y
�z
�6
(�2,
0)�
2c�
5d�
3(�
4,1)
2r�
3s�
9(�
3,�
5)
25.T
he
sum
of
two
nu
mbe
rs i
s 12
.Th
e di
ffer
ence
of
the
sam
e tw
o n
um
bers
is
�4.
Fin
d th
e n
um
bers
.4,
8
26.T
wic
e a
nu
mbe
r m
inu
s a
seco
nd
nu
mbe
r is
�1.
Tw
ice
the
seco
nd
nu
mbe
r ad
ded
to t
hre
e ti
mes
th
e fi
rst
nu
mbe
r is
9.F
ind
the
two
nu
mbe
rs.
1,3
1 � 2
3-2
Lesson 3-2
Cha
pter
317
Gle
ncoe
Alg
ebra
2
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
__P
ER
IOD
____
_
3-2
Copyright ©Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Sol
ve e
ach
sys
tem
of
equ
atio
ns
by
usi
ng
sub
stit
uti
on.
1.2x
�y
�4
2.x
�3y
�9
3.g
�3h
�8
no
3x�
2y�
1(7
,�10
)x
�2y
��
1(3
,�2)
g�
h�
9so
luti
on
4.2a
�4b
�6
infi
nit
ely
5.2m
�n
�6
6.4x
�3y
��
6�
a�
2b�
�3
man
y5m
�6n
�1
(5,�
4)�
x�
2y�
7(�
3,�
2)
7.u
�2v
�8.
x�
3y�
169.
w�
3z�
1
�u
�2v
�5
no
so
luti
on
4x�
y�
9(1
,�5)
3w�
5z�
�4
��,
�
Sol
ve e
ach
sys
tem
of
equ
atio
ns
by
usi
ng
elim
inat
ion
.
10.2
r�
s�
511
.2m
�n
��
112
.6x
�3y
�6
3r�
s�
20(5
,�5)
3m�
2n�
30(4
,9)
8x�
5y�
12(�
1,4)
13.3
j�
k�
1014
.2x
�y
��
4n
o
15.2
g�
h�
64j
�k
�16
(6,8
)�
4x�
2y�
6so
luti
on
3g�
2h�
16(4
,�2)
16.2
t�
4v�
6in
fin
itel
y17
.3x
�2y
�12
18.
x�
3y�
11�
t�
2v�
�3
man
y2x
�y
�14
(6,3
)8x
�5y
�17
(4,3
)
Sol
ve e
ach
sys
tem
of
equ
atio
ns
by
usi
ng
eith
er s
ub
stit
uti
on o
r el
imin
atio
n.
19.8
x�
3y�
�5
20.8
q�
15r
��
4021
.3x
�4y
�12
infi
nit
ely
10x
�6y
��
13�
,�3 �
4q�
2r�
56(1
0,8)
x�
y�
man
y
22.4
b�
2d�
5n
o23
.s�
3y�
424
.4m
�2p
�0
�2b
�d
�1
solu
tio
ns
�1
(1,1
)�
3m�
9p�
5�
,�
25.5
g�
4k�
1026
.0.5
x�
2y�
527
.h�
z�
3n
o�
3g�
5k�
7(6
,�5)
x�
2y�
�8
(�2,
3)�
3h�
3z�
6so
luti
on
SPO
RTS
For
Exe
rcis
es 2
8 an
d 2
9,u
se t
he
foll
owin
g in
form
atio
n.
Las
t ye
ar t
he
voll
eyba
ll t
eam
pai
d $5
per
pai
r fo
r so
cks
and
$17
per
pair
for
sh
orts
on
ato
tal
purc
has
e of
$31
5.T
his
yea
r th
ey s
pen
t $3
42 t
o bu
y th
e sa
me
nu
mbe
r of
pai
rs o
f so
cks
and
shor
ts b
ecau
se t
he
sock
s n
ow c
ost
$6 a
pai
r an
d th
e sh
orts
cos
t $1
8.
28.W
rite
a s
yste
m o
f tw
o eq
uat
ion
s th
at r
epre
sen
ts t
he
nu
mbe
r of
pai
rs o
f so
cks
and
shor
tsbo
ugh
t ea
ch y
ear.
5x�
17y
�31
5,6x
�18
y�
342
29.H
ow m
any
pair
s of
soc
ks a
nd
shor
ts d
id t
he
team
bu
y ea
ch y
ear?
sock
s:12
,sh
ort
s:15
2 � 31 � 3
4 � 34 � 9
1 � 31 � 2
2 � 3
1 � 2
1 � 21 � 2
1 � 2
1 � 3
Prac
tice
S
olv
ing
Sys
tem
s o
f E
qu
atio
ns
Alg
ebra
ical
ly
Answers (Lesson 3-2)
A1-A19 A2-03-873973 5/22/06 8:25 AM Page A7
Chapter 3 A8 Glencoe Algebra 2
Cha
pter
318
Gle
ncoe
Alg
ebra
2
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
__P
ER
IOD
____
_
3-2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Wor
d Pr
oble
m P
ract
ice
So
lvin
g S
yste
ms
of
Eq
uat
ion
s A
lgeb
raic
ally
1.SU
PPLI
ESK
irst
a an
d A
rth
ur
both
nee
dpe
ns
and
blan
k C
Ds.
Th
e eq
uat
ion
th
atre
pres
ents
Kir
sta’
s pu
rch
ases
is
y�
27 �
3x.T
he
equ
atio
n t
hat
repr
esen
ts A
rth
ur’
s pu
rch
ases
is
y�
17 �
x.If
xre
pres
ents
th
e pr
ice
of t
he
pen
s,an
d y
repr
esen
ts t
he
pric
e of
the
CD
s,w
hat
are
th
e pr
ices
of
the
pen
san
d th
e C
Ds?
pen
s ar
e $5
a p
ack,
CD
s ar
e $1
2 a
pac
k
2.W
ALK
ING
Am
y is
wal
kin
g a
stra
igh
tpa
th t
hat
can
be
repr
esen
ted
by t
he
equ
atio
n y
�2x
�3.
At
the
sam
e ti
me
Ken
dra
is w
alki
ng
the
stra
igh
t pa
thth
at h
as t
he
equ
atio
n 3
y�
6x�
6.W
hat
is
the
solu
tion
to
the
syst
em o
feq
uat
ion
s th
at r
epre
sen
ts t
he
path
s th
e tw
o gi
rls
wal
ked?
Exp
lain
.T
her
e is
no
so
luti
on
.Th
eir
pat
hs
nev
er c
ross
.
3.C
AFE
TER
IAT
o fu
rnis
h a
caf
eter
ia,a
sch
ool
can
spe
nd
$520
0 on
tab
les
and
chai
rs.T
able
s co
st $
200
and
chai
rs c
ost
$40.
Eac
h t
able
wil
l h
ave
8 ch
airs
arou
nd
it.H
ow m
any
tabl
es a
nd
chai
rsw
ill
the
sch
ool
purc
has
e?10
tab
les
and
80
chai
rs
4.PR
ICES
At
a st
ore,
toot
hbr
ush
es c
ost
xdo
llar
s an
d ba
rs o
f so
ap c
ost
ydo
llar
s.O
ne
cust
omer
bou
ght
2 to
oth
bru
shes
and
1 ba
r of
soa
p fo
r $1
1.A
not
her
cust
omer
bou
ght
6 to
oth
bru
shes
an
d 5
bars
of
soap
for
$38
.Bot
h a
mou
nts
do
not
in
clu
de t
ax.W
rite
an
d so
lve
asy
stem
of
equ
atio
ns
for
xan
d y.
2x+
y�
116x
+ 5y
�38
x�
4.25
an
d y
�2.
50
GA
MES
For
Exe
rcis
es 5
–7,u
se t
he
foll
owin
g in
form
atio
n.
Mar
k an
d S
teph
anie
are
pla
yin
g a
gam
ew
her
e th
ey t
oss
a da
rt a
t a
gam
e bo
ard
that
is h
angi
ng
on t
he
wal
l.T
he
poin
ts e
arn
edfr
om a
tos
s de
pen
ds o
n w
her
e th
e da
rtla
nds
.Th
e ce
nte
r ar
ea i
s w
orth
mor
e po
ints
than
th
e su
rrou
ndi
ng
area
.Eac
h p
laye
rto
sses
12
dart
s.
5.S
teph
anie
ear
ned
a t
otal
of
66 p
oin
tsw
ith
6 d
arts
lan
din
g in
eac
h a
rea.
Mar
kea
rned
a t
otal
of
56 p
oin
ts w
ith
4 d
arts
lan
din
g in
th
e ce
nte
r ar
ea,a
nd
8 da
rts
lan
din
g in
th
e su
rrou
ndi
ng
area
.Wri
te a
syst
em o
f eq
uat
ion
s th
at r
epre
sen
ts t
he
nu
mbe
r of
dar
ts e
ach
pla
yer
toss
ed i
nto
each
sec
tion
.Use
xfo
r th
e in
ner
cir
cle,
and
yfo
r th
e ou
ter
circ
le.
6x+
6y�
66;
4x+
8y�
56
6.H
ow m
any
poin
ts i
s th
e in
ner
cir
cle
wor
th?
How
man
y po
ints
is
the
oute
rci
rcle
wor
th?
inn
er c
ircl
e �
8 p
oin
ts;
ou
ter
circ
le �
3 p
oin
ts
7.If
a p
laye
r ge
ts 1
0 da
rts
in t
he
inn
erci
rcle
an
d 2
in t
he
oute
r ci
rcle
th
e to
tal
scor
e is
dou
bled
.How
man
y po
ints
wou
ld t
he
play
er e
arn
if
he
or s
he
gets
exac
tly
10 d
arts
in
th
e ce
nte
r?17
2 p
oin
ts
3-2
Lesson 3-2
Cha
pter
319
Gle
ncoe
Alg
ebra
2
NA
ME
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__D
ATE
____
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____
__P
ER
IOD
____
_
3-2
Copyright ©Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Enri
chm
ent
Usi
ng
Co
ord
inat
esF
rom
on
e ob
serv
atio
n p
oin
t,th
e li
ne
of s
igh
t to
a d
own
ed p
lan
e is
giv
en b
y y
�x
�1.
Th
is e
quat
ion
des
crib
es t
he
dist
ance
fro
m t
he
obse
rvat
ion
poi
nt
to t
he
plan
e in
a s
trai
ght
lin
e.F
rom
an
oth
er o
bser
vati
on p
oin
t,th
e li
ne
of
sigh
t is
giv
en b
y x
�3y
�21
.Wh
at a
re t
he
coor
din
ates
of
the
poin
t at
wh
ich
th
e cr
ash
occ
urr
ed?
Sol
ve t
he
syst
em o
f eq
uat
ion
s�y
�x
�1
.x
�3y
�21
x�
3y�
21x
�3(
x�
1) �
21S
ubst
itute
x�
1 fo
r y.
x�
3x�
3 �
214x
�24
x�
6
x�
3y�
216
�3y
�21
Sub
stitu
te 6
for
x.
3y�
15y
�5
Th
e co
ordi
nat
es o
f th
e cr
ash
are
(6,
5).
Sol
ve.
1.T
he
lin
es o
f si
ght
to a
for
est
fire
are
as
foll
ows.
Fro
m R
ange
r S
tati
on A
:3x
�y
�9
Fro
m R
ange
r S
tati
on B
:2x
�3y
�13
Fin
d th
e co
ordi
nat
es o
f th
e fi
re.
(2,3
)
2.A
n a
irpl
ane
is t
rave
lin
g al
ong
the
lin
e x
�y
��
1 w
hen
it
sees
an
oth
er
airp
lan
e tr
avel
ing
alon
g th
e li
ne
5x�
3y�
19.I
f th
ey c
onti
nu
e al
ong
the
sam
e li
nes
,at
wh
at p
oin
t w
ill
thei
r fl
igh
t pa
ths
cros
s?(2
,3)
3.T
wo
min
e sh
afts
are
du
g al
ong
the
path
s of
th
e fo
llow
ing
equ
atio
ns.
x�
y�
1400
2x�
y�
1300
If t
he
shaf
ts m
eet
at a
dep
th o
f 20
0 fe
et,w
hat
are
th
e co
ordi
nat
es o
f th
e po
int
at w
hic
h t
hey
mee
t?(9
00,�
500)
Answers (Lesson 3-2)
A1-A19 A2-03-873973 5/22/06 8:25 AM Page A8
Chapter 3 A9 Glencoe Algebra 2
An
swer
s
Cha
pter
320
Gle
ncoe
Alg
ebra
2
NA
ME
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____
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____
__D
ATE
____
____
____
__P
ER
IOD
____
_
3-3
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Less
onRe
adin
g G
uide
So
lvin
g S
yste
ms
of
Ineq
ual
itie
s by
Gra
ph
ing
Get
Rea
dy
for
the
Less
on
Rea
d t
he
intr
odu
ctio
n t
o L
esso
n 3
-3 i
n y
our
text
boo
k.
Sat
ish
is
37 y
ears
old
.He
has
a b
lood
pre
ssu
re r
eadi
ng
of 1
35/9
9.Is
his
blo
od p
ress
ure
wit
hin
th
e n
orm
al r
ange
? E
xpla
in.
Sam
ple
an
swer
:N
o;
his
sys
tolic
pre
ssu
re is
no
rmal
,bu
t h
is d
iast
olic
pre
ssu
re is
to
o h
igh
.It
sho
uld
be
bet
wee
n 6
0 an
d 9
0.
Rea
d t
he
Less
on
1.W
ith
out
actu
ally
dra
win
g th
e gr
aph
,des
crib
e th
e bo
un
dary
lin
es
⏐x⏐
�3
for
the
syst
em o
f in
equ
alit
ies
show
n a
t th
e ri
ght.
⏐y⏐
�5
Two
das
hed
ver
tica
l lin
es (
x�
3 an
d x
��
3) a
nd
tw
o s
olid
ho
rizo
nta
llin
es (
y�
�5
and
y�
5)
2.T
hin
k ab
out
how
th
e gr
aph
wou
ld l
ook
for
the
syst
em g
iven
abo
ve.W
hat
wil
l be
th
esh
ape
of t
he
shad
ed r
egio
n?
(It
is n
ot n
eces
sary
to
draw
th
e gr
aph
.See
if
you
can
imag
ine
it w
ith
out
draw
ing
anyt
hin
g.If
th
is i
s di
ffic
ult
to
do,m
ake
a ro
ugh
ske
tch
to
hel
p yo
u a
nsw
er t
he
ques
tion
.)a
rect
ang
le
3.W
hic
h s
yste
m o
f in
equ
alit
ies
mat
ches
th
e gr
aph
sh
own
at
th
e ri
ght?
BA
.x
�y
��
2B
.x
�y
��
2x
�y
�2
x�
y�
2
C.
x�
y�
�2
D.
x�
y�
�2
x�
y�
2x
�y
�2
Rem
emb
er W
hat
Yo
u L
earn
ed
4.T
o gr
aph
a s
yste
m o
f in
equ
alit
ies,
you
mu
st g
raph
tw
o or
mor
e bo
un
dary
lin
es.W
hen
you
gra
ph e
ach
of
thes
e li
nes
,how
can
th
e in
equ
alit
y sy
mbo
ls h
elp
you
rem
embe
rw
het
her
to
use
a d
ash
ed o
r so
lid
lin
e?U
se a
das
hed
lin
e if
th
e in
equ
alit
y sy
mb
ol i
s �
or
�,b
ecau
se t
hes
esy
mb
ols
do
no
t in
clu
de
equ
alit
y an
d t
he
das
hed
lin
e re
min
ds
you
th
atth
e lin
e it
self
is n
ot
incl
ud
ed in
th
e g
rap
h.U
se a
so
lid li
ne
if t
he
sym
bo
lis
�o
r
,bec
ause
th
ese
sym
bo
ls in
clu
de
equ
alit
y an
d t
ell y
ou
th
at t
he
line
itse
lf is
incl
ud
ed in
th
e g
rap
h.
x
y
O
Exam
ple
Stud
y G
uide
and
Inte
rven
tion
So
lvin
g S
yste
ms
of
Ineq
ual
itie
s by
Gra
ph
ing
Lesson 3-3
Cha
pter
321
Gle
ncoe
Alg
ebra
2
NA
ME
____
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____
____
____
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____
____
____
__D
ATE
____
____
____
__P
ER
IOD
____
_
3-3
Copyright ©Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Gra
ph
Sys
tem
s o
f In
equ
alit
ies
To s
olve
a s
yste
m o
f in
equa
litie
s,gr
aph
the
ineq
ualit
ies
in t
he s
ame
coor
dina
te p
lane
.The
sol
utio
n se
t is
rep
rese
nted
by
the
inte
rsec
tion
of
the
grap
hs.
Sol
ve t
he
syst
em o
f in
equ
alit
ies
by
grap
hin
g.y
2x
�1
and
y�
�2
Th
e so
luti
on o
f y
�2x
�1
is R
egio
ns
1 an
d 2.
Th
e so
luti
on o
f y
��
2 is
Reg
ion
s 1
and
3.
Th
e in
ters
ecti
on o
f th
ese
regi
ons
is R
egio
n 1
,wh
ich
is
the
solu
tion
set
of
the
syst
em o
f in
equ
alit
ies.
Sol
ve e
ach
sys
tem
of
ineq
ual
itie
s b
y gr
aph
ing.
1.x
�y
�2
2.3x
�2y
��
13.
⏐y⏐
�1
x�
2y�
1x
�4y
��
12x
�2
4.y
��
35.
y�
�2
6.y
��
�1
y�
2xy
��
2x�
1y
�3x
�1
7.x
�y
�4
8.x
�3y
�3
9.x
�2y
�6
2x�
y�
2x
�2y
�4
x�
4y�
�4
x
y
Ox
y
O
x
y
O
x
y
Ox
y
O
x
y
O
x � 4x � 3
x � 2
x
y
Ox
y
Ox
y
O
x � 3
x � 3
x
y
O
Regi
on 3
Regi
on 1
Regi
on 2
Exer
cise
s
Answers (Lesson 3-3)
A1-A19 A2-03-873973 5/22/06 8:25 AM Page A9
Chapter 3 A10 Glencoe Algebra 2
Exam
ple
Cha
pter
322
Gle
ncoe
Alg
ebra
2
NA
ME
____
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____
____
____
____
____
____
____
____
__D
ATE
____
____
____
__P
ER
IOD
____
_
3-3
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Fin
d V
erti
ces
of
a Po
lyg
on
al R
egio
nS
omet
imes
th
e gr
aph
of
a sy
stem
of
ineq
ual
itie
s fo
rms
a bo
un
ded
regi
on.Y
ou c
an f
ind
the
vert
ices
of
the
regi
on b
y a
com
bin
atio
nof
th
e m
eth
ods
use
d ea
rlie
r in
th
is c
hap
ter:
grap
hin
g,su
bsti
tuti
on,a
nd/
or e
lim
inat
ion
.
Fin
d t
he
coor
din
ates
of
the
vert
ices
of
the
figu
re f
orm
ed b
y 5x
�4y
�20
,y�
2x�
3,an
d x
�3y
�4.
Gra
ph t
he
bou
nda
ry o
f ea
ch i
neq
ual
ity.
Th
e in
ters
ecti
ons
of t
he
bou
nda
ry l
ines
are
th
e ve
rtic
es o
f a
tria
ngl
e.T
he
vert
ex (
4,0)
can
be
dete
rmin
ed f
rom
th
e gr
aph
.To
fin
d th
eco
ordi
nat
es o
f th
e se
con
d an
d th
ird
vert
ices
,sol
ve t
he
two
syst
ems
of e
quat
ion
s
y�
2x�
3an
dy
�2x
�3
5x�
4y�
20x
�3y
�4
x
y
O
Stud
y G
uide
and
Inte
rven
tion
(co
ntin
ued)
So
lvin
g S
yste
ms
of
Ineq
ual
itie
s by
Gra
ph
ing
For
th
e fi
rst
syst
em o
f eq
uat
ion
s,re
wri
te t
he
firs
teq
uat
ion
in
sta
nda
rd f
orm
as
2x�
y�
�3.
Th
enm
ult
iply
th
at e
quat
ion
by
4 an
d ad
d to
th
e se
con
deq
uat
ion
.2x
�y
��
3M
ultip
ly b
y 4.
8x�
4y�
�12
5x�
4y�
20(�
)5x
�4y
�20
13x
�8
x�
Th
en s
ubs
titu
te x
�in
on
e of
th
e or
igin
al e
quat
ion
s an
d so
lve
for
y.
2 ���
y�
�3
�y
��
3
y�
Th
e co
ordi
nat
es o
f th
e se
con
d ve
rtex
are
�,4
�.3 � 13
8 � 13
55 � 13
16 � 138 � 13
8 � 13
8 � 13
For
th
e se
con
d sy
stem
of
equ
atio
ns,
use
su
bsti
tuti
on.
Su
bsti
tute
2x
�3
for
yin
th
ese
con
d eq
uat
ion
to
get
x�
3(2x
�3)
�4
x�
6x�
9 �
4�
5x�
13x
��
Th
en s
ubs
titu
te x
��
in t
he
firs
t eq
uat
ion
to
solv
e fo
r y.
y�
2 ����
3
y�
��
3
y�
�
Th
e co
ordi
nat
es o
f th
e th
ird
vert
ex a
re ��
2,�
2�.
1 � 53 � 5
11 � 526 � 5
13 � 5
13 � 5
13 � 5
Th
us,
the
coor
din
ates
of
the
thre
e ve
rtic
es a
re (
4,0)
, �,4
�,an
d ��2
,�2
�.
Fin
d t
he
coor
din
ates
of
the
vert
ices
of
the
figu
re f
orm
ed b
y ea
ch s
yste
m o
fin
equ
alit
ies.
1.y
��
3x�
72.
x�
�3
3.y
��
x�
3
y�
x(2
,1),
(�4,
�2)
,y
��
x�
3(�
3,4)
,y
�x
�1
(�2,
4),(
2,2)
,y
��
2(3
,�2)
y�
x�
1(�
3,�
4),(
3,2)
y�
3x�
10��
3,�
�4 � 5
3 � 5
1 � 2
1 � 31 � 2
1 � 2
1 � 53 � 5
3 � 138 � 13
Exer
cise
s
Skill
s Pr
acti
ceS
olv
ing
Sys
tem
s o
f In
equ
alit
ies
by G
rap
hin
g
Lesson 3-3
Cha
pter
323
Gle
ncoe
Alg
ebra
2
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
__P
ER
IOD
____
_
3-3
Copyright ©Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Sol
ve e
ach
sys
tem
of
ineq
ual
itie
s b
y gr
aph
ing.
1.x
�1
2.x
��
33.
x�
2y
��
1y
��
3x
�4
no
so
luti
on
4.y
�x
5.y
��
4x6.
x�
y�
�1
y�
�x
y�
3x�
23x
�y
�4
7.y
�3
8.y
��
2x�
39.
x�
y�
4x
�2y
�12
y�
x�
22x
�y
�4
Fin
d t
he
coor
din
ates
of
the
vert
ices
of
the
figu
re f
orm
ed b
y ea
ch s
yste
m o
fin
equ
alit
ies.
10.y
�0
11.y
�3
�x
12.x
��
2x
�0
y�
3y
�x
�2
y�
�x
�1
x�
�5
x�
y�
2 (�
2,4)
,(0
,0),
(0,�
1),(
�1,
0)(0
,3),
(�5,
3),(
�5,
8)(�
2,�
4),(
2,0)
x
y
Ox
y
Ox
y
O2
2
x
y
Ox
y
Ox
y
O
x
y
Ox
y
Ox
y
O
Answers (Lesson 3-3)
A1-A19 A2-03-873973 5/22/06 8:25 AM Page A10
Chapter 3 A11 Glencoe Algebra 2
An
swer
s
Cha
pter
324
Gle
ncoe
Alg
ebra
2
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
__P
ER
IOD
____
_
3-3
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Sol
ve e
ach
sys
tem
of
ineq
ual
itie
s b
y gr
aph
ing.
1.y
�1
��
x2.
x�
�2
3.y
�2x
�3
y�
12y
�3x
�6
y�
�x
�2
4.x
�y
��
25.
⏐y⏐
�1
6.3y
�4x
3x�
y�
�2
y�
x�
12x
�3y
��
6
Fin
d t
he
coor
din
ates
of
the
vert
ices
of
the
figu
re f
orm
ed b
y ea
ch s
yste
m o
fin
equ
alit
ies.
7.y
�1
�x
8.x
�y
�2
9.y
�2x
�2
y�
x�
1x
�y
�2
2x�
3y�
6x
�3
x�
�2
y�
4
(1,0
),(3
,2),
(3,�
2)(�
2,4)
,(�
2,�
4),(
2,0)
(�3,
4), �
,1�,(
3,4)
DR
AM
AF
or E
xerc
ises
10
and
11,
use
th
e fo
llow
ing
info
rmat
ion
.T
he
dram
a cl
ub
is s
elli
ng
tick
ets
to i
ts p
lay.
An
adu
lt t
icke
t co
sts
$15
and
a st
ude
nt
tick
et c
osts
$11
.Th
e au
dito
riu
m w
ill
seat
300
tic
ket-
hol
ders
.Th
e dr
ama
clu
b w
ants
to
coll
ect
atle
ast
$363
0 fr
om t
icke
t sa
les.
10.W
rite
an
d gr
aph
a s
yste
m o
f fo
ur
ineq
ual
itie
s th
at
desc
ribe
how
man
y of
eac
h t
ype
of t
icke
t th
e cl
ub
mu
st
sell
to
mee
ts i
ts g
oal.
x�
0,y
�0,
x�
y
300,
15x
�11
y�
3630
11.L
ist
thre
e di
ffer
ent
com
bin
atio
ns
of t
icke
ts s
old
that
sati
sfy
the
ineq
ual
itie
s.S
amp
le a
nsw
er:
250
adu
lt a
nd
50
stu
den
t,20
0 ad
ult
an
d
100
stu
den
t,14
5 ad
ult
an
d 1
48 s
tud
ent
Pla
y T
ickets
Ad
ult
Tic
kets
Student Tickets
100
020
030
040
0
400
350
300
250
200
150
100 50
3 � 2
x
y O
y
xO
y
xO
x
y
Ox
y
Ox
y
O
1 � 2
Prac
tice
S
olv
ing
Sys
tem
s o
f In
equ
alit
ies
by G
rap
hin
g
Lesson 3-3
Cha
pter
325
Gle
ncoe
Alg
ebra
2
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
__P
ER
IOD
____
_
3-3
Copyright ©Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Wor
d Pr
oble
m P
ract
ice
So
lvin
g S
yste
ms
of
Ineq
ual
itie
s by
Gra
ph
ing
1.B
IRD
BA
THM
elis
sa w
ants
to
put
a bi
rdba
th i
n h
er y
ard
at p
oin
t (x
,y),
and
wan
ts i
t to
be
is i
nsi
de t
he
encl
osed
shad
ed a
rea
show
n i
n t
he
grap
h.
Fir
st,s
he
chec
ks t
hat
x�
�3
and
y�
�2.
Wh
at l
inea
r in
equ
alit
y m
ust
sh
ech
eck
to c
oncl
ude
th
at (
x,y)
is
insi
de t
he
tria
ngl
e?x
� y
3
2.SQ
UA
RES
M
att
fin
ds a
blo
t of
in
kco
veri
ng
his
wri
tin
g in
his
not
es f
orm
ath
cla
ss.H
e se
es “A
squ
are
is d
efin
edby
|x|
�8
and
_”.W
rite
an
in
equ
alit
yth
at c
ompl
etes
th
is s
ente
nce
.
A sq
uare
is d
efin
ed b
y |x
| , 8
and
Sam
ple
an
swer
:|y
| 8.
3.H
OLI
DA
YA
man
da r
ecei
ved
pres
ents
and
card
s fr
om f
rien
ds o
ver
the
hol
iday
seas
on.E
very
pre
sen
t ca
me
wit
h o
ne
card
an
d n
one
of h
er f
rien
ds s
ent
her
mor
e th
an o
ne
card
.Les
s th
an 1
0 of
her
frie
nds
sen
t on
ly a
car
d.D
escr
ibe
this
situ
atio
n u
sin
g in
equ
alit
ies.
Let
pb
e th
e n
um
ber
of
pre
sen
tsan
d c
be
the
nu
mb
er o
f ca
rds.
p
can
d c
�p
�10
4.D
ECK
Th
e W
righ
ts a
re b
uil
din
g a
deck
.T
he
deck
is
defi
ned
by
the
ineq
ual
itie
s x
�5,
0.25
x�
y�
�4.
75,y
�5,
and
4.5x
�y
��
17.5
.Gra
ph t
he
ineq
ual
itie
s an
d fi
nd
the
coor
din
ates
of
th
e de
ck’s
cor
ner
s?
(5,5
),(�
5,5)
,(5,
�6)
,(�
3,�
4)TI
CK
ETS
For
Exe
rcis
es 5
an
d 6
,use
th
efo
llow
ing
info
rmat
ion
.A
th
eate
r ch
arge
s $1
0 fo
r ad
ult
s an
d $5
for
chil
dren
12
or u
nde
r.T
he
thea
ter
mak
es a
prof
it i
f th
ey c
an s
ell
mor
e th
an $
600
wor
thof
tic
kets
.Th
e th
eate
r h
as s
eati
ng
for
100
peop
le.
5.W
rite
a s
yste
m o
f li
nea
r in
equ
alit
ies
that
des
crib
es t
he
situ
atio
n.
Let
Ab
e th
e n
um
ber
of
adu
lts
and
Cb
e th
e n
um
ber
of
child
ren
:A
�0,
C�
0,A
�C
10
0,10
A�
5C�
600.
6.G
raph
th
e so
luti
on t
o th
e in
equ
alit
ies.
Can
th
e th
eate
r m
ake
a pr
ofit
if
no
adu
lts
com
e to
th
e pe
rfor
man
ce?
no
A
CO
100
100 50
50
y
xO
5
5
-5
-5
y
xO
5
5
-5
Answers (Lesson 3-3)
A1-A19 A2-03-873973 5/22/06 8:25 AM Page A11
Chapter 3 A12 Glencoe Algebra 2
Cha
pter
326
Gle
ncoe
Alg
ebra
2
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
__P
ER
IOD
____
_
3-3
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Enri
chm
ent
Cre
ativ
e D
esig
ns
A s
yste
m o
f li
nea
r in
equ
alit
ies
can
be
use
d to
def
ine
the
regi
on b
oun
ded
by a
geo
met
ric
shap
e gr
aph
ed o
n a
coo
rdin
ate
plan
e.F
or e
xam
ple,
the
rect
angl
e sh
own
can
be
defi
ned
by
the
syst
em x�
4x
�0
y�
3y
�0.
Th
e tr
ian
gle
show
n c
an b
e de
scri
bed
usi
ng
the
ineq
ual
itie
s
x�
2y�
4x
�0
y�
1.
1.F
ind
a sy
stem
of
lin
ear
ineq
ual
itie
s to
des
crib
e th
e ar
ea b
oun
ded
by t
he
bow
tie
sh
ape
belo
w.T
he
inte
rsec
tion
poi
nts
are
(1,
1),(
1,4)
,(3,
3),(
5,2)
,an
d (5
,5).
if 1
x
3
if 3
x
5
y�
xan
dy
x
x�
2y
9x
�2y
�9
2.F
ind
a sy
stem
of
lin
ear
ineq
ual
itie
s to
des
crib
e th
e ar
ea b
oun
ded
by t
he
basi
c ‘h
ouse
’sh
ape
show
n b
elow
.Th
e in
ters
ecti
on p
oin
ts a
re (
1,1)
,(1,
5),(
3,7)
,(5,
5),a
nd
(5,1
).
x�
1x
5
y
x�
4y
�
x�
10y
� 1
y
xO
y
xO
y
xO
y
xO
Lesson 3-3
Cha
pter
327
Gle
ncoe
Alg
ebra
2
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
__P
ER
IOD
____
_
3-4
Copyright ©Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Less
on R
eadi
ng G
uide
Lin
ear
Pro
gra
mm
ing
Get
Rea
dy
for
the
Less
on
Rea
d t
he
intr
odu
ctio
n t
o L
esso
n 3
-4 i
n y
our
text
boo
k.
Nam
e tw
o or
mor
e fa
cts
that
in
dica
te t
hat
you
wil
l n
eed
to u
se i
neq
ual
itie
s to
mod
el
this
sit
uat
ion
.S
amp
le a
nsw
er:T
he
buoy
ten
der
can
car
ry u
p t
o 8
new
bu
oys.
Th
ere
seem
sto
be
a lim
it o
f 24
ho
urs
on
th
e ti
me
the
crew
has
at
sea.
Th
e cr
ew w
ill w
ant
to r
epai
r o
r re
pla
ce t
he
max
imu
m n
um
ber
of
buoy
s p
oss
ible
.
Rea
d t
he
Less
on
1.C
ompl
ete
each
sen
ten
ce.
a.W
hen
you
fin
d th
e fe
asib
le r
egio
n f
or a
lin
ear
prog
ram
min
g pr
oble
m,y
ou a
re s
olvi
ng
a sy
stem
of
lin
ear
call
ed
.Th
e po
ints
in t
he
feas
ible
reg
ion
are
of
th
e sy
stem
.
b.
Th
e co
rner
poi
nts
of
a po
lygo
nal
reg
ion
are
th
e of
th
e fe
asib
le r
egio
n.
2.A
pol
ygon
al r
egio
n a
lway
s ta
kes
up
only
a l
imit
ed p
art
of t
he
coor
din
ate
plan
e.O
ne
way
to t
hin
k of
th
is i
s to
im
agin
e a
circ
le o
r re
ctan
gle
that
th
e re
gion
wou
ld f
it i
nsi
de.I
n t
he
case
of
a po
lygo
nal
reg
ion
,you
can
alw
ays
fin
d a
circ
le o
r re
ctan
gle
that
is
larg
e en
ough
to c
onta
in a
ll t
he
poin
ts o
f th
e po
lygo
nal
reg
ion
.Wh
at w
ord
is u
sed
to d
escr
ibe
a re
gion
that
can
be
encl
osed
in
th
is w
ay?
Wh
at w
ord
is u
sed
to d
escr
ibe
a re
gion
th
at i
s to
o la
rge
to b
e en
clos
ed i
n t
his
way
?b
ou
nd
ed;
un
bo
un
ded
3.H
ow d
o yo
u f
ind
the
corn
er p
oin
ts o
f th
e po
lygo
nal
reg
ion
in
a l
inea
r pr
ogra
mm
ing
prob
lem
?Yo
u s
olv
e a
syst
em o
f tw
o li
nea
r eq
uat
ion
s.
4.W
hat
are
som
e ev
eryd
ay m
ean
ings
of
the
wor
d fe
asib
leth
at r
emin
d yo
u o
f th
em
ath
emat
ical
mea
nin
g of
th
e te
rm f
easi
ble
regi
on?
Sam
ple
an
swer
:p
oss
ible
or
ach
ieva
ble
Rem
emb
er W
hat
Yo
u L
earn
ed
5.L
ook
up
the
wor
d co
nst
rain
tin
a d
icti
onar
y.If
mor
e th
an o
ne
defi
nit
ion
is
give
n,c
hoo
seth
e on
e th
at s
eem
s cl
oses
t to
th
e id
ea o
f a
con
stra
int
in a
lin
ear
prog
ram
min
g pr
oble
m.
How
can
th
is d
efin
itio
n h
elp
you
to
rem
embe
r th
e m
ean
ing
of c
onst
rain
tas
it
is u
sed
inth
is l
esso
n?
Sam
ple
an
swer
:A
co
nst
rain
tis
a r
estr
icti
on
or
limit
atio
n.T
he
con
stra
ints
in a
lin
ear
pro
gra
mm
ing
pro
ble
m a
re r
estr
icti
on
s o
n t
he
vari
able
s th
at t
ran
slat
e in
to in
equ
alit
y st
atem
ents
.
vert
ices
solu
tio
ns
con
stra
ints
ineq
ual
itie
s
Answers (Lessons 3-3 and 3-4)
A1-A19 A2-03-873973 5/22/06 8:25 AM Page A12
Chapter 3 A13 Glencoe Algebra 2
An
swer
s
Cha
pter
328
Gle
ncoe
Alg
ebra
2
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
__P
ER
IOD
____
_
3-4
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Stud
y G
uide
and
Inte
rven
tion
Lin
ear
Pro
gra
mm
ing
Max
imu
m a
nd
Min
imu
m V
alu
esW
hen
a s
yste
m o
f li
nea
r in
equ
alit
ies
prod
uce
s a
bou
nde
d po
lygo
nal
reg
ion
,th
e m
axim
um
or m
inim
um
valu
e of
a r
elat
ed f
un
ctio
n w
ill
occu
rat
a v
erte
x of
th
e re
gion
.
Gra
ph
th
e sy
stem
of
ineq
ual
itie
s.N
ame
the
coor
din
ates
of
the
vert
ices
of
the
feas
ible
reg
ion
.Fin
d t
he
max
imu
m a
nd
min
imu
m v
alu
es o
f th
efu
nct
ion
f(x
,y)
�3x
�2y
for
this
pol
ygon
al r
egio
n.
y
4y
�
x�
6
y�
x�
y
6x�
4F
irst
fin
d th
e ve
rtic
es o
f th
e bo
un
ded
regi
on.G
raph
th
e in
equ
alit
ies.
Th
e po
lygo
n f
orm
ed i
s a
quad
rila
tera
l w
ith
ver
tice
s at
(0
,4),
(2,4
),(5
,1),
and
(�1,
�2)
.Use
th
e ta
ble
to f
ind
the
max
imu
m a
nd
min
imu
m v
alu
es o
f f(
x,y)
�3x
�2y
.
Th
e m
axim
um
val
ue
is 1
7 at
(5,
1).T
he
min
imu
m v
alu
e is
�7
at (
�1,
�2)
.
Gra
ph
eac
h s
yste
m o
f in
equ
alit
ies.
Nam
e th
e co
ord
inat
es o
f th
e ve
rtic
es o
f th
efe
asib
le r
egio
n.F
ind
th
e m
axim
um
an
d m
inim
um
val
ues
of
the
give
n f
un
ctio
n f
orth
is r
egio
n.
1.y
�2
2.y
��
23.
x�
y�
21
�x
�5
y�
2x�
44y
�x
�8
y�
x�
3x
�2y
��
1y
�2x
�5
f(x,
y) �
3x�
2yf(
x,y)
�4x
�y
f(x,
y) �
4x�
3y
vert
ices
:(1
,2),
(1,4
),ve
rtic
es:
(�5,
�2)
,ve
rtic
es (
0,2)
,(4,
3),
(5,8
),(5
,2);
max
:11
;(3
,2),
(1,�
2);
�,�
�;m
ax:
25;
min
:6
m
in;
�5
max
:10
;m
in:
�18
1 � 37 � 3
x
y
Ox
y
O
x
y
O
(x,y
)3x
�2y
f(x,
y)
(0,
4)3(
0) �
2(4)
8
(2,
4)3(
2) �
2(4)
14
(5,
1)3(
5) �
2(1)
17
(�1,
�2)
3(�
1) �
2(�
2)�
7
x
y O
3 � 21 � 2
Exam
ple
Exer
cise
s
Exam
ple
Lesson 3-4
Cha
pter
329
Gle
ncoe
Alg
ebra
2
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
__P
ER
IOD
____
_
3-4
Copyright ©Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Rea
l-W
orl
d P
rob
lem
sW
hen
sol
vin
g li
nea
r p
rogr
amm
ing
prob
lem
s,u
se t
he
foll
owin
g pr
oced
ure
.
1.D
efin
e va
riab
les.
2.W
rite
a s
yste
m o
f in
equ
alit
ies.
3.G
raph
th
e sy
stem
of
ineq
ual
itie
s.4.
Fin
d th
e co
ordi
nat
es o
f th
e ve
rtic
es o
f th
e fe
asib
le r
egio
n.
5.W
rite
an
exp
ress
ion
to
be m
axim
ized
or
min
imiz
ed.
6.S
ubs
titu
te t
he
coor
din
ates
of
the
vert
ices
in
th
e ex
pres
sion
.7.
Sel
ect
the
grea
test
or
leas
t re
sult
to
answ
er t
he
prob
lem
.
A p
ain
ter
has
exa
ctly
32
un
its
of y
ello
w d
ye a
nd
54
un
its
of g
reen
dye
.He
pla
ns
to m
ix a
s m
any
gall
ons
as p
ossi
ble
of
colo
r A
an
d c
olor
B.E
ach
gall
on o
f co
lor
A r
equ
ires
4 u
nit
s of
yel
low
dye
an
d 1
un
it o
f gr
een
dye
.Eac
hga
llon
of
colo
r B
req
uir
es 1
un
it o
f ye
llow
dye
an
d 6
un
its
of g
reen
dye
.Fin
d t
he
max
imu
m n
um
ber
of
gall
ons
he
can
mix
.
Ste
p 1
Def
ine
the
vari
able
s.x
�th
e n
um
ber
of g
allo
ns
of c
olor
A m
ade
y�
the
nu
mbe
r of
gal
lon
s of
col
or B
mad
e
Ste
p 2
Wri
te a
sys
tem
of
ineq
ual
itie
s.S
ince
th
e n
um
ber
of g
allo
ns
mad
e ca
nn
ot b
en
egat
ive,
x�
0 an
d y
�0.
Th
ere
are
32 u
nit
s of
yel
low
dye
;eac
h g
allo
n o
f co
lor
A r
equ
ires
4 u
nit
s,an
d ea
ch g
allo
n o
f co
lor
B r
equ
ires
1 u
nit
.S
o 4x
�y
�32
.S
imil
arly
for
th
e gr
een
dye
,x�
6y�
54.
Ste
ps
3 an
d 4
Gra
ph t
he
syst
em o
f in
equ
alit
ies
and
fin
d th
e co
ordi
nat
es o
f th
e ve
rtic
es o
fth
e fe
asib
le r
egio
n.T
he
vert
ices
of
the
feas
ible
reg
ion
are
(0,
0),(
0,9)
,(6,
8),a
nd
(8,0
).S
tep
s 5–
7F
ind
the
max
imu
m n
um
ber
of g
allo
ns,
x�
y,th
at h
e ca
n m
ake.
Th
e m
axim
um
nu
mbe
r of
gal
lon
s th
e pa
inte
r ca
n m
ake
is 1
4,6
gall
ons
of c
olor
A a
nd
8 ga
llon
s of
col
or B
.
1.FO
OD
A d
elic
ates
sen
has
12
pou
nds
of
plai
n s
ausa
ge a
nd
10 p
oun
ds o
f sp
icy
sau
sage
.
A p
oun
d of
Bra
twu
rst
A c
onta
ins
pou
nd
of p
lain
sau
sage
an
d po
un
d of
spi
cy
sau
sage
.A p
oun
d of
Bra
twu
rst
B c
onta
ins
pou
nd
of e
ach
sau
sage
.
Fin
d th
e m
axim
um
nu
mbe
r of
pou
nds
of
brat
wu
rst
that
can
be
mad
e.4
po
un
ds
of
Bra
twu
rst
A a
nd
18
po
un
ds
of
Bra
twu
rst
B
2.M
AN
UFA
CTU
RIN
GM
ach
ine
A c
an p
rodu
ce 3
0 st
eeri
ng
wh
eels
per
hou
r at
a c
ost
of $
8pe
r h
our.
Mac
hin
e B
can
pro
duce
40
stee
rin
g w
hee
ls p
er h
our
at a
cos
t of
$12
per
hou
r.T
he
com
pan
y ca
n u
se e
ith
er m
ach
ine
by i
tsel
f or
bot
h m
ach
ines
at
the
sam
e ti
me.
Wh
atis
th
e m
inim
um
nu
mbe
r of
hou
rs n
eede
d to
pro
duce
380
ste
erin
g w
hee
ls i
f th
e co
st m
ust
be n
o m
ore
than
$10
8?6
ho
urs
;6
ho
urs
on
Mac
hin
e A
an
d 5
ho
urs
on
Mac
hin
e B
sim
ult
aneo
usl
y
1 � 2
1 � 43 � 4
Co
lor
A (
gal
lon
s)
Color B (gallons)
510
1520
2530
3540
4550
550
40 35 30 25 20 15 10 5
( 6, 8
)
( 8, 0
)( 0
, 9)
Stud
y G
uide
and
Inte
rven
tion
(co
ntin
ued)
Lin
ear
Pro
gra
mm
ing
Exer
cise
s
Answers (Lesson 3-4)
A1-A19 A2-03-873973 5/22/06 8:25 AM Page A13
Chapter 3 A14 Glencoe Algebra 2
Gra
ph
eac
h s
yste
m o
f in
equ
alit
ies.
Nam
e th
e co
ord
inat
es o
f th
e ve
rtic
es o
f th
efe
asib
le r
egio
n.F
ind
th
e m
axim
um
an
d m
inim
um
val
ues
of
the
give
n f
un
ctio
n f
orth
is r
egio
n.
1.x
�2
2.x
�1
3.x
�0
x�
5y
�6
y�
0y
�1
y�
x�
2y
�7
�x
y�
4f(
x,y)
�x
�y
f(x,
y) �
3x�
yf(
x,y)
�x
�y
max
.:9,
min
.:3
max
.:2,
min
.:�
5m
ax.:
21,m
in.:
0
4.x
��
15.
y�
2x6.
y�
�x
�2
x�
y�
6y
�6
�x
y�
3x�
2f(
x,y)
�x
�2y
y�
6y
�x
�4
f(x,
y) �
4x�
3yf(
x,y)
��
3x�
5y
max
.:13
,no
min
.n
o m
ax.,
min
.:20
max
.:22
,min
.:�
2
7.M
AN
UFA
CTU
RIN
GA
bac
kpac
k m
anu
fact
ure
r pr
odu
ces
an i
nte
rnal
fra
me
pack
an
d an
exte
rnal
fra
me
pack
.Let
xre
pres
ent
the
nu
mbe
r of
in
tern
al f
ram
e pa
cks
prod
uce
d in
one
hou
r an
d le
t y
repr
esen
t th
e n
um
ber
of e
xter
nal
fra
me
pack
s pr
odu
ced
in o
ne
hou
r.T
hen
th
e in
equ
alit
ies
x�
3y�
18,2
x�
y�
16,x
�0,
and
y�
0 de
scri
be t
he
con
stra
ints
for
man
ufa
ctu
rin
g bo
th p
acks
.Use
th
e pr
ofit
fu
nct
ion
f(x
) �
50x
�80
yan
d th
eco
nst
rain
ts g
iven
to
dete
rmin
e th
e m
axim
um
pro
fit
for
man
ufa
ctu
rin
g bo
th b
ackp
acks
for
the
give
n c
onst
rain
ts.
$620
( 1, 5
)
( –1,
–1)
( –3,
1)
x
y
O
( 3, 6
)
( 2, 4
)
x
y
O
( –1,
7)
x
y
O
x
y
O
( 0, 7
)
( 0, 0
)( 7
, 0)
x
y
O
( 1, 6
)
( 1, –
1)
( 8, 6
)
x
y
O
( 2, 4
)
( 2, 1
)( 5
, 1)
( 5, 4
)
Cha
pter
330
Gle
ncoe
Alg
ebra
2
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
__P
ER
IOD
____
_
3-4
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Skill
s Pr
acti
ceL
inea
r P
rog
ram
min
g
3-4
Lesson 3-4
Cha
pter
331
Gle
ncoe
Alg
ebra
2
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
__P
ER
IOD
____
_
3-4
Copyright ©Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Gra
ph
eac
h s
yste
m o
f in
equ
alit
ies.
Nam
e th
e co
ord
inat
es o
f th
e ve
rtic
es o
f th
efe
asib
le r
egio
n.F
ind
th
e m
axim
um
an
d m
inim
um
val
ues
of
the
give
n f
un
ctio
n f
orth
is r
egio
n.
1.2x
�4
�y
2.3x
�y
�7
3.x
�0
�2x
�4
�y
2x�
y�
3y
�0
y�
2y
�x
�3
y�
6f(
x,y)
��
2x�
yf(
x,y)
�x
�4y
y�
�3x
�15
f(x,
y) �
3x�
y
max
.:8,
min
.:�
4m
ax.:
12,m
in.:
�16
max
.:15
,min
.:0
4.x
�0
5.y
�3x
�6
6.2x
�3y
�6
y�
04y
�3x
�3
2x�
y�
24x
�y
��
7x
��
2x
�0
f(x,
y) �
�x
�4y
f(x,
y) �
�x
�3y
y�
0f(
x,y)
�x
�4y
�3
max
.:28
,min
.:0
max
.:,n
o m
in.
no
max
.,m
in.:
PRO
DU
CTI
ON
For
Exe
rcis
es 7
–9,u
se t
he
foll
owin
g in
form
atio
n.
A g
lass
blo
wer
can
for
m 8
sim
ple
vase
s or
2 e
labo
rate
vas
es i
n a
n h
our.
In a
wor
k sh
ift
of n
om
ore
than
8 h
ours
,th
e w
orke
r m
ust
for
m a
t le
ast
40 v
ases
.
7.L
et s
repr
esen
t th
e h
ours
for
min
g si
mpl
e va
ses
and
eth
e h
ours
for
min
g el
abor
ate
vase
s.W
rite
a s
yste
m o
f in
equ
alit
ies
invo
lvin
g th
e ti
me
spen
t on
eac
h t
ype
of v
ase.
s�
0,e
�0,
s �
e
8,8s
�2e
�40
8.If
th
e gl
ass
blow
er m
akes
a p
rofi
t of
$30
per
hou
r w
orke
d on
th
e si
mpl
e va
ses
and
$35
per
hou
r w
orke
d on
th
e el
abor
ate
vase
s,w
rite
a f
un
ctio
n f
or t
he
tota
l pr
ofit
on
th
e va
ses.
f(s,
e) �
30s
�35
e
9.F
ind
the
nu
mbe
r of
hou
rs t
he
wor
ker
shou
ld s
pen
d on
eac
h t
ype
of v
ase
to m
axim
ize
prof
it.W
hat
is
that
pro
fit?
4 h
on
eac
h;
$260
17 � 234 � 5
x
y
O
(3 – 2, 1)
( 0, 2
)
x
y O
(–7 – 5, 9 – 5)
( –2,
0)
x
y
O(–
7 – 4, 0)
( 0, 0
)
( 0, –
7)
( 3, 6
)
( 5, 0
)
( 0, 6
)
( 0, 0
)x
y
O
( 2, –
1)
( 4, 5
)
( 0, –
3)
x
y
O
( –3,
2)
( 3, 2
)
( 0, –
4)
x
y
OPrac
tice
L
inea
r P
rog
ram
min
g
Answers (Lesson 3-4)
A1-A19 A2-03-873973 5/22/06 8:25 AM Page A14
Chapter 3 A15 Glencoe Algebra 2
An
swer
s
Wor
d Pr
oble
m P
ract
ice
Lin
ear
Pro
gra
mm
ing
Cha
pter
332
Gle
ncoe
Alg
ebra
2
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
__P
ER
IOD
____
_
3-4
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
1.R
EGIO
NS
A r
egio
n i
n t
he
plan
e is
form
ed b
y th
e eq
uat
ion
s x
�y
�3,
x�
y�
�3,
and
x�
y�
�3.
Is t
his
regi
on b
oun
ded
or u
nbo
un
ded?
Exp
lain
.T
he
reg
ion
is u
nb
ou
nd
edb
ecau
se it
is o
pen
.Th
e p
oin
ts
(n,n
) ar
e in
th
e re
gio
n f
or
all
po
siti
ve v
alu
es o
f n
.
2.M
AN
UFA
CTU
RIN
GE
igh
ty w
orke
rsar
e av
aila
ble
to a
ssem
ble
tabl
es a
nd
chai
rs.I
t ta
kes
5 pe
ople
to
asse
mbl
e a
tabl
e an
d 3
peop
le t
o as
sem
ble
a ch
air.
Th
e w
orke
rs a
lway
s m
ake
at l
east
as
man
y ta
bles
as
chai
rs b
ecau
se t
he
tabl
esar
e ea
sier
to
mak
e.If
xis
th
e n
um
ber
ofta
bles
an
d y
is t
he
nu
mbe
r of
ch
airs
,th
esy
stem
of
ineq
ual
itie
s th
at r
epre
sen
tw
hat
can
be
asse
mbl
ed i
s x
�0,
y�
0,y
�x,
and
5x�
3y�
80.W
hat
is
the
max
imu
m t
otal
nu
mbe
r of
ch
airs
an
dta
bles
th
e w
orke
rs c
an m
ake?
10 t
able
s an
d 1
0 ch
airs
3.FI
SHA
n a
quar
ium
is
2000
cu
bic
inch
es.
Nat
han
wan
ts t
o po
pula
te t
he
aqu
ariu
mw
ith
neo
n t
etra
s an
d ca
tfis
h.I
t is
reco
mm
ende
d th
at e
ach
neo
n t
etra
be
allo
wed
50
cubi
c in
ches
an
d ea
ch c
atfi
shbe
all
owed
200
cu
bic
inch
es o
f sp
ace.
Nat
han
wou
ld l
ike
at l
east
on
e ca
tfis
hfo
r ev
ery
4 n
eon
tet
ras.
Let
nbe
th
en
um
ber
of n
eon
tet
ra a
nd
cbe
th
en
um
ber
of c
atfi
sh.T
he
foll
owin
gin
equ
alit
ies
form
th
e fe
asib
le r
egio
n f
orth
is s
itu
atio
n:n
�0,
c�
0,4c
�n
,an
d50
n�
200c
�20
00.W
hat
is
the
max
imu
m n
um
ber
of f
ish
Nat
han
can
put
in h
is a
quar
ium
?20
�5
�25
4.EL
EVA
TIO
N A
tra
pezo
idal
par
k is
bu
ilt
on a
sli
ght
incl
ine.
Th
e fu
nct
ion
for
th
egr
oun
d el
evat
ion
abo
ve s
ea l
evel
is
f(x,
y) �
x�
3y�
20 f
eet.
Wh
at a
re t
he
coor
din
ates
of
the
hig
hes
t po
int
in t
he
park
?
(5,2
)
CER
AM
ICS
For
Exe
rcis
es 5
–7,u
se t
he
foll
owin
g in
form
atio
n.
Josh
has
8 d
ays
to m
ake
pots
an
d pl
ates
to
sell
at
a lo
cal
fair
.Eac
h p
ot w
eigh
s 2
pou
nds
and
each
pla
te w
eigh
s 1
pou
nd.
Josh
can
not
carr
y m
ore
than
50
pou
nds
to
the
fair
.Eac
hda
y,h
e ca
n m
ake
at m
ost
5 pl
ates
an
d at
mos
t 3
pots
.He
wil
l m
ake
$12
prof
it f
orev
ery
plat
e an
d $2
5 pr
ofit
for
eve
ry p
ot t
hat
he
sell
s.
5.W
rite
lin
ear
ineq
ual
itie
s to
rep
rese
nt
the
nu
mbe
r of
pot
s p
and
plat
es a
Josh
may
bri
ng
to t
he
fair
.a
�0,
p�
0,2p
�a
50
,a
40,
and
p
24.
6.L
ist
the
coor
din
ates
of
the
vert
ices
of
the
feas
ible
reg
ion
.(0
,0),
(0,2
4),(
40,0
),(2
,24)
,(4
0,5)
wh
ere
the
ho
rizo
nta
l axi
sis
tak
en t
o r
epre
sen
t a
7.H
ow m
any
pots
an
d h
ow m
any
plat
essh
ould
Jos
h m
ake
to m
axim
ize
his
pote
nti
al p
rofi
t?2
pla
tes
and
24
po
ts
y
xO
5
5
Enri
chm
ent
Cha
pter
333
Gle
ncoe
Alg
ebra
2
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
__P
ER
IOD
____
_
3-4
Copyright ©Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Lesson 3-4
Sen
siti
vity
An
alys
isA
lin
ear
prog
ram
min
g m
odel
has
spe
cifi
c ob
ject
ive
coef
fici
ents
.For
exa
mpl
e,if
th
e va
lue
of a
mod
el i
s fo
un
d by
2x
�3y
�5,
the
obje
ctiv
e co
effi
cien
ts a
re {
2,3}
.Wh
at i
f th
ese
coef
fici
ents
wer
e {2
.1,2
.9}
or {
2.5,
3.1}
? H
ow w
ould
th
ese
chan
ges
affe
ct t
he
opti
mal
lin
ear
prog
ram
valu
e? T
his
typ
e of
in
vest
igat
ion
is
call
ed s
ensi
tivi
ty a
nal
ysis
.
In g
ener
al,t
he
obje
ctiv
e fu
nct
ion
in
tw
o-va
riab
le l
inea
r pr
ogra
mm
ing
prob
lem
can
be
wri
tten
as:
max
imiz
e (o
r m
inim
ize)
Ax
�B
y�
C,s
ubj
ect
to a
set
of
con
stra
int
equ
atio
ns.
Ch
ange
s to
th
e pa
ram
eter
sA
and
Bco
uld
ch
ange
th
e sl
ope
of t
he
lin
e.T
his
ch
ange
of
slop
eco
uld
lea
d to
a c
han
ge i
n t
he
opti
mu
m s
olu
tion
to
a di
ffer
ent
corn
er p
oin
t (R
ecal
l,th
eop
tim
um
sol
uti
on o
ccu
rs a
t a
corn
er p
oin
t).
Th
ere
is a
ran
ge i
n t
he
slop
e va
lue
that
wil
l pr
odu
ce t
his
chan
ge,t
hu
s th
ere
is a
ran
ge o
f va
riat
ion
for
bot
h A
and
Bth
at w
ill
keep
th
e op
tim
al s
olu
tion
th
e sa
me
(see
gra
ph).
1.F
ind
the
slop
e of
Ax
�B
y�
Can
d ob
serv
e h
owch
ange
s to
th
e pa
ram
eter
s A
and
Bca
n c
han
ge t
he
slop
e of
th
e li
ne.
Th
e sl
op
e,m
,of
Ax
�B
y�
C is
m�
.
Con
sid
er t
he
Lin
ear
Pro
gram
min
g p
rob
lem
:
Max
imiz
e:C
�2x
�3y
,S
ubj
ect
to:
3x �
y�
21x
�y
�9
y�
xy
�4
Aft
er f
indi
ng
the
inte
rsec
tion
s an
d ev
alu
atin
g th
e ob
ject
ive
equ
atio
n,w
e fi
nd
the
max
imal
solu
tion
is
(5,4
).If
th
e ob
ject
ive
coef
fici
ents
are
ch
ange
d fr
om 2
an
d 3
to A
and
B,t
he
opti
mu
m s
olu
tion
wit
h r
emai
n a
t (5
,4)
wh
ile
the
slop
e re
mai
ns
betw
een
th
e sl
ope
of
x�
y�
9 an
d th
e sl
ope
of 3
x�
y�
21.I
f n
ot,t
hen
th
e n
ew o
ptim
al s
olu
tion
wil
l be
at
(4,4
) or
(6,
3).
2.E
xpre
ss t
he
rela
tion
ship
,th
e sl
ope
of t
he
obje
ctiv
e fu
nct
ion
is
betw
een
th
e sl
ope
of t
he
lin
e x
�y
�9
and
the
slop
e of
th
e li
ne
3x�
y�
21,a
lgeb
raic
ally
.
�3
�1
3.D
eter
min
e th
e ra
nge
on
Aif
Bre
mai
ns
equ
al t
o 3.
�9
A
�
3
A � B
�A
�B
(x,
y)(0
, 0)
(4,
4)(5
, 4)
(6,
3)(7
, 0)
C0
2022
2114
y
xO
Answers (Lesson 3-4)
A1-A19 A2-03-873973 5/22/06 8:25 AM Page A15
Chapter 3 A16 Glencoe Algebra 2
Answers (Lessons 3-4 and 3-5)
Exam
ple
Cha
pter
334
Gle
ncoe
Alg
ebra
2
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
__P
ER
IOD
____
_
3-4
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Gra
ph
th
e sy
stem
x�
3y�
�7,
5x�
y
13,x
�6y
��
9,3x
�2y
��
7,an
d f
(x,y
)�4x
�3y
.Fin
d t
he
coor
din
ates
of
the
feas
ible
regi
on.T
hen
fin
d t
he
max
imu
m a
nd
min
imu
m v
alu
es f
or t
he
syst
em.
Sol
ve e
ach
in
equ
alit
y fo
r y.
En
ter
each
bou
nda
ry e
quat
ion
in
th
e Y
�sc
reen
.Fin
d th
e ve
rtic
es o
f th
e fe
asib
le r
egio
n.T
hen
fin
d th
e va
lues
of f
(x,y
) to
det
erm
ine
the
max
imu
m a
nd
min
imu
m v
alu
es.
Key
stro
kes:
1 3
7 3
5 13
1
6 3
2 3
2 7
2 6
[CA
LC
] 5
[QU
IT]
[ {
]
[Y]
4 3
[Y]
[ }
]
[CA
LC
] 5
[QU
IT]
[EN
TR
Y]
[CA
LC
] 5
[QU
IT]
[EN
TR
Y]
[CA
LC
] 5
[QU
IT]
[EN
TR
Y]
.
The
max
imum
val
ue o
f the
sys
tem
is 1
8 an
d th
e m
inim
um v
alue
is�
10.
EN
TER
2nd
2nd
EN
TER
EN
TER
EN
TER
2nd
GR
AP
HE
NTE
R2n
d2n
dE
NTE
RE
NTE
R
EN
TER
2nd
GR
AP
HE
NTE
R2n
d
2nd
EN
TER
EN
TER
EN
TER
2nd
GR
AP
H
EN
TER
2nd
ALP
HA
—,
ALP
HA
,2n
d2n
dE
NTE
RE
NTE
RE
NTE
R2n
d
ZO
OM
)�
(+
)�
(E
NTE
R)
�(
—)
�(
(–)
EN
TER
+(–
)
EN
TER
)�
(+
)�
(Y
=Gra
phin
g Ca
lcul
ator
Act
ivit
yL
inea
r P
rog
ram
min
gA
gra
phin
g ca
lcu
lato
r ca
n s
tore
th
e x-
an
d y-
coor
din
ates
wh
en u
sin
g th
ein
ters
ect
com
man
d in
th
e [C
AL
C]
men
u.T
his
can
be
disp
laye
d on
th
eh
ome
scre
en a
nd
use
d to
eva
luat
e an
exp
ress
ion
wit
h x
and
yva
riab
les.
Th
ispr
oces
s is
use
ful i
n fi
ndin
g th
e ve
rtic
es o
f th
e fe
asib
le r
egio
n an
d de
term
inin
gth
e m
axim
um
or
min
imu
m v
alu
e fo
r f(
x,y)
.
[�10
, 10]
scl
:1 b
y [�
10, 1
0] s
cl:1
Gra
ph
eac
h s
yste
m.F
ind
th
e co
ord
inat
es o
f th
e ve
rtic
es o
f th
e fe
asib
lere
gion
.Th
en f
ind
th
e m
axim
um
an
d m
inim
um
val
ues
for
th
e sy
stem
.
1.2x
�3y
�6
2.y
�4x
�6
3.y
�16
�x
3x�
2y�
�4
x�
4y�
70
�2y
�17
5x�
y�
152x
�y
�7
2x�
3y�
11f(
x,y)
�x
�3y
x�
6y�
10y
�3x
�1
f(x,
y) �
2x�
yy
�2x
�13
y�
7 �
2 xf(
x,y)
�5x
�6y
[�10
, 10]
scl
:1 b
y [�
10, 1
0] s
cl: 1
[�10
, 10]
scl
:1 b
y [�
10, 1
0] s
cl: 1
[�10
, 10]
scl
:1 b
y [�
10, 1
0] s
cl: 1
(0,2
),(3
,0),
(2,5
);(�
1,2)
,(�
2,�
2),(
3,1)
,(5
.5,0
),(6
.5,0
),(7
.5,8
.5),
min
.�3,
max
.�17
(4,�
1);
min
.��
4,m
ax.�
9(1
.2,4
.6),
(2.5
,2),
(9.6
6,6.
33);
min
.�24
.5,m
ax.�
88.5
Exer
cise
s
Lesson 3-4
Cha
pter
335
Gle
ncoe
Alg
ebra
2
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
__P
ER
IOD
____
_
3-5
Copyright ©Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Less
on R
eadi
ng G
uide
So
lvin
g S
yste
ms
of
Eq
uat
ion
s in
Th
ree
Var
iab
les
Get
Rea
dy
for
the
Less
on
Rea
d t
he
intr
odu
ctio
n t
o L
esso
n 3
-5 i
n y
our
text
boo
k.
At
the
1960
Su
mm
er O
lym
pics
in
Rom
e,It
aly,
the
Un
ited
Sta
tes
won
71
med
als.
Th
e U
.S.
team
won
13
mor
e go
ld m
edal
s th
an s
ilve
r an
d 5
few
er b
ron
ze m
edal
s th
an s
ilve
r.U
sin
g th
esa
me
vari
able
s as
th
ose
in t
he
intr
odu
ctio
n,w
rite
a s
yste
m o
f eq
uat
ion
s th
at d
escr
ibes
th
em
edal
s w
on f
or t
he
1960
Oly
mpi
cs.
g�
s�
b�
71;
g�
s�
13;
b�
s�
5
Rea
d t
he
Less
on
1.T
he
plan
es f
or t
he
equ
atio
ns
in a
sys
tem
of
thre
e li
nea
r eq
uat
ion
s in
th
ree
vari
able
sde
term
ine
the
nu
mbe
r of
sol
uti
ons.
Mat
ch e
ach
gra
ph d
escr
ipti
on b
elow
wit
h t
he
desc
ript
ion
of
the
nu
mbe
r of
sol
uti
ons
of t
he
syst
em.(
Som
e of
th
e it
ems
on t
he
righ
t m
aybe
use
d m
ore
than
on
ce,a
nd
not
all
pos
sibl
e ty
pes
of g
raph
s ar
e li
sted
.)
a.th
ree
para
llel
pla
nes
I.
one
solu
tion
b.
thre
e pl
anes
th
at i
nte
rsec
t in
a l
ine
II.
no
solu
tion
s
c.th
ree
plan
es t
hat
in
ters
ect
in o
ne
poin
t II
I.in
fin
ite
solu
tion
s
d.
one
plan
e th
at r
epre
sen
ts a
ll t
hre
e eq
uat
ion
s
2.S
upp
ose
that
th
ree
clas
smat
es,M
oniq
ue,
Josh
,an
d L
illy
,are
stu
dyin
g fo
r a
quiz
on
th
isle
sson
.Th
ey w
ork
toge
ther
on
sol
vin
g a
syst
em o
f eq
uat
ion
s in
th
ree
vari
able
s,x,
y,an
dz,
foll
owin
g th
e al
gebr
aic
met
hod
sh
own
in
you
r te
xtbo
ok.T
hey
fir
st f
ind
that
z�
3,th
enth
at y
��
2,an
d fi
nal
ly t
hat
x�
�1.
Th
e st
ude
nts
agr
ee o
n t
hes
e va
lues
,bu
t di
sagr
eeon
how
to
wri
te t
he
solu
tion
.Her
e ar
e th
eir
answ
ers:
Mon
iqu
e:(3
,�2,
�1)
Josh
:(�
2,�
1,3)
Lil
ly:(
�1,
�2,
3)
a.H
ow d
o yo
u t
hin
k ea
ch s
tude
nt
deci
ded
on t
he
orde
r of
th
e n
um
bers
in
th
e or
dere
dtr
iple
?S
amp
le a
nsw
er:
Mo
niq
ue
arra
ng
ed t
he
valu
es in
th
e o
rder
inw
hic
h s
he
fou
nd
th
em.J
osh
arr
ang
ed t
hem
fro
m s
mal
lest
to
larg
est.
Lill
y ar
ran
ged
th
em in
alp
hab
etic
al o
rder
of
the
vari
able
s.
b.
Wh
ich
stu
den
t is
cor
rect
?L
illy
Rem
emb
er W
hat
Yo
u L
earn
ed
3.H
ow c
an y
ou r
emem
ber
that
obt
ain
ing
the
equ
atio
n 0
�0
indi
cate
s a
syst
em w
ith
infi
nit
ely
man
y so
luti
ons,
wh
ile
obta
inin
g an
equ
atio
n s
uch
as
0 �
8 in
dica
tes
a sy
stem
wit
h n
o so
luti
ons?
0 �
0 is
alw
ays
tru
e,w
hile
0 �
8 is
nev
er t
rue.
III
I
III
II
A1-A19 A2-03-873973 5/22/06 8:25 AM Page A16
Chapter 3 A17 Glencoe Algebra 2
An
swer
s
Answers (Lesson 3-5)
Cha
pter
336
Gle
ncoe
Alg
ebra
2
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
__P
ER
IOD
____
_
3-5
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Stud
y G
uide
and
Inte
rven
tion
So
lvin
g S
yste
ms
of
Eq
uat
ion
s in
Th
ree
Var
iab
les
Syst
ems
in T
hre
e V
aria
ble
sU
se t
he
met
hod
s u
sed
for
solv
ing
syst
ems
of l
inea
req
uat
ion
s in
tw
o va
riab
les
to s
olve
sys
tem
s of
equ
atio
ns
in t
hre
e va
riab
les.
A s
yste
m o
fth
ree
equ
atio
ns
in t
hre
e va
riab
les
can
hav
e a
un
iqu
e so
luti
on,i
nfi
nit
ely
man
y so
luti
ons,
orn
o so
luti
on.A
sol
uti
on i
s an
ord
ered
tri
ple
.
Sol
ve t
his
sys
tem
of
equ
atio
ns.
3x�
y�
z�
�6
2x�
y�
2z�
84x
�y
�3z
��
21
Ste
p 1
Use
eli
min
atio
n t
o m
ake
a sy
stem
of
two
equ
atio
ns
in t
wo
vari
able
s.3x
�y
�z
��
6F
irst
equa
tion
2x�
y�
2z�
8S
econ
d eq
uatio
n
(�)
2x�
y�
2z�
8S
econ
d eq
uatio
n(�
) 4x
�y
�3z
��
21T
hird
equ
atio
n
5x�
z�
2A
dd t
o el
imin
ate
y.6x
�
z�
�13
Add
to
elim
inat
e y.
Ste
p 2
Sol
ve t
he
syst
em o
f tw
o eq
uat
ion
s.5x
�z
�2
(�)
6x�
z�
�13
11x
��
11A
dd t
o el
imin
ate
z.
x�
�1
Div
ide
both
sid
es b
y 11
.
Su
bsti
tute
�1
for
xin
on
e of
th
e eq
uat
ion
s w
ith
tw
o va
riab
les
and
solv
e fo
r z.
5x�
z�
2E
quat
ion
with
tw
o va
riabl
es
5(�
1) �
z�
2R
epla
ce x
with
�1.
�5
�z
�2
Mul
tiply
.
z�
7A
dd 5
to
both
sid
es.
Th
e re
sult
so
far
is x
��
1 an
d z
�7.
Ste
p 3
Su
bsti
tute
�1
for
xan
d 7
for
zin
on
e of
th
e or
igin
al e
quat
ion
s w
ith
th
ree
vari
able
s.3x
�y
�z
��
6O
rigin
al e
quat
ion
with
thr
ee v
aria
bles
3(�
1) �
y�
7 �
�6
Rep
lace
xw
ith �
1 an
d z
with
7.
�3
�y
�7
��
6M
ultip
ly.
y�
4S
impl
ify.
Th
e so
luti
on i
s (�
1,4,
7).
Sol
ve e
ach
sys
tem
of
equ
atio
ns.
1.2x
�3y
�z
�0
2.2x
�y
�4z
�11
3.x
�2y
�z
�8
x�
2y�
4z�
14x
�2y
�6z
��
112x
�y
�z
�0
3x�
y�
8z�
173x
�2y
�10
z�
113x
�6y
�3z
�24
(4,�
3,�
1)�2,
�5,
�in
fin
itel
y m
any
solu
tio
ns
4.3x
�y
�z
�5
5.2x
�4y
�z
�10
6.x
�6y
�4z
�2
3x�
2y�
z�
114x
�8y
�2z
�16
2x�
4y�
8z�
166x
�3y
�2z
��
123x
�y
�z
�12
x�
2y�
5
�,2
,�5 �
no
so
luti
on
infi
nit
ely
man
yso
luti
on
s
2 � 3
1 � 2
Exam
ple
Exer
cise
s
Exam
ple
Lesson 3-5
Cha
pter
337
Gle
ncoe
Alg
ebra
2
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
__P
ER
IOD
____
_
3-5
Copyright ©Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Rea
l-W
orl
d P
rob
lem
s
Th
e L
ared
o S
por
ts S
hop
sol
d 1
0 b
alls
,3 b
ats,
and
2 b
ases
for
$99
on M
ond
ay.O
n T
ues
day
th
ey s
old
4 b
alls
,8 b
ats,
and
2 b
ases
for
$78
.On
Wed
nes
day
th
ey s
old
2 b
alls
,3 b
ats,
and
1 b
ase
for
$33.
60.W
hat
are
th
e p
rice
s of
1b
all,
1 b
at,a
nd
1 b
ase?
Fir
st d
efin
e th
e va
riab
les.
x�
pric
e of
1 b
all
y�
pric
e of
1 b
atz
�pr
ice
of 1
bas
e
Tra
nsl
ate
the
info
rmat
ion
in
th
e pr
oble
m i
nto
th
ree
equ
atio
ns.
10x
�3y
�2z
�99
4x�
8y�
2z�
782x
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�33
.60
Stud
y G
uide
and
Inte
rven
tion
(co
ntin
ued)
So
lvin
g S
yste
ms
of
Eq
uat
ion
s in
Th
ree
Var
iab
les
Su
btra
ct t
he
seco
nd
equ
atio
n f
rom
th
e fi
rst
equ
atio
n t
o el
imin
ate
z.10
x�
3y�
2z�
99(�
) 4x
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�78
6x�
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21
Mu
ltip
ly t
he
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d eq
uat
ion
by
2 an
dsu
btra
ct f
rom
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e se
con
d eq
uat
ion
.4x
�8y
�2z
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(�)
4x�
6y�
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67.2
02y
�10
.80
y�
5.40
Su
bsti
tute
5.4
0 fo
r y
in t
he
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atio
n
6x�
5y�
21.
6x�
5(5.
40)
�21
6x�
48x
�8
Su
bsti
tute
8 f
or x
and
5.40
for
yin
on
e of
the
orig
inal
equ
atio
ns
to s
olve
for
z.
10x
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�2z
�99
10(8
) �
3(5.
40)
�2z
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80 �
16.2
0 �
2z�
992z
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80z
�1.
40
So
a ba
ll c
osts
$8,
a ba
t $5
.40,
and
a ba
se $
1.40
.
1.FI
TNES
S TR
AIN
ING
Car
ly i
s tr
ain
ing
for
a tr
iath
lon
.In
her
tra
inin
g ro
uti
ne
each
wee
k,sh
e ru
ns
7 ti
mes
as
far
as s
he
swim
s,an
d sh
e bi
kes
3 ti
mes
as
far
as s
he
run
s.O
ne
wee
ksh
e tr
ain
ed a
tot
al o
f 23
2 m
iles
.How
far
did
sh
e ru
n t
hat
wee
k?56
mile
s
2.EN
TERT
AIN
MEN
TA
t th
e ar
cade
,Rya
n,S
ara,
and
Tim
pla
yed
vide
o ra
cing
gam
es,p
inba
ll,
and
air
hoc
key.
Rya
n s
pen
t $6
for
6 r
acin
g ga
mes
,2 p
inba
ll g
ames
,an
d 1
gam
e of
air
hock
ey.S
ara
spen
t $1
2 fo
r 3
raci
ng g
ames
,4 p
inba
ll g
ames
,and
5 g
ames
of
air
hock
ey.T
imsp
ent
$12.
25 f
or 2
rac
ing
gam
es,7
pin
ball
gam
es,a
nd 4
gam
es o
f ai
r ho
ckey
.How
muc
h di
dea
ch o
f th
e ga
mes
cos
t?ra
cin
g g
ame:
$0.5
0;p
inb
all:
$0.7
5;ai
r h
ock
ey:
$1.5
0
3.FO
OD
A n
atu
ral
food
sto
re m
akes
its
ow
n b
ran
d of
tra
il m
ix o
ut
of d
ried
app
les,
rais
ins,
and
pean
uts
.On
e po
un
d of
th
e m
ixtu
re c
osts
$3.
18.I
t co
nta
ins
twic
e as
mu
ch p
ean
uts
by w
eigh
t as
app
les.
On
e po
un
d of
dri
ed a
pple
s co
sts
$4.4
8,a
pou
nd
of r
aisi
ns
$2.4
0,an
da
pou
nd
of p
ean
uts
$3.
44.H
ow m
any
oun
ces
of e
ach
in
gred
ien
t ar
e co
nta
ined
in
1 p
oun
dof
th
e tr
ail
mix
?3
oz
of
app
les,
7 o
z o
f ra
isin
s,6
oz
of
pea
nu
ts
Exer
cise
s
A1-A19 A2-03-873973 5/22/06 8:25 AM Page A17
Chapter 3 A18 Glencoe Algebra 2
Answers (Lesson 3-5)
Skill
s Pr
acti
ceS
olv
ing
Sys
tem
s o
f E
qu
atio
ns
in T
hre
e V
aria
ble
sS
olve
eac
h s
yste
m o
f eq
uat
ion
s.
1.2a
�c
��
10(5
,�5,
�20
)2.
x�
y�
z�
3(0
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a�
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c�
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no
so
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on
5x�
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3x�
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10.5
m�
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63m
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y�
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n�
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8
11.2
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nit
ely
man
y12
.x�
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z�
4(1
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y�
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6
13.3
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��
12in
fin
itel
y m
any
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x�
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z�
39�
3x�
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12
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x�
y�
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1)16
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x�
y�
z�
�3
x�
2y�
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��
125x
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2x�
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nit
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y3x
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x�
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z�
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19.T
he
sum
of
thre
e n
um
bers
is
18.T
he
sum
of
the
firs
t an
d se
con
d n
um
bers
is
15,
and
the
firs
t n
um
ber
is 3
tim
es t
he
thir
d n
um
ber.
Fin
d th
e n
um
bers
.9,
6,3
Cha
pter
338
Gle
ncoe
Alg
ebra
2
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
__P
ER
IOD
____
_
3-5
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Lesson 3-5
Cha
pter
339
Gle
ncoe
Alg
ebra
2
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
__P
ER
IOD
____
_
3-5
Copyright ©Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Sol
ve e
ach
sys
tem
of
equ
atio
ns.
1.2x
�y
�2z
�15
2.x
�4y
�3z
��
273.
a�
b�
3�
x�
y�
z�
32x
�2y
�3z
�22
�b
�c
�3
3x�
y�
2z�
184z
��
16a
�2c
�10
(3,1
,5)
(1,4
,�4)
(2,1
,4)
4.3m
�2n
�4p
�15
5.2g
�3h
�8j
�10
6.2x
�y
�z
��
8m
�n
�p
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g�
4h�
14x
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��
3m
�4n
�5p
�0
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�5
�3x
�y
�2z
�5
(3,3
,3)
no
so
luti
on
(�2,
�3,
1)
7.2x
�5y
�z
�5
8.2x
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�2
9.p
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��
73x
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84x
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x�
5y�
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q�
r�
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2)(1
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2)
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x�
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811
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f�
012
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93x
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e�
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infi
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y(1
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2)(2
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4)
13.5
x�
9y�
z�
2014
.2x
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315
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2x�
y�
z�
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3x�
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55x
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5x�
2y�
2z�
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�9
3x�
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3z�
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(�7,
6,1)
infi
nit
ely
man
y(1
,3,�
2)
16.2
u�
v�
w�
217
.x�
5y�
3z�
�18
18.x
�2y
�z
��
1�
3u�
2v�
3w�
73x
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�z
�6
�u
�v
�2w
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�28
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(0,�
1,3)
(1,�
2,3)
no
so
luti
on
19.2
x�
2y�
4z�
�2
20.x
�y
�9z
��
2721
.2x
�5y
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�7
3x�
3y�
6z�
�3
2x�
4y�
z�
�1
�4x
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y�
2z�
6�
2x�
3y�
z�
73x
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6x�
15y
�z
��
19(4
,5,0
)(2
,2,�
3)in
fin
itel
y m
any
22.T
he
sum
of
thre
e n
um
bers
is
6.T
he
thir
d n
um
ber
is t
he
sum
of
the
firs
t an
d se
con
dn
um
bers
.Th
e fi
rst
nu
mbe
r is
on
e m
ore
than
th
e th
ird
nu
mbe
r.F
ind
the
nu
mbe
rs.
4,�
1,3
23.T
he
sum
of
thre
e n
um
bers
is
�4.
Th
e se
con
d n
um
ber
decr
ease
d by
th
e th
ird
is e
qual
to
the
firs
t.T
he
sum
of
the
firs
t an
d se
con
d n
um
bers
is
�5.
Fin
d th
e n
um
bers
.�
3,�
2,1
24.S
POR
TSA
lexa
ndr
ia H
igh
Sch
ool
scor
ed 3
7 po
ints
in
a f
ootb
all
gam
e.S
ix p
oin
ts a
reaw
arde
d fo
r ea
ch t
ouch
dow
n.A
fter
eac
h t
ouch
dow
n,t
he
team
can
ear
n o
ne
poin
t fo
r th
eex
tra
kick
or
two
poin
ts f
or a
2-p
oin
t co
nve
rsio
n.T
he
team
sco
red
one
few
er 2
-poi
nt
con
vers
ion
s th
an e
xtra
kic
ks.T
he
team
sco
red
10 t
imes
du
rin
g th
e ga
me.
How
man
yto
uch
dow
ns
wer
e m
ade
duri
ng
the
gam
e?5
Prac
tice
S
olv
ing
Sys
tem
s o
f E
qu
atio
ns
in T
hre
e V
aria
ble
s
A1-A19 A2-03-873973 5/22/06 8:25 AM Page A18
Chapter 3 A19 Glencoe Algebra 2
An
swer
s
Answers (Lesson 3-5)
Wor
d Pr
oble
m P
ract
ice
So
lvin
g S
yste
ms
of
Eq
uat
ion
s in
Th
ree
Var
iab
les
Cha
pter
340
Gle
ncoe
Alg
ebra
2
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
__P
ER
IOD
____
_
3-5
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
1.SI
BLI
NG
SA
my,
Kar
en,a
nd
Nol
an a
resi
blin
gs.T
hei
r ag
es i
n y
ears
can
be
repr
esen
ted
by t
he
vari
able
s A
,K,a
nd
N,r
espe
ctiv
ely.
Th
ey h
ave
live
d a
tota
lof
22
year
s co
mbi
ned
.Kar
en h
as l
ived
twic
e as
man
y ye
ars
as A
my,
and
Nol
anh
as l
ived
6 y
ears
lon
ger
than
Am
y.U
seth
e eq
uat
ion
s A
�K
�N
�22
,K�
2A,
and
N�
A�
6 to
fin
d th
e ag
e of
eac
hsi
blin
g.A
my
is 4
,Kar
en is
8,a
nd
No
lan
is10
.
2.H
OC
KEY
Bob
by H
ull
sco
red
Ggo
als,
Aas
sist
s,an
d P
poin
ts i
n h
is N
HL
care
er.B
y de
fin
itio
n,P
�G
�A
.He
scor
ed 5
0 m
ore
goal
s th
an a
ssis
ts.H
adh
e sc
ored
15
mor
e go
als
and
15 m
ore
assi
sts,
he
wou
ld h
ave
scor
ed 1
200
poin
ts.H
ow m
any
goal
s,as
sist
s,an
dpo
ints
did
Bob
by H
ull
sco
re?
610
go
als,
560
assi
sts,
and
11
70 p
oin
ts
3.EX
ERC
ISE
Lar
ry,C
amil
le,a
nd
Sim
one
are
keep
ing
trac
k of
how
far
th
ey w
alk
each
day
.At
the
end
of t
he
wee
k,th
eyco
mbi
ned
th
eir
dist
ance
s an
d fo
un
d th
atth
ey h
ad w
alke
d 34
mil
es i
n t
otal
.Th
eyal
so l
earn
ed t
hat
Cam
ille
wal
ked
twic
eas
far
as
Lar
ry,a
nd
that
Lar
ry w
alke
d 2
mor
e m
iles
th
an S
imon
e.H
ow f
ar d
idea
ch p
erso
n w
alk?
Cam
ille
wal
ked
18
mi,
Lar
ry w
alke
d 9
mi,
and
S
imo
ne
wal
ked
7 m
i.
DIS
TAN
CES
For
Exe
rcis
es 5
an
d 6
,use
the
foll
owin
g in
form
atio
n.
Let
cbe
th
e di
stan
ce b
etw
een
Car
lisl
e an
dW
elle
sley
,let
bbe
th
e di
stan
ce b
etw
een
Car
lisl
e an
d S
ton
ebri
dge,
and
let
abe
th
edi
stan
ce b
etw
een
Wel
lesl
ey a
nd
Sto
neb
ridg
e.·I
f yo
u d
id a
cir
cuit
,tra
veli
ng
from
Car
lisl
e to
Wel
lesl
ey t
o S
ton
ebri
dge
and
back
to
Car
lisl
e,yo
u w
ould
tra
vel
73m
iles
.·S
ton
ebri
dge
is 1
2 m
iles
far
ther
th
anW
elle
sley
is
from
Car
lisl
e.·I
f yo
u d
rove
fro
m S
ton
ebri
dge
to C
arli
sle
and
back
to
Sto
neb
ridg
e,an
d th
enco
nti
nu
ed t
o W
elle
sley
th
en b
ack
toS
ton
ebri
dge,
you
wou
ld t
rave
l 10
2 m
iles
.
5.W
rite
a s
yste
m o
f li
nea
r eq
uat
ion
s to
repr
esen
t th
e si
tuat
ion
.1.
a�
b�
c�
732.
b�
c�
123.
2a�
2b�
102
6.S
olve
th
e sy
stem
of
equ
atio
ns.
Exp
lain
the
mea
nin
g of
th
e so
luti
on i
n t
he
con
text
of
the
situ
atio
n.
a�
17,b
�34
,c�
22;
It is
17
mile
s fr
om
Wel
lesl
ey t
oS
ton
ebri
dg
e,34
mile
s fr
om
Sto
neb
rid
ge
to C
arlis
le,a
nd
22
mile
s fr
om
Car
lisle
to
Wel
lesl
ey.
ab
c
Ston
ebrid
ge
Wel
lesl
ey
Carli
sle
Not d
raw
nto
sca
le
Lesson 3-5
Cha
pter
341
Gle
ncoe
Alg
ebra
2
NA
ME
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__D
ATE
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__P
ER
IOD
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_
3-5
Copyright ©Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Enri
chm
ent
Ho
mo
gen
ou
s S
yste
ms
A s
yste
m o
f eq
uat
ion
s is
cal
led
hom
ogen
eou
s if
it
is o
f th
e fo
rm:
gz�
hy
�kz
�0
dx
�ey
�fz
�0
ax�
by�
cz�
0
Hom
ogen
eou
s sy
stem
s h
ave
som
e u
niq
ue
char
acte
rist
ics
that
set
th
em a
part
fro
m g
ener
alsy
stem
s of
equ
atio
ns.
Th
e fo
llow
ing
exer
cise
s w
ill
expl
ore
som
e of
th
ese
un
iqu
ech
arac
teri
stic
s.
1.E
valu
ate
the
foll
owin
g st
atem
ent.
Is t
his
sta
tem
ent
alw
ays,
som
etim
es,o
r n
ever
tru
e?E
xpal
in y
our
reas
onin
g.E
very
hom
ogen
eou
s sy
stem
of
equ
atio
ns
wil
l h
ave
at l
east
on
e tr
ivia
l so
luti
on:(
0,0,
0).
Alw
ays;
Sam
ple
an
swer
:if
yo
u r
epla
ce e
ach
var
iab
le w
ith
0,y
ou
will
alw
ays
get
eac
h e
qu
atio
n e
qu
al t
o 0
.
2.F
ind
a n
on-t
rivi
al s
olu
tion
to
the
foll
owin
g h
omog
enou
s sy
stem
of
equ
atio
ns.
x�
y�
5z�
02x
�y
�7z
�0
x�
2z�
0
x�
�2z
and
y�
�3z
;C
ho
ose
a n
um
ber
fo
r z,
for
exam
ple
,z�
1.S
o,a
sam
ple
an
swer
is (
�2,
�3,
1).
3.M
ult
iply
th
e so
luti
on y
ou f
oun
d in
Exe
rcis
e 2
by 3
.Is
the
new
ord
ered
tri
ple
a so
luti
on t
oth
e sy
stem
?ye
s
4.M
ult
iply
th
e so
luti
on y
ou f
oun
d in
Exe
rcis
e 2
by �
6.Is
th
e n
ew o
rder
ed t
ripl
e a
solu
tion
to t
he
syst
em?
yes
5.M
ake
a co
nje
ctu
re a
bou
t an
y m
ult
iple
of
a gi
ven
sol
uti
on t
o a
hom
ogen
eou
s sy
stem
of
equ
atio
ns.
Sam
ple
an
swer
:an
y m
ult
iple
of
a g
iven
so
luti
on
to
a h
om
og
eneo
us
syst
em o
f eq
uat
ion
s w
ill a
lso
be
a so
luti
on
.
6.M
ake
a co
nje
ctu
re a
bou
t th
e n
um
ber
of s
olu
tion
s th
at a
hom
ogen
eou
s sy
stem
of
equ
atio
ns
wil
l h
ave
if i
t h
as a
t le
ast
one
non
-tri
vial
sol
uti
on.
Sam
ple
an
swer
:it
will
hav
e an
infi
nit
e n
um
ber
of
solu
tio
ns.
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