Sec1 Mod2 Systems Tnh 71713

8/13/2019 Sec1 Mod2 Systems Tnh 71713

http://slidepdf.com/reader/full/sec1-mod2-systems-tnh-71713 1/88

! #$%# &'()*+'(,-. /,.,01 2304*-( 5 &! 2

61 7'3(1*3.),7 8,() ()* 9(') :('(* ;<<,-* 0< =>?-'(,01

@,-*1.*> ?1>*3 ()* A3*'(,B* A0++01. C((3,D?(,01EF01A0++*3-,'GE:)'3*CG,H* IJ$ 91703(*> G,-*1.*J

Secondary One Mathematics:

An Integrated Approach

Module 2 HonorsSystems

By

The Mathematics Vision Project:

Scott Hendrickson, Joleigh Honey,

Barbara Kuehl, Travis Lemon, Janet Sutoriuswww.mathematicsvisionproject.org

In partnership with the

Utah State Office of Education

1

Systems 1



! #$%# &'()*+'(,-. /,.,01 2304*-( 5 &! 2

61 7'3(1*3.),7 8,() ()* 9(') :('(* ;<<,-* 0< =>?-'(,01

!"#$%&$' )%'$* +,$ -*$.+"/$ -0110%& 2++*"3)+"0%450%-011$*#".647,.*$26"8$ 9:; <%=0*+$' 6"#$%&$:

&0>?@* # A :B.(*+. 0< =C?'(,01. '1> 61*C?'@,(,*.

!"#$$%&&' )#$*+ >$+ 7"++$*&4 2 ?$/$60= <%'$*&+.%'"%@ A.&8

!" $"%&'()*%$'" %' &+,&+-+"%$". *'"-%&/$"%- 0$%1 -2-%+3- '4 $"+5)/6$%$+- B,-!./-0C

12#345 6275 8& 9&'2:&%*+ 7D&+$1& E:F

!"#$$%&&' )#$*+ A00 G"@ 0* 50+ A00 G"@H A,.+ "& +,$ I)$&+"0% 4 2 706"'"JD <%'$*&+.%'"%@ A.&8

7&$%$". /"( .&/,1$". 6$"+/& $"+5)/6$%$+- $" %0' 8/&$/96+- B,-!./-;5 ,-1.<-=;C

12#345 6275 8& 9&'2:&%*+ 7D&+$1& E:E

!"#$$%&&' )#$*+ 701$ 0J K%$H 50%$ 0J +,$ K+,$*L 2 706"'"JD <%'$*&+.%'"%@ A.&8

7&$%$". /"( -'68$". +5)/%$'"- $" %0' 8/&$/96+- B,-!./-;5 ,-!./->C

12#345 6275 8& 9&'2:&%*+ 7D&+$1& E:9

!"#$$%&&' )#$*+ >.1=$*"%@ .%' M$$'"%@ A"1$ L 2 >*.#+"#$ <%'$*&+.%'"%@ A.&8

7&$%$". /"( .&/,1$". $"+5)/6$%$+- $" %0' 8/&$/96+- %' &+,&+-+"% *'"-%&/$"%- B,-!./-;5 ,-!./-05 ,-1.<-=;C

12#345 6275 8& 9&'2:&%*+ 7D&+$1& E:N

!"#$$%&&' )#$*+ 266 J0* K%$H K%$ J0* 266 L 2 706"'"JD <%'$*&+.%'"%@ A.&8

:&/,1$". %1+ -'6)%$'" -+% %' / 6$"+/& -2-%+3 '4 $"+5)/6$%$+- B,-!./-05 ,-1.<-=;C

12#345 6275 8& 9&'2:&%*+ 7D&+$1& E:O

!"#$$%&&' )#$*+ P$+ +0 +,$ >0"%+ L 2 706"'"JD <%'$*&+.%'"%@ A.&8

;'68$". -2-%+3- '4 6$"+/& +5)/%$'"- $" %0' 8/&$/96+- B,-1.<-?C

12#345 6275 8& 9&'2:&%*+ 7D&+$1& E:Q

!"#$$%&&' )#$*+ 7,0=="%@ J0* -.+& .%' ?0@& L 2 ?$/$60= <%'$*&+.%'"%@ A.&8

!" $"%&'()*%$'" %' -'68$". -2-%+3- '4 6$"+/& +5)/%$'"- 92 +6$3$"/%$'" B,-1.<-@5 ,-1.<-?C12#345 6275 8& 9&'2:&%*+ 7D&+$1& E:R

!"#$$%&&' )#$*+ -.% S0) P$+ +0 +,$ >0"%+H A00T L 2 706"'"JD <%'$*&+.%'"%@ A.&8

;'68$". -2-%+3- '4 6$"+/& +5)/%$'"- 92 +6$3$"/%$'" B,-1.<-@5 ,-1.<-?C

12#345 6275 8& 9&'2:&%*+ 7D&+$1& E:U

!"#$$%&&' )#$*+ M00' J0* M"'0 .%' M6)JJD L 2 706"'"JD <%'$*&+.%'"%@ A.&8

;'68$". -2-%+3- '4 6$"+/& $"+5)/6$%$+- &+,&+-+"%$". *'"-%&/$"%- B,-!./-0C

12#345 6275 8& 9&'2:&%*+ 7D&+$1& E:V

!"#$$%&&' )#$*+ A.8$% K)+ 0J -0%+$W+ L 2 >*.#+"#$ <%'$*&+.%'"%@ A.&8

7'&<$". 0$%1 -2-%+3- '4 6$"+/& +5)/%$'"-= $"*6)($". $"*'"-$-%+"% /"( (+,+"(+"% -2-%+3- B,-1.<-?C

12#345 6275 8& 9&'2:&%*+ 7D&+$1& E:F;

2

Systems 2



! #$%# &'()*+'(,-. /,.,01 2304*-( 5 &! 2

61 7'3(1*3.),7 8,() ()* 9(') :('(* ;<<,-* 0< =>?-'(,01

!"#$%&$' )%'$* +,$ -*$.+"/$ -0110%& 2++*"3)+"0%450%-011$*#".647,.*$26"8$ 9:; <%=0*+$' 6"#$%&$:

!"#$$%&&' )#$*+ X0*$ A,"%@& A.8$% K)+ 0J -0%+$W+ L 2 >*.#+"#$ <%'$*&+.%'"%@ A.&8

7'&<$". 0$%1 -2-%+3- '4 6$"+/& $"+5)/6$%$+- /"( %1+$& 9')"(/&$+- B,-1.<-=;C

12#345 6275 8& 9&'2:&%*+ 7D&+$1& E:FF

!"#$$%&&' )#$*+ >$+ 7"++$*& Y$/"&"+$' L 2 ?$/$60= <%'$*&+.%'"%@ A.&8

>-$". -2-%+3- '4 6$"+/& +5)/%$'"- /"( $"+5)/6$%$+- $" / 3'(+6$". *'"%+?% B9ABC 6DC&&" E&32"AFB

67#F3#%3C

12#345 6275 8& 9&'2:&%*+ 7D&+$1& E:FE

9&F&%$ !"#$$%&&' )#$*+ A0 X.*8$+ Z"+, X.+*"#$& L 2 706"'"JD <%'$*&+.%'"%@ A.&8

!" $"%&'()*%$'" %' -'68$". -2-%+3- '4 6$"+/& +5)/%$'"- )-$". 3/%&$*+- BG) 9&F&%$ 67#F3#%3C

12#345 6275 8& 9&'2:&%*+ 7D&+$1& E:F9[

9&F&%$ !"#$$%&&' )#$*+ 706/"%@ 7D&+$1& Z"+, X.+*"#$& L 2 >*.#+"#$ <%'$*&+.%'"%@ A.&8

;'68$". -2-%+3- '4 6$"+/& +5)/%$'"- )-$". 3/%&$*+- BG) 9&F&%$ 67#F3#%3C

12#345 6275 8& 9&'2:&%*+ 7D&+$1& E:FN[

3

Systems 3



! #$%# &'()*+'(,-. /,.,01 2304*-( 5 &! 2

61 7'3(1*3.),7 8,() ()* 9(') :('(* ;<<,-* 0< =>?-'(,01!"#$%&$' )%'$* +,$ -*$.+"/$ -0110%& 2++*"3)+"0%450%-011$*#".647,.*$26"8$ 9:; <%=0*+$' 6"#$%&$:

>

? ; @ ? A A

A B

6 " # 8 * # 0 1 C = , 0 + 0 & C ' ) D & = *

#@% 2*( :,((*3. ! #$%$&'( )*+$,-./*+0*1 2/-3

A)* &'3(,1*B (8,1.C D'3E0. '1> DE'3,('C '3* (3F,1G (0 <,1> ' 8'F (0

+'H* +01*F >?3,1G .?++*3 I'-'(,01@ J)*1 ()*F 0I*3)*'3 ()*,3'?1( -0+7E',1,1G 'K0?( )08 >,<<,-?E( ,( ,. (0 <,1> .0+*01* (0 -'3* <03

)*3 7*(. 8),E* .)* 8,EE K* '8'F 01 ' (3,7C D'3E0. '1> DE'3,(' H108 ()*F )'I* <0?1> ()* 7*3<*-(

.0E?(,01@ L0( 01EF >0 ()*F )'I* ' E'3G*C ?1?.*> .(03'G* .)*> 01 ()*,3 7307*3(F 8)*3* ()*F -'1

)0?.* '1,+'E.C ()*F 'E.0 )'I* ' .7'-,0?. <*1-*> K'-HF'3> 8)*3* ()* 7*(. -'1 7E'F@

D'3E0. '1> DE'3,(' '3* +'H,1G ' E,.( 0< .0+* 0< ()* ,..?*. ()*F 1**> (0 -01.,>*3 '. 7'3( 0< ()*,3

K?.,1*.. 7E'1 (0 -'3* <03 -'(. '1> >0G. 8),E* ()*,3 081*3. '3* 01 I'-'(,01@

• 4(/5$M D'( 7*1. 8,EE 3*N?,3* O <( # 0< .7'-*C 8),E* >0G 3?1. 3*N?,3* #P <( #@ D'3E0. '1> DE'3,(')'I* ?7 (0 QO$ <( # 'I',E'KE* ,1 ()* .(03'G* .)*> <03 7*1. '1> 3?1.C 8),E* .(,EE E*'I,1G *10?G)

300+ (0 +0I* '30?1> ()* -'G*.@

• 4./,.67( 8'-.-M D'3E0. '1> DE'3,(' 7E'1 (0 ,1I*.( +?-) 0< ()* R%#S$ ()*F *'31*> <30+ ()*,3E'.( K?.,1*.. I*1(?3* (0 7?3-)'.* -'( 7*1. '1> >0G 3?1.@ 6( 8,EE -0.( RQ# <03 *'-) -'( 7*1'1> RS$ <03 *'-) >0G 3?1@

;< -0?3.*C D'3E0. '1> DE'3,(' 8'1( (0 +'H* '. +?-) +01*F '. 70..,KE* <30+ ()*,3 K?.,1*..C .0 ()*F

'3* (3F,1G (0 >*(*3+,1* )08 +'1F 0< *'-) (F7* 0< 7*( ()*F .)0?E> 7E'1 (0 '--0++0>'(*@ A)*F 7E'1

(0 -)'3G* RS 7*3 >'F <03 K0'3>,1G *'-) -'( '1> R#$ 7*3 >'F <03 *'-) >0G@

T<(*3 .?3I*F,1G ()* -0++?1,(F 3*G'3>,1G ()* 7*( K0'3>,1G 1**>.C D'3E0. '1> DE'3,(' '3* -01<,>*1(()'( ()*F -'1 H**7 'EE 0< ()*,3 K0'3>,1G .7'-*. <,EE*> <03 ()* .?++*3@

:0 ()* N?*.(,01 ,.C )08 +'1F 0< *'-) (F7* 0< 7*( .)0?E> ()*F 73*7'3* <03U A)*,3 >'> )'. .?GG*.(*>

()* .'+* 1?+K*3 0< *'-)C 7*3)'7. %# -'(. '1> %# >0G.@ D'3E0. (),1H. ()*F .)0?E> 7E'1 <03 +03*

>0G.C .,1-* ()*F -'1 -)'3G* +03*@ DE'3,(' (),1H. ()*F .)0?E> 7E'1 <03 +03* -'(. .,1-* ()*F ('H* E*..

.7'-* '1> (,+*C '1> ()*3*<03* ()*F -'1 K0'3> +03*@

J)'( >0 F0? (),1HU J)'( 3*-0++*1>'(,01. 80?E> F0? G,I* (0 D'3E0. '1> DE'3,('C '1> 8)'(

'3G?+*1( 80?E> F0? ?.* (0 -01I,1-* ()*+ ()'( F0?3 3*-0++*1>'(,01 ,. 3*'.01'KE*U

4

Systems 4



! #$%# &'()*+'(,-. /,.,01 2304*-( 5 &!2

61 7'3(1*3.),7 8,() ()* 9(') :('(* ;<<,-* 0< =>?-'(,01!"#$%&$' )%'$* +,$ -*$.+"/$ -0110%& 2++*"3)+"0%450%-011$*#".647,.*$26"8$ 9:; <%=0*+$' 6"#$%&$:

#@% 2*( :,((*3. V A*'-)*3 L0(*. ! #$%$&'( )*+$,-./*+0*1 2/-3

"#$%&'(M T. .(?>*1(. 803H 8,() ()* -01(*W( 0< +'H,1G 3*-0++*1>'(,01. <03 )08 +'1F >0G. '1>

-'(. D'3E0. '1> DE'3,(' .)0?E> 7E'1 (0 '--0++0>'(*C ()*F 8,EE .?3<'-* +'1F ,>*'.C .(3'(*G,*. '1>3*73*.*1('(,01. 3*E'(*> (0 .0EI,1G .F.(*+. 0< *N?'(,01. '1> ,1*N?'E,(,*.@ X03 *W'+7E*C ()*F 8,EE

*W7E03* ()* 10(,01 0< 5'*-.,/0*.- .,1-* ,1 (),. ('.H ()* 1?+K*3 0< *'-) (F7* 0< 7*( ()'( -'1 K*

'--0++0>'(*> ,. E,+,(*> KF .7'-* '1> +01*FC K?( +'1F >,<<*3*1( -0+K,1'(,01. 0< >0G. '1> -'(.

'3* 70..,KE*@ A)*F +'F -01.,>*3 ()* 10(,01 0< ' -9-.$: '; $<7/.0'*- .,1-* *'-) -01.(3',1( Y.7'-*C

.('3(Z?7 -0.(.[ 'EE08. <03 ' >,<<*3*1( .*( 0< 70..,K,E,(,*.\' 7'3(,-?E'3 -0+K,1'(,01 0< >0G. '1> -'(.

+'F .'(,.<F 01* -01.(3',1( K?( 10( '10()*3\.0 K0() -01.(3',1(. +?.( K* -01.,>*3*>

.,+?E('1*0?.EF@ X,1'EEFC ()*F +'F .?3<'-* ()* 10(,01 0< / -9-.$: '; 0*$<7/&0.0$- .,1-* D'3E0. '1>

DE'3,(' >01]( )'I* (0 ?.* ?7 'EE 0< ()* 'I',E'KE* .7'-* 03 +01*FC ,+7EF,1G ()'( *'-) -01.(3',1( +'F

K* 3*73*.*1(*> KF '1 ,1*N?'E,(F@

*&$( +,-./-$/' 0&1#'M

23*4536 ^*73*.*1( -01.(3',1(. KF *N?'(,01. 03 ,1*N?'E,(,*.C '1> KF .F.(*+. 0< *N?'(,01. '1>_03

,1*N?'E,(,*.C '1> ,1(*373*( .0E?(,01. '. I,'KE* 03 101ZI,'KE* 07(,01. ,1 ' +0>*E,1G -01(*W(@

7(8-,(/ +,-./-$/'9 :3;3<= 2374>3?<

@-#.1A BCA&8( *8-''D9

T<(*3 3*'>,1G '1> >,.-?..,1G ()* `2*( :,((*3.a .-*1'3,0C -)'EE*1G* .(?>*1(. (0 -0+* ?7 8,() '

-0+K,1'(,01 0< >0G. '1> -'(. ()'( 80?E> F,*E> ()* ),G)*.( >',EF ,1-0+*@ b,I* .(?>*1(. ' <*8

+,1?(*. (0 803H ,1>*7*1>*1(EF (0 <,1> ()* ,1-0+* <03 ' 7'3(,-?E'3 -0+K,1'(,01 0< >0G. '1> -'(.@T<(*3 ' <*8 +,1?(*. )'I* .(?>*1(. -0+7'3* ()*,3 >',EF ,1-0+* 8,() 0()*3. '1> ()*1 803H 8,() '

7'3(1*3 (0 ,+730I* ()*,3 ,1,(,'E G?*..*.

4E%8&$( B+F-88 G$&#%D9

X03 .(?>*1(. 8)0 >01]( H108 8)*3* (0 K*G,1C )'I* ()*+ >*(*3+,1* ()* >',EF ,1-0+* ,< D'3E0. '1>

DE'3,(' <0EE08 ()*,3 >'>]. '>I,-* 0< K0'3>,1G `()* .'+* 1?+K*3 0< *'-)C 7*3)'7. %# -'(. '1> %#

>0G.@a T<(*3 -'E-?E'(,1G ()* ,1-0+*C '.H ,< D'3E0. '1> DE'3,(' )'I* *10?G) .7'-* '1> +01*F (0

'--0++0>'(* %# -'(. '1> %# >0G. *'-) >'F@

6( ,. '1(,-,7'(*> ()'( +'1F .(?>*1(. 8,EE ?.* ' G?*.. '1> -)*-H .(3'(*GFM 7,-H ' 7'3(,-?E'3-0+K,1'(,01 0< >0G. '1> -'(.C -)*-H ()'( ()* 7'3(,-?E'3 -0+K,1'(,01 .'(,.<,*. K0() -01.(3',1(.C '1>

()*1 -'E-?E'(* ()* ,1-0+*@ T<(*3 <,1>,1G ' -0+K,1'(,01 ()'( 803H.C .(?>*1(. +'F -01.,>*3 8)'(

)'77*1. '. ()* 1?+K*3 0< >0G.C -'(. 03 K0() ,. ,1-3*'.*>@



! #$%# &'()*+'(,-. /,.,01 2304*-( 5 &!2

61 7'3(1*3.),7 8,() ()* 9(') :('(* ;<<,-* 0< =>?-'(,01!"#$%&$' )%'$* +,$ -*$.+"/$ -0110%& 2++*"3)+"0%450%-011$*#".647,.*$26"8$ 9:; <%=0*+$' 6"#$%&$:

:(?>*1(. 8)0 <0-?. 01 4?.( 01* -01.(3',1( '1> <,1> -0+K,1'(,01. 0< -'(. '1> >0G. ()'( .'(,.<F ()'(

-01.(3',1( )'I* 1'(?3'EEF .,+7E,<,*> ()* E'3G*3 730KE*+ ,1(0 ' +03* +'1'G*'KE* ('.H@ A)*F 8,EE

G',1 I'E?'KE* ,1.,G)( ,1(0 ()* +'()*+'(,-'E 803H 0< (),. +0>?E*C '1> 1**> 10( K* 73*..*> '( (),.

(,+* (0 -01.,>*3 K0() -01.(3',1(.C 03 ()* K,GG*3 ,..?* 0< +'W,+,B,1G ()* 730<,(@ 23*.. <03 ()* 803H

F0? <**E F0?3 .(?>*1(. -'1 )'1>E*C '1> ?.* (),. ('.H (0 '..*.. 8)'( ,>*'.C .(3'(*G,*. '1>

3*73*.*1('(,01. 8,EE K* 'I',E'KE* <03 <?(?3* 803H ,1 (),. +0>?E*@ X03 *W'+7E*C ' .(?>*1( 8)0I,.?'E,B*. ()* .7'-* -01.(3',1( KF >3'8,1G 70..,KE* E'F0?(. 0< )08 ()* .)*> +,G)( K* 0--?7,*> 8,()

-'( 7*1. '1> >0G 3?1. ,. 10(,-,1G ()'( ()*3* '3* +'1F .?-) -0+K,1'(,01. ()'( ?.* ?7 ()* (0('E

'I',E'KE* .7'-*@

J'(-) <03 .(?>*1(. 8)0 <0-?. 01 ()* <0EE08,1G ,>*'.C .,1-* ()*F ?1>*37,1 *..*1(,'E .(3'(*G,*. ,1

<?(?3* ('.H.M YA),. E,.( ,. .*N?*1-*> <30+ +0.( E,H*EF (0 E*.. E,H*EF (0 0--?3c >01]( K* -01-*31*> ,< 10(

'EE 0< ()*.* ,>*'. '3* 73*.*1( ,1 F0?3 -E'..C .,1-* <?(?3* ('.H. 8,EE *E,-,( *'-) 0< ()*.* 8'F. 0<

(),1H,1G 'K0?( ()* .F.(*+ 0< -01.(3',1(.@[

• :(?>*1(. 8)0 -'E-?E'(* ()* ,1(*3-*7(. 0< ()* -01.(3',1(.c ()'( ,.C .(?>*1(. 8)0 -01.,>*3 )08

()*F +,G)( ?.* ?7 'EE 0< ()* +01*F 03 'EE 0< ()* .7'-*C KF K0'3>,1G 4?.( -'(. 03 4?.( >0G.@• :(?>*1(. 8)0 +'H* 10(* 0< '1 `*W-)'1G* 3'(*a K*(8**1 -'(. '1> >0G. ,1 (*3+. 0< *,()*3 0<

()* -01.(3',1(.@ X03 *W'+7E*C <0?3 -'(. ?.* ()* .'+* .7'-* '. 01* >0G@

• :(?>*1(. 8)0 -3*'(* -)'3(. (0 H**7 (3'-H 0< ()* -0+K,1'(,01. 0< >0G. '1> -'(. ()*F )'I*

(3,*>@ :?-) -)'3(. 8,EE 730K'KEF ,1-E?>* -0E?+1. (0 (3'-H ()* 1?+K*3 0< >0G.C ()* 1?+K*3 0<

-'(.C '1> ()* +01*F *'31*>@ :(?>*1(. 8,EE 'E.0 )'I* (0 H**7 (3'-H 0< 8)*()*3 ' 7'3(,-?E'3

-0+K,1'(,01 0< >0G. '1> -'(. .'(,.<,*. *'-) 0< ()* -01.(3',1(.@

• :(?>*1(. 8)0 7E0( -0+K,1'(,01. 01 ' -003>,1'(* G3,> (0 H**7 (3'-H 0< ()* -0+K,1'(,01. ()*F

)'I* (*.(*>@

• :(?>*1(. 8)0 (3F (0 83,(* *N?'(,01. 03 ,1*N?'E,(,*. (0 3*73*.*1( ()* -01.(3',1(.@

5H'1#'' BCA&8( *8-''D9

d*G,1 8,() ' -0+K,1'(,01 0< >0G. '1> -'(. ()'( 803H*>C ' .*-01> -0+K,1'(,01 ()'( 803H*> '1>

3*.?E(*> ,1 ' ),G)*3 730<,(C <0EE08*> KF ' -0+K,1'(,01 ()'( >,>1]( 803H K*-'?.* ,( >,>1]( .'(,.<F 01*

03 +03* 0< ()* -01.(3',1(.@ D0EE*-( -0+K,1'(,01. ()'( 803H ?.,1G ' ('KE* 03 -003>,1'(* G3,>C '.

.?GG*.(*> KF .(?>*1( 803H@ e'I* .(?>*1(. .)'3* ()*,3 ,>*'. 'K0?( )08 ()*F -'1 ,+730I* ()* >',EF

,1-0+* 8),E* .'(,.<F,1G ()* -01.(3',1(.C -011*-(,1G ()*,3 ,>*'. (0 ()* 03G'1,B'(,01'E

3*73*.*1('(,01. 0< ('KE*. '1>_03 70,1(. 7E0((*> 01 ' -003>,1'(* G3,>@

28HI.(/ 7(-/J= +(,= G&9 +J',(F' <3?



Name: Systems 2.1

© 2012 Mathematics Vision Project | M P

In partnership with the Utah State Office of Education

Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license

Ready, Set, Go!

ReadyTopic: Determine if given value is a solution and solve systems of equations

Substitute the given points into the equations to determine which ordered pair satisfies the

system of linear equations, then graph both equations and label the point of intersection.

1. 2.

a. () a. (2, 7)

b. ( ) b. (-7, 11)

c. ( ) c. (0, 5)

Solve the following systems by graphing. Check the solution by evaluating both equations atthe point of intersection.

3. 4.

© 2012 www.flickr.com/photos/dugspr

5

Systems 5



Name: Systems 2.1




Set Topic: Determining possible solutions

5. A theater wants to take in at least $2000 for a certain matinee. Children’s tickets cost $5 each and

adult tickets cost $10 each. The theater can seat up to 350 people. Find five combinations ofchildren and adult tickets that will make their goal.

Go

Topic: graphing linear equations and determining if a given value is a solution

Graph each equation below, then determine if the point (3,5) is a solution to the equation.

Name two additional points that are solutions to the equation and show these points on the

graph.

6. 7.

8. 9.

Need help? Check out this related video:

https://www.youtube.com/watch?v=vo-CXaCf1I4

6

Systems 6






! #$%# &'()*+'(,-. /,.,01 2304*-( 5 &!2

61 7'3(1*3.),7 8,() ()* 9(') :('(* ;<<,-* 0< =>?-'(,01!"#$%&$' )%'$* +,$ -*$.+"/$ -0110%& 2++*"3)+"0%450%-011$*#".647,.*$26"8$ 9:; <%=0*+$' 6"#$%&$:

#@# A00 B,C 03 D0( A00 B,CE A)'( ,.()* F?*.(,01

! #$%&'&() *+',-./0+'&+1 20.3

G. H'3I0. ,. -01.,>*3,1C ()* '+0?1( 0< +01*J 'K',I'LI* <03 7?3-)'.,1C-'( 7*1. '1> >0C 3?1. M.** L*I08N )* 3*'I,O*. ()'( ),. <'()*3P..?CC*.(,01 0< L0'3>,1C Q()* .'+* 1?+L*3 0< *'-)E 7*3)'7. %# -'(. '1>%# >0C.R ,. (00 L,C@ S)JT

• #/0-/456 7$./.U H'3I0. '1> HI'3,(' 7I'1 (0 ,1K*.( +?-) 0< ()* V%#W$ ()*J *'31*> <30+ ()*,3I'.( L?.,1*.. K*1(?3* (0 7?3-)'.* -'( 7*1. '1> >0C 3?1.@ 6( 8,II -0.( VX# <03 *'-) -'( 7*1'1> VW$ <03 *'-) >0C 3?1@

%@

Y,1> '( I*'.( Z +03* -0+L,1'(,01. 0< -'(. '1> >0C. ()'( 80?I> L* Q(00 L,CR L'.*> 01 (),. #/0-/456 7$./ "#$%&'()$&* 2I0( *'-) 0< ()*.* -0+L,1'(,01. '. 70,1(. 01 ' -003>,1'(* C3,> ?.,1C ()*.'+* -0I03 <03 *'-) 70,1(@

#@ Y,1> '( I*'.( Z -0+L,1'(,01. 0< -'(. '1> >0C. ()'( 80?I> 10( L* Q(00 L,CR L'.*> 01 (),. #/0-/456

7$./ -01.(3',1(@ 2I0( *'-) 0< ()*.* -0+L,1'(,01. '. 70,1(. 01 ' -003>,1'(* C3,> ?.,1C ' >,<<*3*1(-0I03 <03 ()* 70,1(. ()'1 J0? ?.*> ,1 [%@

X@ Y,1> '( I*'.( Z -0+L,1'(,01. 0< -'(. '1> >0C. ()'( 80?I> L* Q4?.( 3,C)(R L'.*> 01 (),. #/0-/456

7$./ -01.(3',1(@ A)'( ,.E <,1> -0+L,1'(,01. 0< -'( 7*1. '1> >0C 3?1. ()'( 80?I> -0.( *\'-(IJV%#W$@ 2I0( *'-) 0< ()*.* -0+L,1'(,01. '. 70,1(. 01 ' -003>,1'(* C3,> ?.,1C ' (),3> -0I03@

]@ S)'( >0 J0? 10(,-* 'L0?( ()*.* ()3** >,<<*3*1( -0II*-(,01. 0< 70,1(.T

Z@ S3,(* '1 *^?'(,01 <03 ()* I,1* ()'( 7'..*. ()30?C) ()* 70,1(. 3*73*.*1(,1C -0+L,1'(,01. 0< -'(7*1. '1> >0C 3?1. ()'( -0.( *\'-(IJ V%#W$@ S)'( >0*. ()* .I07* 0< (),. I,1* 3*73*.*1(T

H'3I0. '1> HI'3,(' >01P( )'K* (0 .7*1> 'II 0< ()*,3 +01*J 01 -'( 7*1. '1> >0C 3?1.E ?1I*.. ,( 8,II)*I7 ()*+ +'\,+,O* ()*,3 730<,(@

_@ :)'>* 'II 0< ()* 70,1(. 01 J0?3 -003>,1'(* C3,> ()'( %(&)%,- ()* #/0-/456 7$./. -01.(3',1(@

`@ S3,(* ' +'()*+'(,-'I 3?I* (0 3*73*.*1( ()* 70,1(. .)'>*> ,1 [_@ A)'( ,.E 83,(* '1 ,1*^?'I,(J8)0.* %#./&)#$ %0& ,. ()* -0II*-(,01 0< 70,1(. ()'( .'(,.<J ()* #/0-/456 7$./. -01.(3',1(@

7

Systems 7



! #$%# &'()*+'(,-. /,.,01 2304*-( 5 &!2

61 7'3(1*3.),7 8,() ()* 9(') :('(* ;<<,-* 0< =>?-'(,01!"#$%&$' )%'$* +,$ -*$.+"/$ -0110%& 2++*"3)+"0%450%-011$*#".647,.*$26"8$ 9:; <%=0*+$' 6"#$%&$:

61 '>>,(,01 (0 ./0-/456 8$./.9 H'3I0. 1**>. (0 -01.,>*3 )08 +?-) .7'-* )* )'. 'K',I'LI*E L'.* 01 ()*<0II08,1CU

• #608,U H'( 7*1. 8,II 3*^?,3* _ <(

#

0< .7'-*E 8),I* >0C 3?1. 3*^?,3* #] <( #

@ H'3I0. '1> HI'3,(')'K* ?7 (0 X_$ <( # 'K',I'LI* ,1 ()* .(03'C* .)*> <03 7*1. '1> 3?1.E 8),I* .(,II I*'K,1C *10?C)300+ (0 +0K* '30?1> ()* -'C*.@

W@ S3,(* '1 ,1*^?'I,(J (0 3*73*.*1( ()* .0I?(,01 .*( <03 ()* .608, -01.(3',1( : :)'>* ()* .0I?(,01 .*(<03 (),. ,1*^?'I,(J 01 ' >,<<*3*1( -003>,1'(* C3,>@

8

Systems 8



! #$%# &'()*+'(,-. /,.,01 2304*-( 5 &!2

61 7'3(1*3.),7 8,() ()* 9(') :('(* ;<<,-* 0< =>?-'(,01!"#$%&$' )%'$* +,$ -*$.+"/$ -0110%& 2++*"3)+"0%450%-011$*#".647,.*$26"8$ 9:; <%=0*+$' 6"#$%&$:

#@# A00 B,C 03 D0( A00 B,CE A)'( ,. ()* F?*.(,01 aA*'-)*3 D0(*.

! #$%&'&() *+',-./0+'&+1 20.3

120")(. 3#&0 &# 40("50'%6 A)*3* '3* (80 70..,LI* 8'J. J0? +'J -01.,>*3 ,+7I*+*1(,1C (),. ('.b@A)* <,3.( '7730'-) '..?+*. ()'( .(?>*1(. 8,II 10( .(3?CCI* 8,() 83,(,1C *^?'(,01. 0< ()* L0?1>'3JI,1*c730L'LIJ ,1 .I07*d,1(*3-*7( <03+c3*73*.*1(,1C ()* (80 -01.(3',1(.E '1> ()'( ()* 1*8 ,..?*<03 ()*.* .(?>*1(. ,. <0-?.,1C 01 ()* )'I<d7I'1* ()'( 3*73*.*1(. ()* .0I?(,01 .*( (0 ()* ,1*^?'I,(J'..0-,'(*> 8,() ()* -01.(3',1(@ A)* .*-01> '7730'-) (0 ()* ('.b 80?I> L* (0 .7I,( ()* 803b ,1(0 (807'3(.@ A)* <,3.( 7'3( *1>,1C 8,() .(?>*1(. -01.,>*3,1C )08 (0 83,(* ()* *^?'(,01 0< ()* I,1*3*73*.*1(,1C ()* 70,1(. <03 8),-) ()* -0+L,1'(,01 0< -'(. '1> >0C. -0.(. *\'-(IJ V%#W$ M^?*.(,01ZN@ B*<03* 730-**>,1C (0 ()* 1*\( 7'3( 0< ()* ('.bE J0? -0?I> )'K* .(?>*1(. 803b 01 ()* 1*\( ('.bQ:0+* 0< ;1*E D01* 0< ()* ;()*3R 8)*3* ()*J 8,II *\'+,1* .7*-,<,- .(3'(*C,*. <03 83,(,1C ()*

*^?'(,01. 0< ()* I,1*. 3*73*.*1(,1C ()* -01.(3',1(.@ A)* .(?>*1(. -'1 ()*1 3*(?31 (0 ()* 3*+',1>*30< (),. ('.b (0 -01.,>*3 ()* <'-( ()'( ()* -01.(3',1(. '3* '-(?'IIJ 83,((*1 ?.,1C (*3+. ()'( .?CC*.(()*J '3* ,1*^?'I,(,*.E 10( *^?'(,01.@ 6< J0? ?.* ()* <,3.( '7730'-) (),. ('.bE J0? .)0?I> .(,II <0II08 ,(8,() Q:0+* 0< ;1*E D01* 0< ()* ;()*3R (0 .0I,>,<J ()* '>>,(,01'I ,>*'. 'L0?( *^?,K'I*1( <03+. 0< ()*L0?1>'3J I,1* *^?'(,01. ()'( '3* 73*.*1(*> ()*3*@

7/'2#%0U G. .(?>*1(. -01(,1?* (0 803b 8,() ()* 7*( .,((*3 -01(*\( ()*J 8,II *\'+,1* '1> *\(*1>+'1J ,>*'.E .(3'(*C,*. '1> 3*73*.*1('(,01. 3*I'(*> (0 .0IK,1C .J.(*+. 0< *^?'(,01. '1> ,1*^?'I,(,*.@Y03 *\'+7I*E *'-) -01.(3',1( M.7'-*E .('3(d?7 -0.(.N -'1 L* 3*73*.*1(*> LJ '1 ,1*^?'I,(J .,1-*H'3I0. '1> HI'3,(' >01P( )'K* (0 ?.* ?7 'II 0< ()* 'K',I'LI* .7'-* 03 +01*J@ A)* .0I?(,01 .*( 0< '1,1*^?'I,(J M,1 ^?'>3'1( 6N -01.,.(. 0< 'II 0< ()* 70,1(. ()'( .'(,.<J ()* -01.(3',1(@ A)* L0?1>'3JL*(8**1 70,1(. ()'( .'(,.<J ()* -01.(3',1( '1> ()0.* ()'( >0 10( ,. ' I,1*@ A)* .I07* 0< ()* I,1*

3*73*.*1(. ()* Q*\-)'1C* 3'(*R L*(8**1 -'( 7*1. '1> >0C 3?1. M*@C@E <03 *K*3J # <*8*3 >0C 3?1. 8*L?JE 8* -'1 7?3-)'.* Z +03* -'( 7*1.NE '1> ()* ;4 '1> )4,1(*3-*7(. 0< ()* I,1* 3*73*.*1( 7?3-)'.,1C*,()*3 'II -'( 7*1. '1> 10 >0C 3?1.E 03 'II >0C 3?1. '1> 10 -'( 7*1.@ G .)'>*> )'I<d7I'1* ,1 ()*-003>,1'(* C3,> ,. L*,1C ?.*> (0 3*73*.*1( ()* .0I?(,01 .*( <03 '1 ,1*^?'I,(J ,1 (80dK'3,'LI*.@ 61 (),.-'.*E ()* 70,1(. 01 ()* L0?1>'3J I,1* '3* 'I.0 ,1-I?>*> ,1 ()* .0I?(,01 .*(E ,1>,-'(*> LJ >3'8,1C '.0I,> I,1* <03 ()* L0?1>'3J@

8#'0 1&($9('9% :#"/%U

;*8<=*> H3*'(* *^?'(,01. ,1 (80 03 +03* K'3,'LI*. (0 3*73*.*1( 3*I'(,01.),7. L*(8**1 ^?'1(,(,*.eC3'7) *^?'(,01. 01 -003>,1'(* '\*. 8,() I'L*I. '1> .-'I*.@

;*?<@*A> f3'7) ()* .0I?(,01. (0 ' I,1*'3 ,1*^?'I,(J ,1 (80 K'3,'LI*. '. ' )'I<d7I'1* M*\-I?>,1C ()*L0?1>'3J ,1 ()* -'.* 0< ' .(3,-( ,1*^?'I,(JN

?0.(&09 1&($9('9%6 3*B*>C ;*11<*A(C ;*?<@*ADC ;*8<=*>C :*E<*AFC :*E<*G



! #$%# &'()*+'(,-. /,.,01 2304*-( 5 &!2

61 7'3(1*3.),7 8,() ()* 9(') :('(* ;<<,-* 0< =>?-'(,01!"#$%&$' )%'$* +,$ -*$.+"/$ -0110%& 2++*"3)+"0%450%-011$*#".647,.*$26"8$ 9:; <%=0*+$' 6"#$%&$:

E(/$"5 HI5#.0 8.(%%J6 !"#!$

g*'> ()30?C) ()* <,3.( <0?3 ^?*.(,01. 01 ()* 803b.)**( '1> K*3,<J ()'( .(?>*1(. 3*-0C1,O* ()'( ()*J'3* (0 ,>*1(,<J ()3** >,<<*3*1( .*(. 0< 70,1(.c()0.* ()'( '3* (00 L,CE (00 .+'II '1> 4?.( 3,C)(E 3*I'(,K*(0 ()* .('3(d?7 -0.(. -01.(3',1(c'1> (0 ?.* ' >,<<*3*1( -0I03 (0 7I0( *'-) .*( 0< 70,1(. 01 '-003>,1'(* C3,>@ h,.-?.. ()* +*'1,1C 0< ()* 803> Q-01.(3',1(R M.)081 ,1 L0I> (J7* 01 ()* )'1>0?(N@i*'K* C3'7),1C >*-,.,01.c.?-) '. 8),-) '\,. (0 I'L*I Q-'(.R '1> 8),-) (0 I'L*I Q>0C.RE 03 8)'( (0

?.* '. '1 '773073,'(* .-'I* <03 *'-) '\,.c?7 (0 .(?>*1(. (0 +'b*@

<K2.#'0 H1L(.. M'#/2J6

:(?>*1(. +'J )'K* 'I3*'>J <0?1> .0+* 70,1(. ()'( 803b 03 >0 10( 803b <03 ()* .('3(d?7 -0.(. <30+()* 73*K,0?. ('.b@ A0>'J ()*J 8,II -0I03d-0>* ()*.* 70,1(. '1> (3J (0 ,>*1(,<J ()* L0?1>'3J ()'(.*7'3'(*. 70,1(. ()'( 803b <30+ ()0.* ()'( >0 10(@ i,.(*1 <03 )08 .(?>*1(. '3* >*(*3+,1,1C 8),-)-0I03 .)0?I> L* ?.*> 8)*1 7I0((,1C 70,1(.E '1> ()'( ()*J '3* 730>?-,1C ' -0II*-(,01 0< 70,1(. ()'(.('3( (0 Q<,II ?7R ()* 3*C,01. 01 *,()*3 .,>* 0< ()* L0?1>'3J M*@C@E ,( ,. L*.( ,< .0+*E L?( 10( 'II 70,1(.E'3* -I?.(*3*> '30?1> ()* L0?1>'3Jc)*I7 .(?>*1(. .** ()'( ()*3* '3* 0LK,0?. 70,1(. ()'( I,* ,1*,()*3 .*(E '. 8*II '. 70,1(. <03 8),-) ()*J 1**> (0 -'3*<?IIJ -)*-b (0 .** ,< ()*J .'(,.<J ()*-01.(3',1(N@

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

:?CC*.( ()'( .(?>*1(. (),1b 'L0?( 8)'( ()*J b108 'L0?( ()* L0?1>'3J I,1* ()'( 8,II )*I7 ()*+83,(* ,(. *^?'(,01@

=)%"/%% HI5#.0 8.(%%J6Y0-?. ()* <,3.( 7'3( 0< ()* >,.-?..,01 01 ^?*.(,01 ]E QS)'( >0 J0? 10(,-* 'L0?( ()* ()3** >,<<*3*1(-0II*-(,01. 0< 70,1(.TR :(?>*1(. .)0?I> L* 'LI* (0 >*.-3,L* ()'( ()* 70,1(. ()'( I,* 01 ()* L0?1>'3J<03+ ' I,1*E '1> ()'( 70,1(. ()'( I,* L*I08 ()* I,1* 3*73*.*1( -0+L,1'(,01. 0< >0C. '1> -'(. <038),-) ()* (8,1. -'1 '<<03> (0 7?3-)'.* 3?1. '1> 7*1.E '1> ()'( 70,1(. ()'( I,* 'L0K* ()* I,1*3*73*.*1( -0+L,1'(,01. 0< -'(. '1> >0C. ()'( 80?I> L* (00 *\7*1.,K*@ j'K* .(?>*1(. 73*.*1(70,1(. ()'( ()*J <0?1> 8),-) I,* 01 ()* L0?1>'3J I,1* '1> *\7I',1 )08 ()*J b108 ()*,3 70,1(. I,* 01()* L0?1>'3J@

MD0(*U ,< .(?>*1(. '3* 10( 3*'>J (0 '3(,-?I'(* ()*.* ,>*'.E J0? 8,II 1**> (0 >0 +03* .7*-,<,- 803bL*<03* J0? -'1 <0-?. 01 ^?*.(,01 ]@ j'K* .(?>*1(. 'C3** 01 -0I03. (0 ?.* <03 *'-) -0II*-(,01 0<70,1(.E )08 (0 I'L*I ()* ;4'1> ) d'\,.E '1> 8)'( .-'I* (0 ?.* 01 *'-) '\,. .0 ()'( 70,1(. ,>*1(,<,*> LJ,1>,K,>?'I C30?7. -'1 L* -0II*-(*> 01(0 ' .,1CI* -I'.. C3'7)@ Y,3.( -0II*-( .*K*3'I 70,1(. ()'(803b*>E ()*1 .*K*3'I 70,1(. ()'( >,>1P( 803bE '1> <,1'IIJE 70,1(. 01 ()* L0?1>'3J@ 9.,1C ()* -I'..C3'7)E +0K* 01 (0 ' >,.-?..,01 0< ^?*.(,01 ]@N

&0K* (0 ' >,.-?..,01 0< ^?*.(,01 Z LJ '.b,1C 8)'( .(?>*1(. b108 'L0?( ()* L0?1>'3J I,1* ()'( 8,II)*I7 ()*+ 83,(* ,(. *^?'(,01@ :(?>*1(. +,C)( ?.* ' K'3,*(J 0< .(3'(*C,*. (0 83,(* ()* *^?'(,01 0< ()*



! #$%# &'()*+'(,-. /,.,01 2304*-( 5 &!2

61 7'3(1*3.),7 8,() ()* 9(') :('(* ;<<,-* 0< =>?-'(,01!"#$%&$' )%'$* +,$ -*$.+"/$ -0110%& 2++*"3)+"0%450%-011$*#".647,.*$26"8$ 9:; <%=0*+$' 6"#$%&$:

L0?1>'3J I,1*@ S'(-) <03 .(3'(*C,*. ()'( ?.* ,1<03+'(,01 <30+ ()* C3'7) M*@C@E .I07*E K*3(,-'I,1(*3-*7(E 03 L0() ,1(*3-*7(.N '. 8*II '. .(3'(*C,*. L'.*> 01 ()* 803>,1C 0< ()* -01.(3',1(E I*'>,1C (0'1 *^?'(,01 0< ()* I,1* ,1 .('1>'3> <03+@

G. 70..,LI*E )'K* 73*.*1('(,01. <30+ .(?>*1(. 8)0 -'1 '3(,-?I'(* ()* +*'1,1C 0< ()* .I07* '. ()*Q*\-)'1C* 3'(*R L*(8**1 -'( 7*1. '1> >0C 3?1.@ 6>*1(,<J .(?>*1(. 8)0 -'1 '3(,-?I'(* ()'( ()* ; d '1>

)4,1(*3-*7(. 3*73*.*1(. .,(?'(,01. 8)*3* ()*3* '3* *,()*3 10 -'(. 03 10 >0C.@ GI.0 I00b <03 .(?>*1(.8)0 )'K* 83,((*1 ()* *^?'(,01 0< ()* I,1* ,1 .('1>'3> <03+ <30+ ()* I'1C?'C* 0< ()* -01.(3',1(@

D0(*U 6< J0?3 .(?>*1(. '3* ?1'LI* (0 83,(* ()* *^?'(,01 0< ()* L0?1>'3J I,1* '( (),. (,+*E +0K* 01(0()* 1*\( ('.b Q:0+* 0< ;1*E D01* 0< ()* ;()*3R 8),-) .0I,>,<,*. 8'J. 0< 83,(,1C ()* *^?'(,01 0< ()*L0?1>'3J I,1* ,1 L0() .I07*d,1(*3-*7( <03+ '1> .('1>'3> <03+@ A)*1 3*(?31 (0 7'3( # 0< (),. ('.b@

E(/$"5 HI5#.0 8.(%%J6 !&#!'

i'?1-) (),. 703(,01 0< ()* ('.b LJ )'K,1C .(?>*1(. -01.,>*3 ()* 803>,1C ,1 ()* .('3(d?7 -0.(-01.(3',1( ()'( .?CC*.( ()'( H'3I0. '1> HI'3,(' +'J 10( 1**> (0 .7*1> 'II 0< ()*,3 +01*J 01 >0C 3?1.'1> -'( 7*1.@ G.b 8)J (),. +,C)( L* ()* -'.*@ ;1-* .(?>*1(. )'K* ,>*1(,<,*> ()'( ()* '+0?1( 0<.7'-* 'K',I'LI* <03 3?1. '1> 7*1. +'J '<<*-( )08 +?-) ()*J .7*1>E )'K* ()*+ 803b 01 ()* I'.(

703(,01 0< (),. ('.b@

<K2.#'0 H1L(.. M'#/2J6

D0(* .(?>*1(. ?.* 0< K'3,0?. 3*73*.*1('(,01. (0 ,1>,-'(* 0%% 0< ()* -0+L,1'(,01. 0< -'(. '1> >0C.()'( 803b <03 *'-) -01.(3',1(E ,1-I?>,1C ()* .)'>,1C 0< ' )'I<d7I'1* M03 .)'>,1C >,.-3*(* 70,1(. ,1()* <,3.( ^?'>3'1( L*I08 ()* L0?1>'3J I,1*E .,1-* 8* -'1P( '-(?'IIJ L0'3> ' 7'3(,'I -'( 03 >0CNE '1>()* ?.* 0< ,1*^?'I,(J 10('(,01 (0 >*10(* ()* .0I?(,01 .*( 'IC*L3',-'IIJ@

=)%"/%% HI5#.0 8.(%%J6

D0(* ()* ?.* 0< ,1*^?'I,(J 10('(,01 ,1.(*'> 0< '1 *^?'(,01 (0 'IC*L3',-'IIJ ,>*1(,<J 0%% 0< ()*-0+L,1'(,01. 0< -'(. '1> >0C. ()'( 803bE ()'( ,.E ()* 70,1(. ()'( I,* L*I08 ()* L0?1>'3J I,1*@ G.b.(?>*1(. (0 >*.-3,L* )08 ()*J b1*8 8)*()*3 (0 ?.* ' C3*'(*3 ()'1 .,C1 03 ' I*.. ()'1 .,C1 ,1 ()*,3

,1*^?'I,(J@ &'b* ' 70,1( 0< >*<,1,1C ()* L0I><'-*> 803>. Q.'(,.<JR '1> Q.0I?(,01 .*(R ,1 (*3+. 0<()*.* ,1*^?'I,(,*.@ G( (),. 70,1( J0? .)0?I> >,.(,1C?,.) L*(8**1 ()* .0I?(,01 .*( (0 ()* ,1*^?'I,(JM()* )'I<d7I'1* IJ,1C L*I08 ()* L0?1>'3J I,1*N '1> ()* .0I?(,01 .*( (0 ()* -01.(3',1( M8),-) 01IJ,1-I?>*. ()* I'((,-* 70,1(. L*I08 ()* L0?1>'3J I,1* ()'( I,* ,1 ^?'>3'1( %N@

;.)N$09 ?0(9-C 10&C M#6 1-%&0L% >*>



Name: Systems 2.2


In partnership with the Utah State Office of EducationLicensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license

. f l i

r .

t

Ready, Set, Go!

ReadyTopic: Determining if given values are solutions to an equation

Identify which of the given points are solutions to the following linear equations.

1. 3x + 2y = 12 2. 5x – y = 10

a. (2, 4) a. (2, 0)

b. (3, 2) b. (3, 0)

c. (4, 0) c. (0, -10)

d. (0, 6) d. (1, 1)

Find the value that will make each ordered pair a solution to the given equations.

3. x + y = 6 4. 2x + 4y = 8 5. 3x – y = 8

a. (2, __) a. (2, __) a. (2, __)

b. (0, __) b. (0, __) b. (0, __)

c. (__, 0) c. (__, 0) c. (__, 0)

Set

Topic: Graph linear inequalities

Graph the following inequalities on the coordinate plane. Name one point that is a solution

to the inequality and one point that is not a solution. Show algebraically and graphically that

your points are correct.

6. 7.

8.

9.

9

Systems 9



Name: Systems 2.2



Go Topic: Solving inequalities

Follow the directions for each problem below. (Show your work!)

10. a. Solve for x. Then graph the solution on the number line.

b. Select an x-value from your graph of the solution of the inequality. Replace x in the original

inequality with your chosen value. Does the inequality hold true?

c. Select an x-value that is outside of the solution set on your graph. Replace x in the original

inequality with your chosen value. Does the inequality still hold true?

11.

a. Solve for y.

b. Now imagine that your inequality is an equation. In other words, your solution will say y = ,

instead of y > or y <. With the equal sign, it should be the equation of a line. Graph your equation.

c. Find the y – intercept.

d. Find the slope.

e. Select a point that is above the line. ( , )

Replace the x and y–

values in the inequality

Is the inequality still true?

f. Select a point that is below the line. ( , )

Replace the x and y – values in the inequality

Is the inequality still true?

g. Explain which side of the line should be shaded.

h. Decide whether the line should be solid or dotted. Justify your decision.

Need help? Check out these related videos:

http://www.khanacademy.org/math/algebra/linear-equations-and-inequalitie/v/graphing-linear-inequalities-in-two-variables-2

10

Systems 10






! #$%# &'()*+'(,-. /,.,01 2304*-( 5 &!2

61 7'3(1*3.),7 8,() ()* 9(') :('(* ;<<,-* 0< =>?-'(,01!"#$%&$' )%'$* +,$ -*$.+"/$ -0110%& 2++*"3)+"0%450%-011$*#".647,.*$26"8$ 9:; <%=0*+$' 6"#$%&$:

> ? ; @ ? A A A

B 6 " # 8 * # 0 1 C = , 0 + 0 & C ' ) D & = *

#@A :0+* 0< ;1*B C01* 0< ()* ;()*3 ! #$%&'&() *+',-./0+'&+1 20.3

D'3E0. '1> DE'3,(' '3* -0+7'3,1F .(3'(*F,*. <03 83,(,1F *G?'(,01.

0< ()* H0?1>'3I E,1*. <03 ()* J2*( :,((*3K -01.(3',1(.@ L)*I '3*>,.-?..,1F ()*,3 803M 01 ()* .405, -01.(3',1(@

• #405,N D'( 7*1. 8,EE 3*G?,3* O <( # 0< .7'-*B 8),E* >0F 3?1. 3*G?,3* #P <( #@ D'3E0. '1> DE'3,(')'Q* ?7 (0 AO$ <( # 'Q',E'HE* ,1 ()* .(03'F* .)*> <03 7*1. '1> 3?1.B 8),E* .(,EE E*'Q,1F *10?F)300+ (0 +0Q* '30?1> ()* -'F*.@

D'3E0.R &*()0>N J6 +'>* ' ('HE*@ 6< 6 >01R( )'Q* '1I >0F.B ()*1 6 )'Q* 300+ <03 O$ -'(.@ 6< 6 ?.* .0+*0< ()* .7'-* <03 % >0FB ()*1 6 -'1 )'Q* SO -'(.@ T,() # >0F.B 6 -'1 H0'3>. S# -'(.@ U03 *'-)'>>,(,01'E >0FB 6 -'1 H0'3> P <*8*3 -'(.@ U30+ +I ('HE* 6 M108 ()* )6,1(*3-*7( 0< +I E,1* ,. O$ '1>()* .E07* ,. VPB .0 +I *G?'(,01 ,. ) W VP 7 X O$@K

DE'3,('R. &*()0>N J6 E*( 7 3*73*.*1( ()* 1?+H*3 0< >0F.B '1> ) ()* 1?+H*3 0< -'(.@ :,1-* >0F 3?1.3*G?,3* #P <( #B #P 7 3*73*.*1(. ()* '+0?1( 0< .7'-* ?.*> HI >0F.@ :,1-* -'( 7*1. 3*G?,3* O <( #B O ) 3*73*.*1(. ()* .7'-* ?.*> HI -'(.@ :0 +I *G?'(,01 ,.#P 7 X O ) W AO$@K

%@ :,1-* H0() *G?'(,01. 3*73*.*1( ()* .'+* ,1<03+'(,01B ()*I +?.( H* *G?,Q'E*1( (0 *'-) 0()*3@

'@ :)08 ()* .(*7. I0? -0?E> ?.* (0 (?31 DE'3,('R. *G?'(,01 ,1(0 D'3E0.R *G?'(,01@ =Y7E',18)I I0? -'1 >0 *'-) .(*7@

H@ :)08 ()* .(*7. I0? -0?E> ?.* (0 (?31 D'3E0.R *G?'(,01 ,1(0 DE'3,('R.@ =Y7E',1 8)I I0?

-'1 >0 *'-) .(*7@

#@ 9.* H0() D'3E0.R '1> DE'3,('R. +*()0>. (0 83,(* ()* *G?'(,01 0< ()* H0?1>'3I E,1* <03 ()* ./0-/6

84 5$./. -01.(3',1(@

• #/0-/684 9$./.N D'3E0. '1> DE'3,(' 7E'1 (0 ,1Q*.( +?-) 0< ()* Z%#[$ ()*I *'31*> <30+ ()*,3E'.( H?.,1*.. Q*1(?3* (0 7?3-)'.* -'( 7*1. '1> >0F 3?1.@ 6( 8,EE -0.( ZA# <03 *'-) -'( 7*1'1> Z[$ <03 *'-) >0F 3?1@

A@ :)08 ()* .(*7. I0? -0?E> ?.* (0 (?31 DE'3,('R. ./0-/684 5$./. *G?'(,01 ,1(0 D'3E0.R *G?'(,01@=Y7E',1 8)I I0? -'1 >0 *'-) .(*7@

P@ :)08 ()* .(*7. I0? -0?E> ?.* (0 (?31 D'3E0.R ./0-/684 5$./. *G?'(,01 ,1(0 DE'3,('R.@ =Y7E',1 8)II0? -'1 >0 *'-) .(*7@

11

Systems 11



! #$%# &'()*+'(,-. /,.,01 2304*-( 5 &!2

61 7'3(1*3.),7 8,() ()* 9(') :('(* ;<<,-* 0< =>?-'(,01!"#$%&$' )%'$* +,$ -*$.+"/$ -0110%& 2++*"3)+"0%450%-011$*#".647,.*$26"8$ 9:; <%=0*+$' 6"#$%&$:

61 '>>,(,01 (0 83,(,1F '1 *G?'(,01 0< ()* H0?1>'3I E,1*.B D'3E0. '1> DE'3,(' 1**> (0 F3'7) ()*,3 E,1*.

01 ' -003>,1'(* F3,>@

D'3E0.R *G?'(,01. '3* 83,((*1 ,1 "#$%&'()*&+,&%* .$+/@ DE'3,('R. *G?'(,01. '3* 83,((*1 ,1 "*0)10+1

.$+/@ \0() <03+. '3* 8'I. 0< 83,(,1F #()&0+ &230*($)"@

\0() D'3E0. '1> DE'3,(' M108 ()*I 01EI 1**> (0 7E0( (80 70,1(. ,1 03>*3 (0 F3'7) ' E,1*@

D'3E0.R .(3'(*FIN ]08 +,F)( D'3E0. ?.* ),. .E07*V,1(*3-*7( <03+B ) W VP 7 X O$B (0 7E0( (80 70,1(. 01

),. E,1*^

DE'3,('R. .(3'(*FIN ]08 +,F)( DE'3,(' ?.* )*3 .('1>'3> <03+B #P 7 X O ) W AO$B (0 7E0( (80 70,1(. 01

)*3 E,1*^ _DE'3,(' ,. 3*'EEI -E*Q*3B .0 .)* E00M. <03 ()* (80 *'.,*.( 70,1(. .)* -'1 <,1>@`

12

Systems 12



! #$%# &'()*+'(,-. /,.,01 2304*-( 5 &!2

61 7'3(1*3.),7 8,() ()* 9(') :('(* ;<<,-* 0< =>?-'(,01

!"#$%&$' )%'$* +,$ -*$.+"/$ -0110%& 2++*"3)+"0%450%-011$*#".647,.*$26"8$ 9:; <%=0*+$' 6"#$%&$:

#@A :0+* 0< ;1*B C01* 0< ()* ;()*3 a L*'-)*3 C0(*. ! #$%&'&() *+',-./0+'&+1 20.3

43+%$"&N b. .(?>*1(. *Y'+,1* D'3E0.R '1> DE'3,('R. +*()0>. <03 83,(,1F '1> F3'7),1F *G?'(,01. (0

<,( ()* J2*( :,((*3K -01.(3',1(. .(?>*1(. 8,EE .0E,>,<I ()* <0EE08,1F +'()*+'(,-.N

• L)* 7 V '1> ) V,1(*3-*7(. 0< ' E,1* 3*73*.*1( .,(?'(,01. 8)*3* 01* 0< ()* G?'1(,(,*.

3*73*.*1(*> HI '1 03>*3*>V7',3 _ 7 B ) ,. $@ L)* .E07* 0< ' E,1*'3 *G?'(,01 3*73*.*1(. ' 3'(* 0<

-)'1F*B .?-) '. ()* *Y-)'1F* 3'(* H*(8**1 -'(. '1> >0F.@ U03 *Y'+7E*B ,1 ()* .7'-*

-01.(3',1( ()* .E07* ,. VP .,1-* 8* E0.* .7'-* <03 P -'( 7*1. *'-) (,+* 8* ,1-3*'.* ()*

1?+H*3 0< >0F 3?1. HI %@ L)* .('1>'3> <03+ 0< ' *G?'(,01 0< ' E,1* -'1 H* 83,((*1 '. '

E,1*'3 -0+H,1'(,01 0< (*3+.B .?-) '. ()* -0.( -01.(3',1(B [$ 7 X A# ) W %#[$B 8)*3* ()* <,3.(

(*3+ 3*73*.*1(. ()* -0.( 0< 7 >0F 3?1.B ()* .*-01> (*3+ 3*73*.*1(. ()* -0.( 0< ) -'( 7*1.B

'1> ()* .?+ 03 -0+H,1'(,01 0< (*3+. F,Q*. ()* (0('E -0.(@ L),. .'+* 3*E'(,01.),7 -'1 H*

3*73*.*1(*> 8,() ' E,1*'3 *G?'(,01 ,1 .E07*V,1(*3-*7( <03+B B 8),-) -'7(?3*.

(80 ,+703('1( <'-(. 'H0?( (),. 3*E'(,01.),7N 8* -'1 '<<03> S <*8*3 -'( 7*1. <03 *Q*3I # >0F

3?1. 8* 7?3-)'.*B '1> ,< 8* 7?3-)'.* 10 >0F 3?1. 8* -'1 '<<03> P$ -'( 7*1.@

• 9.,1F 7307*3(,*. 0< *G?'E,(IB ' E0F,-'E .*G?*1-* 0< *G?,Q'E*1( *G?'(,01. -'1 H* -3*'(*> (0

+0Q* H*(8**1 ' E,1*'3 *G?'(,01 ,1 .E07*V,1(*3-*7( <03+B ) W :7 X ;B '1> .('1>'3> <03+B !7 X

<) W 9 @ L0 <,1> ()* 76,1(*3-*7( 0< ' E,1*'3 *G?'(,01 ,1 *,()*3 <03+B E*( ) W $ '1> .0EQ* ()*

*G?'(,01 <03 7 @ L0 <,1> ()* )6,1(*3-*7(B E*( 7 W $ '1> .0EQ* ()* *G?'(,01 <03 ) @ :,1-* (80

70,1(. >*(*3+,1* ' E,1*B ()* F3'7) 0< ' E,1* -'1 H* 0H(',1*> HI 7E0((,1F H0() ()* 7 V'1> ) V

,1(*3-*7(. <30+ .('1>'3> <03+B 03 ()* )6,1(*3-*7( '1> ' .*-01> 70,1( <0?1> HI -0?1(,1F ()*

3'(,0 0< 3,.* (0 3?1 '. ,1>,-'(*> HI ()* .E07*@

5$+& 6*0)10+1" 7$,3"N

895:;9< D3*'(* *G?'(,01. ,1 (80 03 +03* Q'3,'HE*. (0 3*73*.*1( 3*E'(,01.),7. H*(8**1 G?'1(,(,*.c

F3'7) *G?'(,01. 01 -003>,1'(* 'Y*. 8,() E'H*E. '1> .-'E*.@

895:;9= d*'33'1F* <03+?E'. (0 ),F)E,F)( ' G?'1(,(I 0< ,1(*3*.(B ?.,1F ()* .'+* 3*'.01,1F '. ,1

.0EQ,1F *G?'(,01.@

>&#0*&1 6*0)10+1"? 8966:9@0A 89>:B9@CA 79D:9@E

D03),F GHF$#& 5#0""I?

]'Q* .(?>*1(. 3*'> H0() D'3E0.R '1> DE'3,('R. +*()0>. <03 83,(,1F ()* *G?'(,01. 0< ()* E,1*

3*73*.*1(,1F ()* J2*( :,((*3.K .7'-* -01.(3',1(@ &'M* .?3* ()*I ?1>*3.('1> *'-) +*()0>B '1> -'1

*Y7E',1 >*(',E. 0< ()* +*()0>.B .?-) '. )08 D'3E0. <0?1> ()* .E07* '1> ) V,1(*3-*7( <30+ ),. ('HE*B'1> )08 DE'3,(' ,. <,1>,1F *'-) 0< ()* (*3+. ,1 )*3 E,1*'3 -0+H,1'(,01@ DE'3,<I,1F ()* >*(',E. 0<

()*.* (80 .(3'(*F,*. 8,EE 'EE08 .(?>*1(. (0 803M ,1>*7*1>*1(EI 03 -0EE'H03'(,Q*EI 01 ()* 3*+',1>*3

0< ()* ('.M@ 20,1( 0?( (0 .(?>*1(. ()'( ()*,3 803M 01 G?*.(,01 % ,. (0 .)08 )08 ()*I -'1 +0Q* <30+

01* <03+ 0< ()* *G?'(,01 0< ()* .7'-* -01.(3',1( E,1* (0 ()* 0()*3@



! #$%# &'()*+'(,-. /,.,01 2304*-( 5 &!2

61 7'3(1*3.),7 8,() ()* 9(') :('(* ;<<,-* 0< =>?-'(,01

!"#$%&$' )%'$* +,$ -*$.+"/$ -0110%& 2++*"3)+"0%450%-011$*#".647,.*$26"8$ 9:; <%=0*+$' 6"#$%&$:

:J%#$+& G6/0## K+$3%I?

:('3(,1F 8,() DE'3,('R. *G?'(,01 '1> +0Q,1F (0 D'3E0.R *G?'(,01 ,. ' +'((*3 0< .0EQ,1F DE'3,('R.

*G?'(,01 <03 )= L),. .)0?E> 10( H* >,<<,-?E( <03 .(?>*1(.B H?( F0,1F ()* 0()*3 >,3*-(,01 +'I 730Q*

+03* >,<<,-?E( ?1(,E ()*I 3*'E,e* ()*I -'1 4?.( 3*Q*3.* ()* .(*7.@ f?*.(,01. #VP 3*,1<03-* ()* 803M

.(?>*1(. >,> 73*Q,0?.EI HI '77EI,1F ()* .'+* .(3'(*F,*. (0 ()* -0.( -01.(3',1(@ 6< .(?>*1(. '3*

)'Q,1F >,<<,-?E(I 8,() ()* 'EF*H3'B )'Q* ()*+ +0Q* (0 ()* F3'7),-'E 703(,01 0< ()* ('.M@ L)*I .)0?E>H* <'+,E,'3 8,() ()* .(3'(*FI 0< 7E0((,1F 70,1(. 01 ' E,1* F,Q*1 ()* .E07* '1> ' .('3(,1F 70,1(B .?-) '.

()* ) V,1(*3-*7(@ 23*.. .(?>*1(. (0 <,1> ()* 7 V '1> ) V,1(*3-*7(. 0< ()* E,1* ,1 .('1>'3> <03+ HI '.M,1F

8)'( ()*I M108 'H0?( ()* -003>,1'(*. 0< ' 70,1( ()'( E,*. 01 ()* 7 V'Y,. 03 ()* ) V'Y,.@ ;1-* .(?>*1(.

<,1> ' .(3'(*FI <03 <,1>,1F H0() ,1(*3-*7(. 8)*1 ()* *G?'(,01 0< ()* E,1* ,. ,1 .('1>'3> <03+B )'Q*

()*+ .** )08 ()*.* 73,1-,7E*. -0?E> H* '77E,*> (0 <,1>,1F H0() ,1(*3-*7(. 8)*1 ()* *G?'(,01 0< ()*

E,1* ,. ,1 .E07*V,1(*3-*7( <03+@

b. I0? 73*7'3* <03 ()* 8)0E* F30?7 >,.-?..,01B ,>*1(,<I .(?>*1(. 8)0 -'1 83,(* ()* E,1*'3 *G?'(,01

<03 ()* .('3(V?7 -0.( -01.(3',1( ,1 H0() .('1>'3> <03+ '1> .E07*V,1(*3-*7( <03+B '1> .(?>*1(. 8)0

-'1 +0Q* 'EF*H3',-'EEI H*(8**1 ()* (80 <03+.@ bE.0 ,>*1(,<I .(?>*1(. 8)0 -'1 *Y7E',1 ()*

'EF03,()+ <03 <,1>,1F 7 V '1> ) V,1(*3-*7(.@

;(",3"" GHF$#& 5#0""I?

\*F,1 ()* >,.-?..,01 HI <0-?.,1F 01 ()* E'.( 7'3( 0< ()* ('.MB D'3E0.R '1> DE'3,('R. .(3'(*F,*. <03

7E0((,1F (80 70,1(. 01 ()* E,1* ?.,1F *,()*3 .E07*V,1(*3-*7( 03 .('1>'3> <03+@ bEE08 ()* .(3'(*FI 0<

<,1>,1F ()* ,1(*3-*7(. HI .*((,1F 01* 0< ()* Q'3,'HE*. *G?'E (0 e*30 (0 *+*3F*@ &'M* .?3* .(?>*1(.

-'1 '77EI (),. .(3'(*FI (0 H0() <03+. 0< ()* E,1*@

U0EE08,1F ()* F3'7),-'E >,.-?..,01 'H0?( ,1(*3-*7(.B 3*(?31 (0 ()* 'EF*H3' 0< f#VfP@ ]'Q* .(?>*1(.

73*.*1( ()* (80 <03+. 0< ()* *G?'(,01 <03 ()* .('3(V?7 -0.( -01.(3',1(B '1> .)08 ()*I '3* *G?,Q'E*1(

HI +0Q,1F H'-M '1> <03() H*(8**1 ()* (80 <03+. 'EF*H3',-'EEI@ L)*1 )'Q* .(?>*1(. <,1> ()* 7 V '1>

) V,1(*3-*7(. 0< ()* F3'7) 0< (),. E,1*B 7E0( ()* F3'7) ?.,1F ()*.* (80 70,1(.B '1> Q*3,<I ()'( ()* .E07*

0< ()* E,1* ,. ()* .'+* '. ()'( 73*>,-(*> HI ()* .E07*V,1(*3-*7( *G?'(,01@

8#(L)&1 >&01MA 6&*A K$? 6M"*&/" <9N



Name: Systems 2.3




Ready, Set, Go!

ReadyTopic: Determining points that satisfy equations and solving systems of equations

Three points are given. Each point is a solution to at least one of the equations. Find the

point that satisfies both equations. (This is the solution to the system!) Justify that the point

is a solution to both equations and that the others are not.

1. {

a. (-2 , 5)

b. (2 , 1)

c. (4 , 5)

2. {

a. (-1 , 0)

b. (6 , -3)

c. (0 , 3)

3. {

a. (7 , 2)

b. (2 , -14)

c. (-2 , 2)

4. {

a. (1 , 6)

b. (-3 , -2)

c. (-3 , 2)

SetTopic: Graphing linear equations from standard form using intercepts

Graph the following equations by finding the intercepts.

5. 6.


13

Systems 13



Name: Systems 2.3




7. 8.

Go

Topic: Adding and multiplying fractions

Add. Reduce your answers but leave as improper fractions when applicable.

Multiply. Reduce your answers but leave as improper fractions when applicable.

13.

14.

15.

16.

Need help? Check out these video lessons.

http://www.youtube.com/watch?v=cuNpXve18Pc

http://www.youtube.com/watch?v=6zixwWZ88tk

http://www.youtube.com/watch?v=oHNR0FK_lDE

9. 3

4+1

810.

3

5+

7

1011.

2

3+1

412.

4

7+

8

21

14

Systems 14












! #$%# &'()*+'(,-. /,.,01 2304*-( 5 &! 2

61 7'3(1*3.),7 8,() ()* 9(') :('(* ;<<,-* 0< =>?-'(,01

!"#$%&$' )%'$* +,$ -*$.+"/$ -0110%& 2++*"3)+"0%450%-011$*#".647,.*$26"8$ 9:; <%=0*+$' 6"#$%&$:

> ?

; @ ? A

A A : B 6 " # 8 * : # 0 1 C = , 0 + 0 & C 6 0 ) % D $ * " $

#@A 2'+7*3,1B '1> C**>,1B D,+* ! #$%&'(&) *+,)$-'%+,(+. /%-0

E'3F0. '1> EF'3,(' )'G* H**1 8033,*> 'H0?( .7'-* '1> .('3(I?7

-0.(. <03 ()*,3 7*( .,((*3. H?.,1*..J H?( ()*K 3*'F,L* ()*K 'F.0 )'G* 'F,+,( 01 ()* '+0?1( 0< (,+* ()*K )'G* <03 ('M,1B -'3* 0< ()* '1,+'F.

()*K H0'3>@ D0 M**7 (),1B. <',3J ()*K )'G* 'B3**> 01 ()* <0FF08,1B

(,+* -01.(3',1(.@

• 1)),(+. /(2)N E'3F0. '1> EF'3,(' *.(,+'(* ()'( -'(. 8,FF 3*O?,3* P +,1?(*. (8,-* ' >'KQ

+031,1B '1> *G*1,1BQ(0 <**> '1> -F*'1 ()*,3 F,((*3 H0R*.J <03 ' (0('F 0< %# +,1?(*. 7*3 >'K

<03 *'-) -'(@ S0B. 8,FF 3*O?,3* %$ +,1?(*. (8,-* ' >'K (0 <**> '1> 8'FMJ <03 ' (0('F 0< #$

+,1?(*. 7*3 >'K <03 *'-) >0B@ E'3F0. -'1 .7*1> ?7 (0 T )0?3. *'-) >'K <03 ()* +031,1B '1>

*G*1,1B <**>,1B.J H?( 1**>. ()* +,>>F* 0< ()* >'K 0<< <03 H'.*H'FF 73'-(,-* '1> B'+*.@

• #%23)$(+. /(2)N D)* (8,1. 7F'1 (0 .7*1> %P +,1?(*. *'-) >'K H3?.),1B '1> 7*((,1B *'-)

-'(J '1> #$ +,1?(*. *'-) >'K H'(),1B 03 7F'K,1B 8,() *'-) >0B@ EF'3,(' 1**>. (,+* 0<< ,1 ()*+031,1B <03 .8,+ (*'+ '1> *G*1,1B <03 )*3 '3( -F'..J H?( .)* -'1 .7*1> ?7 (0 T )0?3.

>?3,1B ()* +,>>F* 0< ()* >'K (0 7'+7*3 '1> 7F'K 8,() ()* 7*(.@

U3,(* ,1*O?'F,(,*. <03 *'-) 0< ()*.* '>>,(,01'F (,+* -01.(3',1(.@ :)'>* ()* .0F?(,01 .*( <03 *'-)

-01.(3',1( 01 .*7'3'(* -003>,1'(* B3,>.@

15

Systems 15



! #$%# &'()*+'(,-. /,.,01 2304*-( 5 &!2

61 7'3(1*3.),7 8,() ()* 9(') :('(* ;<<,-* 0< =>?-'(,01

!"#$%&$' )%'$* +,$ -*$.+"/$ -0110%& 2++*"3)+"0%450%-011$*#".647,.*$26"8$ 9:; <%=0*+$' 6"#$%&$:

#@A 2'+7*3,1B '1> C**>,1B D,+* V D*'-)*3 W0(*. ! #$%&'(&) *+,)$-'%+,(+. /%-0

"#$%&'(N 61 ()* -01(*R( 0< *R'+,1,1B -0+H,1'(,01. 0< -'(. '1> >0B. ()'( .'(,.<K ()*.* '>>,(,01'F

X2*( :,((*3Y -01.(3',1(. .(?>*1(. 8,FF 73'-(,-* ()* <0FF08,1B +'()*+'(,-.N ZW0(* ()'( *'-) 0< ()*.*,>*'. )'G* .?3<'-*> '1> )'G* H**1 *R'+,1*> '1> .0F,>,<,*> ,1 73*G,0?. ('.M.@ D),. ,. '1

07703(?1,(K <03 .(?>*1(. (0 73'-(,-* ()*.* ,>*'. ,1 ' 1*8 .*((,1B@[

• E01.(3',1(. -'1 H* 3*73*.*1(*> HK ,1*O?'F,(,*.@ D)* .0F?(,01 .*( 0< '1 ,1*O?'F,(K -01.,.(. 0<

' .*( 0< 70,1(. ()'( F,* ,1 ' )'F<I7F'1* H0?1>*> HK ()* *O?'(,01 '..0-,'(*> 8,() ()*

,1*O?'F,(K@

• :(?>*1(. .)0?F> 'F3*'>K )'G* 8*FFI73'-(,-*> 730-*>?3*. <03 83,(,1B *O?'(,01. 0< F,1*.

()30?B) B,G*1 70,1(. <30+ ' B3'7),-'F 7*3.7*-(,G*@ 61 '>>,(,01J ()*K .)0?F> 73'-(,-* 83,(,1B

()* *O?'(,01 0< ()* H0?1>'3K F,1* ?.,1B ()* (*3+. >*<,1*> HK ()* F'1B?'B* 0< ()*

-01.(3',1(.@ :(?>*1(. 8,FF >*(*3+,1* 8),-) )'F<I7F'1* (0 .)'>* (0 3*73*.*1( ()* .0F?(,01 (0

()* ,1*O?'F,(K HK ?.,1B ()*,3 .7*-,<,- X(*.( 70,1(.Y 03 HK ,>*1(,<K,1B 8),-) 70,1(. '3* ,+7F,*>HK ()* ,1*O?'F,(K .,B1@

• \ .)'>*> )'F<I7F'1* ,1 ()* -003>,1'(* B3,> ,. ' 3*73*.*1('(,01 0< ()* .0F?(,01 .*( <03 '1

,1*O?'F,(K ,1 (80IG'3,'HF*.@ D)* 70,1(. 01 ()* H0?1>'3K F,1* '3* 'F.0 ,1-F?>*> ,1 ()*

.0F?(,01 .*(Q,1>,-'(*> HK >3'8,1B ' .0F,> F,1* <03 ()* H0?1>'3KQ?1F*.. ()* H0?1>'3K ,. '

.(3,-( ,1*O?'F,(K '1> ()*1 ()* H0?1>'3K ,. 3*73*.*1(*> HK ' >0((*> F,1*@

)&$( +,-./-$/' 0&1#'N

23)4536 E3*'(* *O?'(,01. ,1 (80 03 +03* G'3,'HF*. (0 3*73*.*1( 3*F'(,01.),7. H*(8**1 O?'1(,(,*.]

B3'7) *O?'(,01. 01 -003>,1'(* 'R*. 8,() F'H*F. '1> .-'F*.@

23)4537 ^*73*.*1( -01.(3',1(. HK *O?'(,01. 03 ,1*O?'F,(,*.J '1> HK .K.(*+. 0< *O?'(,01. '1>_03

,1*O?'F,(,*.J '1> ,1(*373*( .0F?(,01. '. G,'HF* 03 101IG,'HF* 07(,01. ,1 ' +0>*F,1B -01(*R(@

238493:6 `3'7) ()* .0F?(,01. (0 ' F,1*'3 ,1*O?'F,(K ,1 (80 G'3,'HF*. '. ' )'F<I7F'1* Z*R-F?>,1B ()*

H0?1>'3K ,1 ()* -'.* 0< ' .(3,-( ,1*O?'F,(K[ @ @ @

8(;-,(/ +,-./-$/'<

=-#.1> ?@>&;( );-''A<

:(?>*1(. .)0?F> H* 'HF* (0 B*( .('3(*> 01 (),. ('.M 8,()0?( '>>,(,01'F .?7703(J .,1-* ,( ,. .,+,F'3 ,1

1'(?3* (0 ()* 803M ()*K >,> 01 XD00 a,B 03 W0( D00 a,BJ D)'( ,. ()* b?*.(,01@Y \.M .(?>*1(. (0

3*F'(* ()* 803M 0< 73*G,0?. ('.M. ,1 (),. +0>?F* (0 ()* 803M ()*K '3* B0,1B (0 >0 (0>'K@



! #$%# &'()*+'(,-. /,.,01 2304*-( 5 &!2

61 7'3(1*3.),7 8,() ()* 9(') :('(* ;<<,-* 0< =>?-'(,01

!"#$%&$' )%'$* +,$ -*$.+"/$ -0110%& 2++*"3)+"0%450%-011$*#".647,.*$26"8$ 9:; <%=0*+$' 6"#$%&$:

4B%;&$( ?+C-;; D$&#%A<

D)*3* ,. .0+* *R(3'1*0?. F'1B?'B* ,1 ()* 803>,1B 0< ()*.* -01.(3',1(.J .0 8'(-) <03 .(?>*1(. 8)0

'3* B*((,1B )?1B ?7 01 ()* '-(?'F ,1<03+'(,01 1**>*> (0 >*<,1* *'-) -01.(3',1(@ &'M* .?3* ()'(

.(?>*1(. '3* '((*1>,1B (0 ()* F'1B?'B* ,1 ()*.* -01.(3',1(. ()'( .?BB*.( ()*K '3* ,1*O?'F,(,*.J X?7 (0

T )0?3. *'-) >'KY@ :,1-* (,+* ,. B,G*1 ,1 H0() +,1?(*. '1> )0?3.J .(?>*1(. 8,FF )'G* (0 -01G*3(

H*(8**1 ?1,(. 0< (,+*@ 6< )0?3. '3* ?.*>J ()* -0*<<,-,*1(. 0< ()* (*3+. ,1 ()* ,1*O?'F,(,*. 8,FF H*<3'-(,01.@

U'(-) <03 .(?>*1(. 8)0 )'G* 10( K*( 730-*>?3'F,L*> ()* 730-*.. 0< 83,(,1B F,1*'3 ,1*O?'F,(,*. ,1

(80 G'3,'HF*. 03 <03 >*(*3+,1,1B ()* )'F<I7F'1* <03+*> HK ()* H0?1>'3K F,1* ,1 8),-) ()* .0F?(,01

.*( .)0?F> H* .)'>*>@ :(?>*1(. .)0?F> H* +0G,1B '8'K <30+ (*.(,1B ,1>,G,>?'F 70,1(. '1> (08'3>.

()* +03* *<<,-,*1( .(3'(*BK 0< ?.,1B ()* ,1*O?'F,(K 3*F'(,01.),7 (0 >*(*3+,1* ()* )'F<I7F'1* ,1 8),-)

()* .0F?(,01 .*( F,*.@

5E'1#'' ?@>&;( );-''A<

6>*1(,<K ,..?*. ()'( 80?F> 8'33'1( ' 8)0F* -F'.. >,.-?..,01J .?-) '. 83,(,1B ()* ,1*O?'F,(,*. ,1*,()*3 .('1>'3> <03+ 03 .F07*I,1(*3-*7( <03+J '1> )'G* +03* *R7*3( .(?>*1(. .)'3* ()*,3 803M 01

*'-) 0< ()* -01.(3',1(.@ 6( +,B)( H* ()'( ()* '3,()+*(,- 0< 803M,1B 8,() <3'-(,01'F -0*<<,-,*1(. ,.

+03* 0< ()* ,..?* ,1 (),. ('.M ()'1 ()* .0F?(,01 .*( (0 '1 ,1*O?'F,(K@

2;EF.(/ 8(-/GH +(,H D&< +G',(C' 63I



Name: Systems 2.4




Ready, Set, Go!

Ready

Topic: Substitution and Solving Equations

Determine whether is a solution to each problem.

1. ( )

2. ( )

3.

4.

5.

Determine the value of x that makes each equation true.

6. 7. ( ) 8.

Set

Topic: Creating equations, solving real world problems, solve systems of equations

A phone company offers a choice of three text-messaging plans. Plan A gives you unlimited text

messages for $10 a month; Plan B gives you 60 text messages for $5 a month and then charges you

$0.05 for each additional message; and Plan C has no monthly fee but charges you $0.10 per

message.

9. Write an equation for the monthly cost of each of the three plans.

10. If you send 30 messages per month, which plan is cheapest?

11. What is the cost of each of the three plans if you send 50 messages per month?

12. Determine the values for which each plan is the cheapest?

© 2012 www.flickr.com/photos/loungerie

16

Systems 16



Name: Systems 2.4




Go

Topic: Solve literal equations

Re-write each of the following equations for the indicated variable.

13. 3 x + 5y = 30 for y 14. 24 x + 6y = 360 for x

15. dfor32

801280c

d

16. Ffor)32(9

5 F C

17. =mx+b for b 18. x+ By=C for y

Need help? Check out these related videos.

What does it mean to be a solution?http://patrickjmt.com/an-intro-to-solving-linear-equations-what-does-it-mean-to-be-a-solution/

http://patrickjmt.com/solving-linear-equations/

Solving for a variable.

http://www.khanacademy.org/math/algebra/solving-linear-equations/v/solving-for-a-variable

17

Systems 17

http://patrickjmt.com/an-intro-to-solving-linear-equations-what-does-it-mean-to-be-a-solution/











! #$%# &'()*+'(,-. /,.,01 2304*-( 5 &!2

61 7'3(1*3.),7 8,() ()* 9(') :('(* ;<<,-* 0< =>?-'(,01

!"#$%&$' )%'$* +,$ -*$.+"/$ -0110%& 2++*"3)+"0%450%-011$*#".647,.*$26"8$ 9:; <%=0*+$' 6"#$%&$:

> ? ;

@ ? A A A : B 6 " # 8 * : # 0 1 C = , 0 + 0 & C C ' ) D & = *

#@A BCC D03 ;1*E ;1* D03 BCC ! #$%&'&() *+',-./0+'&+1 20.3

F'3C0. '1> FC'3,(' )'G* <0?1> ' 8'H (0 3*73*.*1( -0+I,1'(,01. 0<

-'(. '1> >0J. ()'( .'(,.<H *'-) 0< ()*,3 ,1>,G,>?'C K2*( :,((*3L

-01.(3',1(.E I?( ()*H 3*'C,M* ()'( ()*H 1**> (0 <,1> -0+I,1'(,01.

()'( .'(,.<H 'CC 0< ()* -01.(3',1(. .,+?C('1*0?.CH@ N)HO

%@ P*J,1 IH C,.(,1J ()* "#"$%& () *+%,-./*$*%" H0? )'G* 83,((*1 (0 3*73*.*1( ()* ./0-/456 7$./.

'1> .607, K2*( :,((*3L -01.(3',1(.@

#@ D,1> '( C*'.( A -0+I,1'(,01. 0< -'(. '1> >0J. ()'( 80?C> .'(,.<H I0() 0< ()* -01.(3',1(.

3*73*.*1(*> IH (),. .H.(*+ 0< ,1*Q?'C,(,*.@ R08 >0 H0? S108 ()*.* -0+I,1'(,01. 803SO

T@ D,1> '( C*'.( A -0+I,1'(,01. 0< -'(. '1> >0J. ()'( 80?C> .'(,.<H 01* 0< ()* -01.(3',1(.E I?(

10( ()* 0()*3@ D03 *'-) -0+I,1'(,01E *U7C',1 )08 H0? S108 ,( 803S. <03 01* 0< ()*

,1*Q?'C,(,*.E I?( 10( <03 0()*3O

V@ :)'>* ' 3*J,01 01 ' -003>,1'(* J3,> ()'( 80?C> 3*73*.*1( ()* "(/-$*(+ "%$ $( $0% "#"$%& ()

*+%,-./*$*%"1 =U7C',1 )08 H0? <0?1> ()* 3*J,01 (0 .)'>*@

A@ W*83,(* H0?3 .H.(*+. 0< ,1*Q?'C,(,*. (0 ,1-C?>* ()* '>>,(,01'C -01.(3',1(. <03 (,,'&+1 /&8,

'1> 6086,-&+1 /&8,@

X@ D,1> '( C*'.( A -0+I,1'(,01. 0< -'(. '1> >0J. ()'( 80?C> .'(,.<H 'CC 0< ()* -01.(3',1(.

3*73*.*1(*> IH (),. 1*8 .H.(*+ 0< ,1*Q?'C,(,*.@ R08 >0 H0? S108 ()*.* -0+I,1'(,01.

803SO

Y@

D,1> '( C*'.( A -0+I,1'(,01. 0< -'(. '1> >0J. ()'( 80?C> .'(,.<H .0+* 0< ()* -01.(3',1(.E I?(10( 'CC 0< ()*+@ D03 *'-) -0+I,1'(,01E *U7C',1 )08 H0? S108 ,( 803S. <03 .0+* ,1*Q?'C,(,*.E

I?( 10( <03 0()*3.O

Z@ :)'>* ' 3*J,01 0< ' -003>,1'(* J3,> ()'( 80?C> 3*73*.*1( ()* .0C?(,01 .*( (0 ()* .H.(*+ 0<

,1*Q?'C,(,*. -01.,.(,1J 0< 'CC V K2*( :,((*3L -01.(3',1(.@ =U7C',1 )08 H0? <0?1> ()* 3*J,01 (0

.)'>*@

[@ :)'>* ' 3*J,01 ,1 Q?'>3'1( % 0< ' -003>,1'(* J3,> ()'( 80?C> 3*73*.*1( 'CC 70..,IC*

-0+I,1'(,01. 0< -'(. '1> >0J. ()'( .'(,.<H ()* V K2*( :,((*3L -01.(3',1(.@ \),. .*( 0< 70,1(. ,.

3*<*33*> (0 '. ()* )%."*2/% 3%4*(+ .,1-* F'3C0. '1> FC'3,(' -'1 <*'.,ICH I0'3> '1H 0< ()*

-0+I,1'(,01. 0< -'(. '1> >0J. 3*73*.*1(*> IH ()* 70,1(. ,1 (),. 3*J,01 8,()0?( *U-**>,1J

'1H 0< ()*,3 -01.(3',1(. 01 (,+*E +01*H 03 .7'-*@

%$@ R08 ,. ()* <*'.,IC* 3*J,01 .)'>*> ,1 ][ >,<<*3*1( <30+ ()* .0C?(,01 .*( (0 ()* .H.(*+ 0<

,1*Q?'C,(,*. .)'>*> ,1 ]ZO

18

Systems 18



! #$%# &'()*+'(,-. /,.,01 2304*-( 5 &!2

61 7'3(1*3.),7 8,() ()* 9(') :('(* ;<<,-* 0< =>?-'(,01

!"#$%&$' )%'$* +,$ -*$.+"/$ -0110%& 2++*"3)+"0%450%-011$*#".647,.*$26"8$ 9:; <%=0*+$' 6"#$%&$:

#@A BCC <03 ;1*E ;1* <03 BCC^\*'-)*3 _0(*. ! #$%&'&() *+',-./0+'&+1 20.3

56%7*./ 8($% $( 9%.70%3": \)*3* '3* (80 70..,IC* 8'H. (0 ,+7C*+*1( (),. ('.S@ \)* <,3.( G*3.,01 ,.

(0 )'G* .(?>*1(. 803S 01 <,1>,1J ()* .0C?(,01 .*( (0 ' .H.(*+ 0< ,1*Q?'C,(,*. ()'( ,1G0CG*. (80 0< ()*K2*( :,((*3L -01.(3',1(.E ()*1 +0G* (0 803S,1J 01 <,1>,1J ()* .0C?(,01 .*( (0 ()* .H.(*+ 0<

,1*Q?'C,(,*. ()'( ,1G0CG* 'CC V -01.(3',1(.@ B. .(?>*1(. 803S (0 <,1> ()* <*'.,IC* 3*J,01 <03 'CC V

-01.(3',1(.E ()*H 8,CC 1**> (0 <,1> ()* 70,1(. 0< ,1(*3.*-(,01 0< ()* I0?1>'3H C,1*.@ 6( ,. .?<<,-,*1( (0

C*( .(?>*1(. *.(,+'(* ()* 70,1(. 0< ,1(*3.*-(,01 <30+ ()* J3'7)E 03 ()*H -'1 ?.* 73*G,0?.CH

>*G*C07*> 'CJ*I3',- .(3'(*J,*. (0 <,1> ()* 70,1(. 0< ,1(*3.*-(,01E ,< .?-) .(3'(*J,*. '3* 'G',C'IC*@ \)*

1*U( ('.SE K`*( (0 ()* 20,1(L <0-?.*. 01 'CJ*I3',- 730-*..*. <03 <,1>,1J ()* 70,1(. 0< ,1(*3.*-(,01@ 6<

H0? 80?C> 73*<*3E H0? -'1 ,+7C*+*1( (),. ('.S ,1 (80 7'3(.@ 61 7'3( %aQ?*.(,01. %bVcE )'G* .(?>*1(.

803S 01 <,1>,1J ()* .0C?(,01 .*( (0 ()* .H.(*+ 0< ,1*Q?'C,(,*. ()'( ,1G0CG*. 4?.( (80 0< ()* K2*(

:,((*3L -01.(3',1(.@ P*<03* 730-**>,1J (0 7'3( #E .(?>*1(. -0?C> 803S 01 ()* 1*U( ('.S K`*( (0 ()*

20,1(L 8)*3* ()*H 8,CC *U'+,1* .7*-,<,- .(3'(*J,*. <03 <,1>,1J ()* 70,1(. 0< ,1(*3.*-(,01. 0< ()*

I0?1>'3H C,1*.@ \)* .(?>*1(. -'1 ()*1 3*(?31 (0 7'3( # 0< (),. ('.S aQ?*.(,01. Ab%$c (0 -01.,>*3 ()*

.H.(*+ 0< ,1*Q?'C,(,*. ()'( ,1-C?>*. 'CC <0?3 K2*( :,((*3L -01.(3',1(.@ 6< H0? ?.* ()* <,3.( G*3.,01 0<(),. ('.SE H0? 1**> (0 >*-,>* ,< H0?3 .(?>*1(. 8,CC I*1*<,( <30+ ()* J3'7),-'CE 1?+*3,-'C '1>

'CJ*I3',- 803S 0< Q?*.(,01 % 0< K`*( (0 ()* 20,1(dE 03 ,< ()*H 'C3*'>H )'G* ()* 1*-*..'3H

?1>*3.('1>,1J <03 'CJ*I3',-'CCH .0CG,1J ' .H.(*+ 0< *Q?'(,01.@ 6< ()*H )'G* 10( 'C3*'>H <0?1> ()*

70,1(. 0< ,1(*3.*-(,01 <03 .0+* 0< ()* 7',3. 0< -01.(3',1(. ?.,1J '1 'CJ*I3',- +*()0>E Q?*.(,01. #

'1> T 0< K`*( (0 ()* 20,1(dL 8,CC )'G* ()*+ >0 .0@

;-36("%e 61 ()* -01(*U( 0< *U'+,1,1J -0+I,1'(,01. 0< -'(. '1> >0J. ()'( .'(,.<H 'CC 0< ()* K2*(

:,((*3L -01.(3',1(. .(?>*1(. 8,CC .0C,>,<H ()* <0CC08,1J +'()*+'(,-.e

• \)* .0C?(,01 .*( (0 ' .H.(*+ 0< ,1*Q?'C,(,*. 3*73*.*1(. ()* .*( 0< 70,1(. ()'( .'(,.<H 'CC 0< ()*

,1*Q?'C,(,*. .,+?C('1*0?.CH@ \)'( ,.E ' 70,1( ,1 ()* .0C?(,01 .*( 0< ' .H.(*+ 0< ,1*Q?'C,(,*.

+'S*. 'CC 0< ()* ,1*Q?'C,(,*. (3?*@ F01(*U(.E .?-) '. K2*( :,((*3.L +'H ,+7CH '>>,(,01'C

-01.(3',1(.E .?-) '. ) ! $ 03 9 ! $E ()'( '3* 10( *U7C,-,(CH .('(*>@

• B .)'>*> 3*J,01 01 ()* -003>,1'(* J3,> ,. ?.*> (0 3*73*.*1( ()* .0C?(,01 .*( <03 ' .H.(*+ 0<

,1*Q?'C,(H ,1 (80bG'3,'IC*.@ 61 (),. -'.*E ()* 70,1(. 01 ()* I0?1>'3H C,1*. ()'( (3'-* 0?( ()*

3*J,01 '3* 'C.0 ,1-C?>*> ,1 ()* .0C?(,01 .*(E ,1>,-'(*> IH >3'8,1J ' .0C,> C,1* <03 ()*

I0?1>'3,*.@ B .)'>*> 70CHJ01'C 3*J,01 ,. ?.*> (0 3*73*.*1( ()* .0C?(,01 (0 ' .H.(*+ 0<

-01.(3',1(.@

<(3% 5$.+=.3=" >(7-"e

?1<@A1B W*73*.*1( -01.(3',1(. IH *Q?'(,01. 03 ,1*Q?'C,(,*.E '1> IH .H.(*+. 0< *Q?'(,01. '1>f03

,1*Q?'C,(,*.E '1> ,1(*373*( .0C?(,01. '. G,'IC* 03 101bG,'IC* 07(,01. ,1 ' +0>*C,1J -01(*U(@

?1C@D1EF `3'7) ()* .0C?(,01. (0 ' C,1*'3 ,1*Q?'C,(H ,1 (80 G'3,'IC*. '. ' )'C<b7C'1* a*U-C?>,1J ()*

I0?1>'3H ,1 ()* -'.* 0< ' .(3,-( ,1*Q?'C,(HcE '1> J3'7) ()* .0C?(,01 .*( (0 ' .H.(*+ 0< C,1*'3

,1*Q?'C,(,*. ,1 (80 G'3,'IC*. '. ()* ,1(*3.*-(,01 0< ()* -033*.701>,1J )'C<b7C'1*.@



! #$%# &'()*+'(,-. /,.,01 2304*-( 5 &!2

61 7'3(1*3.),7 8,() ()* 9(') :('(* ;<<,-* 0< =>?-'(,01

!"#$%&$' )%'$* +,$ -*$.+"/$ -0110%& 2++*"3)+"0%450%-011$*#".647,.*$26"8$ 9:; <%=0*+$' 6"#$%&$:

C%/.$%= 5$.+=.3=": 81G1EH 81G1FH ?1<@A1F

I.-+70 JK0(/% </.""L: "#$"% &'( )*(+ #,

61(30>?-* ()* -01-*7( 0< ' .H.(*+ 0< ,1*Q?'C,(,*. '. ' .*( 0< ,1*Q?'C,(,*. ()'( 3*73*.*1( -01>,(,01.

()'( )'G* (0 I* .'(,.<,*> .,+?C('1*0?.CH@ 61G,(* .(?>*1(. (0 -3*'(* ' .H.(*+ 0< ,1*Q?'C,(,*. (0

3*73*.*1( ()* ./0-/456 7$./. '1> .607, -01.(3',1(.@ :(?>*1(. 8,CC )'G* 83,((*1 ()* ,1>,G,>?'C

,1*Q?'C,(,*. 3*73*.*1(,1J ()*.* (80 -01.(3',1(. '. 7'3( 0< ()*,3 -C'.. 803S <03 K\00 P,J 03 _0( \00P,JL@ `,G* ()*+ ' -0?7C* 0< +,1?(*. (0 -0CC*-( ()*,3 ,1>,G,>?'C ,1*Q?'C,(,*. ,1(0 ' .H.(*+ 0<

,1*Q?'C,(,*.@

:(?>*1(. +'H )'G* 83,((*1 ()*,3 ,1*Q?'C,(,*. ,1 .C07*b,1(*3-*7( <03+ 03 ,1 .('1>'3> <03+@ :*C*-(

.(?>*1(. (0 73*.*1( >,<<*3*1( <03+. 0< ()* ,1*Q?'C,(,*. ()'( -0?C> I* ,1-C?>*> ,1 ()* .H.(*+E 7*3)'7.

,1-C?>,1J ' .H.(*+ ()'( -01(',1. +03* ()'1 01* <03+ <03 83,(,1J ()* ,1*Q?'C,(,*.@ 6< ()* .'+*

-01.(3',1( ,. 3*73*.*1(*> IH ,1*Q?'C,(,*. 83,((*1 ,1 >,<<*3*1( <03+.E '.S ,< I0() ,1*Q?'C,(,*. .)0?C>

I* ,1-C?>*> ,1 ()* .H.(*+@

@M6/(3% J5&.// N3(-6L:

N'(-) <03 .(3'(*J,*. <03 >*(*3+,1,1J ,< ' -0+I,1'(,01 <,(. I0() -01.(3',1(.@ g0 .(?>*1(. J?*.. '1>

-)*-S 70,1(. ,1 I0() ,1*Q?'C,(,*.E 03 >0 ()*H C00S <03 0G*3C'77,1J 3*J,01.O R08 >0 ()*H +'S* ?.* 0<

()* 70,1(. 0< ,1(*3.*-(,01 I*(8**1 ()* I0?1>'3H C,1*. 3*73*.*1(,1J ()* -01.(3',1(.O N'(-) <03

*<<03(. (0 >*(*3+,1* 8)*3* ()* 70,1( 0< ,1(*3.*-(,01 ,. C0-'(*>E I?( 10(* ()'( .7*-,<,- 'CJ*I3',-

+*()0>. <03 >0,1J .0 8,CC I* (3*'(*> ,1 <?(?3* ('.S.E .0 ' G,.?'C *.(,+'(* 0< ()* -003>,1'(*. 0< ()*

70,1( 0< ,1(*3.*-(,01 8,CC 803S <03 ()* (,+* I*,1J@ :0+* .(?>*1(. +'H J?*.. ()* -003>,1'(*. 0< ()*

70,1( 0< ,1(*3.*-(,01 '1> ()*1 (*.( (0 .** ,< ()* 70,1( '-(?'CCH .'(,.<,*. I0() -01.(3',1(.@

23*.. .(?>*1(. (0 <,1> 0%% 0< ()* 70,1(. ()'( 803SE 10( 4?.( ()0.* 0IG,0?. 01*. ()'( C,* <'3 '8'H <30+

()* I0?1>'3H C,1*. '1> -C0.*3 (0 ()* 03,J,1@ h0? +,J)( 3*+,1> .(?>*1(. ()'( ()* *G*1(?'C J0'C ,. (0

-0CC*-( ()* +0.( >',CH ,1-0+*E .0 ()*H 8,CC 8'1( (0 ,>*1(,<H -0+I,1'(,01. 0< -'(. '1> >0J. ()'( 10(

01CH 803S <03 I0() -01.(3',1(.E I?( 8,CC 'C.0 I3,1J ,1 ()* +0.( ,1-0+*^70,1(. 8),-) ,1(?,(,G*CH C,*<'3()*3 '8'H <30+ ()* 03,J,1 '1> 1*'3*3 (0 ()* I0?1>'3H C,1*.@

A*"7-"" JK0(/% </.""L:

\),. >,.-?..,01 .)0?C> <0-?. 01 .)'>,1J ()* 0G*3C'77,1J 3*J,01. 0< ()* (80 )'C<b7C'1*. 8),-)

3*73*.*1( ()* .0C?(,01 .*(. 0< ()* ,1>,G,>?'C ,1*Q?'C,(,*.E '1> ()'( (),. .)'>*> 3*J,01 ,. ()* .0C?(,01

.*( <03 ()* .H.(*+ 0< ,1*Q?'C,(,*.@ 20,1( 0?( ()'( ()* .0C?(,01 (0 ()* .H.(*+ 0< -01.(3',1(. C,* 01CH ,1

()* <,3.( Q?'>3'1(E '1> '3* ,1 <'-( 4?.( ' <,1,(* .*( 0< >,.-3*(* 70,1(. ,1 ()* <,3.( Q?'>3'1(E .,1-* 8*

-'1i( )'G* ' <3'-(,01 0< ' -'( 03 ' >0J@ D03 -01G*1,*1-*E 8* 8,CC .)'>* ' 70CHJ01'C 3*J,01 ,1 ()* <,3.(

Q?'>3'1( (0 3*73*.*1( ()* .0C?(,01 .*( (0 ()* .H.(*+ 0< -01.(3',1(.E .,1-* (),. 3*J,01 >0*. ,1-C?>*

()* 70,1(. 0< ,1(*3*.(@

I.-+70 JK0(/% </.""L: "-$"#. &'( )*(+ /,61G,(* .(?>*1(. (0 -3*'(* ' 1*8 .H.(*+ 0< ,1*Q?'C,(,*. (0 3*73*.*1( 'CC <0?3 K2*( :,((*3L -01.(3',1(.E

,1-C?>,1J (,,'&+1 /&8, '1> 6086,-&+1 /&8,@ `,G* ()*+ ' -0?7C* 0< +,1?(*. (0 83,(* (),. 1*8 .H.(*+

0< ,1*Q?'C,(,*.@



! #$%# &'()*+'(,-. /,.,01 2304*-( 5 &!2

61 7'3(1*3.),7 8,() ()* 9(') :('(* ;<<,-* 0< =>?-'(,01

!"#$%&$' )%'$* +,$ -*$.+"/$ -0110%& 2++*"3)+"0%450%-011$*#".647,.*$26"8$ 9:; <%=0*+$' 6"#$%&$:

@M6/(3% J5&.// N3(-6L:

N'(-) <03 .(3'(*J,*. <03 >*(*3+,1,1J ,< ' -0+I,1'(,01 <,(. 'CC V -01.(3',1(.@ g0 .(?>*1(. J?*.. '1>

-)*-S 70,1(. ,1 'CC <0?3 ,1*Q?'C,(,*.E 03 >0 ()*H C00S <03 0G*3C'77,1J 3*J,01.O B. .(?>*1(. 803S 01

(),. 7'3( 0< ()* ('.S ()*H .)0?C> 10(,-* ()'( 'CC <0?3 0< ()* -01.(3',1(. '3* 10( 1*-*..'3H^()*

(,,'&+1 /&8, -01.(3',1( C,*. 0?(.,>* 0< ()* 3*J,01 >*(*3+,1*> IH ()* 0()*3 ()3** -01.(3',1(.@

A*"7-"" JK0(/% </.""L::)'3* .?--*..<?C .(3'(*J,*. <03 <,1>,1J ()* 70CHJ01'C <*'.,IC* 3*J,01@ \)* <*'.,IC* 3*J,01 C,*. ,1

Q?'>3'1( %E '1> ,. ()*3*<03* ' .?I.*( 0< ()* '-(?'C .0C?(,01 .*( <03 ()* .H.(*+ 0< ,1*Q?'C,(,*. ()'(

*U(*1>. ,1(0 Q?'>3'1(. 66E 666 '1> 6/@ &'S* .?3* ()'( I0() ()* .0C?(,01 .*( (0 ()* .H.(*+ 0<

,1*Q?'C,(,*. 0+' ()* .0C?(,01 .*( <03 ()* -01.(3',1(. a,@*@E <*'.,IC* 3*J,01c '3* ,>*1(,<,*>@ P3,1J 0?(

()'( ()* <*'.,IC* 3*J,01 ,. ' .0C?(,01 .*( (0 ()* .H.(*+ 0< ,1*Q?'C,(,*. ()'( ,1-C?>*. 9 ! $ '1> ) ! $ '.

,+7C,*> -01.(3',1(.@

?/*4+%= C%.=#H 5%$H N(: 5#"$%&" F1O



Name: Systems 2.5




Ready, Set, Go!

Ready

Topic: Graphing two variable inequalities

For each inequality and graph, pick a point and use it to determine which half-plane should

be shaded, then shade the correct half-plane.

1.1

5 x 4 2. 3 x+5

3. 5 x 2 y 10 4. 3 x+4 y ³ 24

© 2012 www.flickr.com/photos//dugspr

19

Systems 19



Name: Systems 2.5




Set

Topic: Writing two variable inequalities

Given the graph with the regions that are shaded write the inequality or system of

inequalities.

5. 6.

7. 8.

-10

-10

10

10

-10

-10

10

10

-10

-10

10

10

-10

-10

10

10

20

Systems 20



Name: Systems 2.5




Go

Topic: Proportional relationships

For each proportional relationship below, one representation is provided. Show the

remaining representations and explain any connections you notice between

representations.

9. Equation: Graph

Table

Days Cost123

4

81624

32

Create a context

10. Equation: Graph Table

Create a context

Claire earns $9 per week allowance.

21

Systems 21



Name: Systems 2.5




11. Equation: y = 3 x Graph

Table

Create a context

12. Equation: Graph

Table

Days Cost

Create a context

Need Help? Check out these related videos.


22

Systems 22






! #$%# &'()*+'(,-. /,.,01 2304*-( 5 &! 2

61 7'3(1*3.),7 8,() ()* 9(') :('(* ;<<,-* 0< =>?-'(,01!"#$%&$' )%'$* +,$ -*$.+"/$ -0110%& 2++*"3)+"0%450%-011$*#".647,.*$26"8$ 9:; <%=0*+$' 6"#$%&$:

> ?;@? AAA:B6"#8*:#01C=,0+0&C=,0+0&+$/$@;@

#@A B*( (0 ()* 20,1(C ! #$%&'&() *+',-./0+'&+1 20.3

D'3E0. '1> DE'3,(' 1**> (0 -E*'1 ()* .(03'F* .)*> 8)*3* ()*G 7E'1

(0 H0'3> ()* 7*(.@ I)*G )'J* >*-,>*> (0 ),3* ' -0+7'1G (0 -E*'1()* 8,1>08.@ K<(*3 -0EE*-(,1F ()* <0EE08,1F ,1<03+'(,01L ()*G )'J* -0+* (0 G0? <03 )*E7 >*-,>,1F8),-) 8,1>08 -E*'1,1F -0+7'1G ()*G .)0?E> ),3*@

• #4+.5&+, 678-,.. 9&+'$: ;%,0+,-. -)'3F*. MN$ <03 *'-) .*3J,-* -'EEL 7E?. M%$ 7*3 8,1>08@

• O<0+,=%,.. 9&+'$: ;%,0+,-. -)'3F*. M#N <03 *'-) .*3J,-* -'EEL 7E?. M%N 7*3 8,1>08@

%@ P),-) -0+7'1G 80?E> G0? 3*-0++*1>L '1> 8)GQ 23*7'3* '1 '3F?+*1( (0 -01J,1-* D'3E0. '1>DE'3,(' ()'( G0?3 3*-0++*1>'(,01 ,. 3*'.01'HE*@ R6( ,. 'E8'G. +03* -01J,1-,1F ,< G0? -'1.?7703( G0?3 -E',+ ,1 +?E(,7E* 8'G.@ S08 +,F)( G0? .?7703( G0?3 3*-0++*1>'(,01 ?.,1F '

('HE*Q K F3'7)Q KEF*H3'QT

U0?3 73*.*1('(,01 (0 D'3E0. 3*+,1>. ),+ 0< .0+*(),1F )* )'. H**1 (),1V,1F 'H0?(W)08 (0 <,1> ()*-003>,1'(*. 0< ()* 70,1(. 8)*3* ()* H0?1>'3G E,1*. ,1 ()* O2*( :,((*3X -01.(3',1(. ,1(*3.*-(@ S*80?E> E,V* (0 >0 (),. 'EF*H3',-'EEG .,1-* )* (),1V. F?*..,1F ()* -003>,1'(*. <30+ ' F3'7) +,F)( H*E*.. '--?3'(*@

#@ P3,(* *Y?'(,01. <03 ()* <0EE08,1F (80 -01.(3',1(.@

• #80>,

• #/0-/?48 ;$./.

Z,1> 8)*3* ()* (80 E,1*. ,1(*3.*-( 'EF*H3',-'EEG@ [*-03> *10?F) .(*7. .0 ()'( .0+*01* *E.*-'1 <0EE08 G0?3 .(3'(*FG@

\@ ]08 <,1> ()* 70,1( 0< ,1(*3.*-(,01 <03 ()* (80 (,+* -01.(3',1(.@

• @,,'&+1 2&A,

• <0A8,-&+1 2&A,

23

Systems 23



! #$%# &'()*+'(,-. /,.,01 2304*-( 5 &!2

61 7'3(1*3.),7 8,() ()* 9(') :('(* ;<<,-* 0< =>?-'(,01!"#$%&$' )%'$* +,$ -*$.+"/$ -0110%& 2++*"3)+"0%450%-011$*#".647,.*$26"8$ 9:; <%=0*+$' 6"#$%&$:

#@A B*( (0 ()* 20,1(C ^ I*'-)*3 ]0(*. ! #$%&'&() *+',-./0+'&+1 20.3

"#$%&'(_ I),. ('.V ,. >*.,F1*> (0 .0E,>,<G F3'7),-'EL 1?+*3,-'E '1> 'EF*H3',- .(3'(*F,*. <03 .0EJ,1F '

.G.(*+ 0< (80 E,1*'3 *Y?'(,01.@ P),E* ()* 70,1( 0< ,1(*3.*-(,01 01 ' F3'7) 3*73*.*1(. ()* .0E?(,01(0 ()* .G.(*+L ,( -'1 H* >,<<,-?E( (0 ,>*1(,<G ()* *`'-( -003>,1'(*. 0< (),. 70,1( 0< ,1(*3.*-(,01@ K('HE* -'1 730J,>* '1 *<<,-,*1( OF?*.. '1> -)*-VX .(3'(*FG <03 -E0.,1F ,1 01 ()* -003>,1'(*. 0< ' 70,1(0< ,1(*3.*-(,01 8)*1 ()* -003>,1'(*. '3* 10( ,1(*F*3.@ K ('HE* +,F)( 'E.0 .?FF*.( '1 'EF*H3',-.(3'(*FG_ .,1-* 8* '3* E00V,1F <03 '1 ,17?( J'E?* <03 8),-) H0() 0?(7?( J'E?*. '3* ()* .'+*L 8* -'1.*( ()* (80 *Y?'(,01. *Y?'E (0 *'-) 0()*3 (0 <,1> ()* 7 aJ'E?* <03 8),-) ()* (80 )?J'E?*. '3* *Y?'E@;1-* (),. O.*( ()* *Y?'(,01. *Y?'E (0 *'-) 0()*3X .(3'(*FG ,. *.('HE,.)*>L ()* 10(,01 0< .?H.(,(?(,1F01* 0< ()* *`73*..,01. <03 ) ,1(0 ()* 0()*3 *Y?'(,01 -'1 H* .?FF*.(*>L .,1-* .*((,1F H0() *Y?'(,01.*Y?'E (0 *'-) 0()*3 ,. *Y?,J'E*1( (0 .?H.(,(?(,1F 01* *`73*..,01 <03 ) ,1(0 ()* 0()*3 *Y?'(,01@

)&$( +,-./-$/' 0&1#'_

2345637 :0EJ* .G.(*+. 0< E,1*'3 *Y?'(,01. *`'-(EG '1> '7730 ,+'(*EG R*@F@L 8,() F3'7).TL <0-?.,1F01 7',3. 0< E,1*'3 *Y?'(,01. ,1 (80 J'3,'HE*.@

4(8-,(/ +,-./-$/'9 :3;3<= 23)5>3<= 23)5>3<= 234563?@

A-#.1B CDB&8( )8-''E9 !"

[*'> ()* ,1,(,'E 8,1>08a-E*'1,1F -01(*`( 8,() ()* -E'..L '1> 70,1( 0?( ()* .('(*+*1( ()'( ()*.(?>*1(. .)0?E> 730J,>* +?E(,7E* 3*73*.*1('(,01. (0 4?.(,<G ()*,3 3*-0++*1>'(,01 '. (0 8),-)8,1>08 -E*'1,1F -0+7'1G D'3E0. '1> DE'3,(' .)0?E> ?.*@

5F%8&$( C+G-88 H$&#%E9

U0?3 .(?>*1(. +'G 'E3*'>G H* <'+,E,'3 8,() .(3'(*F,*. <03 <,1>,1F ()* 70,1( 0< ,1(*3.*-(,01 0< ()*(80 E,1*. <30+ ()* 8,1>08 -E*'1,1F -0+7'1G -01.(3',1(.@ I)* -01(*`( 70,1(. 0?( '1 ,+703('1(,..?*W8),-) -0+7'1G ()*G .)0?E> .*E*-( >*7*1>. 01 ()* 1?+H*3 0< 8,1>08. ,1 ()* .(03'F* .)*>@:,1-* 8* >01b( V108 (),. 1?+H*3L ()* 3*-0++*1>'(,01 .)0?E> H* +'>* ,1 (*3+. 0< (),. ?1V1081'+0?1(@ 61 (),. -'.*L H0() -0+7'1,*. -)'3F* M%$$ (0 8'.) N 8,1>08.@ B<0+,=%,.. 9&+'$: ;%,0+,-. ,. -)*'7*3 ,< ()* (8,1. )'J* <*8*3 ()'1 N 8,1>08. (0 H* -E*'1*>L '1> #4+.5&+, 678-,.. ,. -)*'7*3 ,<()*G )'J* +03*@

23*.. .(?>*1(. (0 *`7E03* F3'7),-'EL 1?+*3,-'E '1> 'EF*H3',- .0E?(,01. '1> (0 H* 'HE* (0 >*.-3,H*()* -011*-(,01. H*(8**1 *'-) R.** 7?370.* .('(*+*1( 'H0J*T@

>I'1#'' CDB&8( )8-''E9

c*F,1 HG )'J,1F ' F3'7) 0< ()* .-*1'3,0 73*.*1(*> ,1 03>*3 (0 ,>*1(,<G ()'( ()* 70,1( 0< ,1(*3.*-(,01>*(*3+,1*. ()* 1?+H*3 0< 8,1>08. <03 8),-) H0() -0+7'1,*. 80?E> -0.( ()* .'+*L '1> ()'( ()*F3'7) -'1 H* ?.*> (0 >*(*3+,1* 8),-) -0+7'1G ,. -)*'7*3 01 *,()*3 .,>* 0< ()* 70,1( 0<,1(*3.*-(,01@



! #$%# &'()*+'(,-. /,.,01 2304*-( 5 &!2

61 7'3(1*3.),7 8,() ()* 9(') :('(* ;<<,-* 0< =>?-'(,01!"#$%&$' )%'$* +,$ -*$.+"/$ -0110%& 2++*"3)+"0%450%-011$*#".647,.*$26"8$ 9:; <%=0*+$' 6"#$%&$:

]*`( *`'+,1* ' ('HE* .)08,1F '1 ,17?( -0E?+1 <03 ()* 1?+H*3 0< 8,1>08.L '1> (80 -0E?+1. <03 ()*'+0?1( -)'3F*> HG *'-) -0+7'1G (0 -E*'1 ()'( 1?+H*3 0< 8,1>08.@ D011*-( ()* 308 8)*3* ()*0?(7?(. '3* ()* .'+* (0 ()* 70,1( 0< ,1(*3.*-(,01 01 ()* F3'7)@ K>> ' 308 (0 (),. ('HE* (0 3*73*.*1(()* F*1*3'E -'.*L '. <0EE08.@ ]0(*L 'E.0L ()* ) a,1(*3-*7( 0< ()* E,1*. 3*73*.*1(. ()* ,1,(,'E -)'3F* <03*'-) -0+7'1GL ,< 10 8,1>08. '3* -E*'1*>@

]?+H*3 0< P,1>08. #4+.5&+, 678-,.. D0.( B<0+,=%,.. D0.(

$ N$ #N

% A$ d$

# e$ NN

\ f$ e$

d g$ fN

N %$$ %$$

A %%$ %%N

e %#$ %\$

C N$ h %$C #N h %NC

9.* (),. ('HE* (0 >,.-?.. )08 .*((,1F ()* (80 *`73*..,01. DE F GEC '1> HD F GDC *Y?'E (0 *'-) 0()*380?E> H* *Y?,J'E*1( (0 <,1>,1F ()* 308 8)*3* H0() -0+7'1,*. -)'3F* ()* .'+* '+0?1(@ I),. ,. 'E.0' .?H.(,(?(,01 +*()0> ,< ()* *`73*..,01. '3* (3*'(*> '. 7'3(. 0< ()* *Y?'(,01. ; I DE F GEC '1>; I HD F GDC @ I)*1 ()* *`73*..,01 DE F GEC -'1 H* .?H.(,(?(*> ,1(0 ()* *Y?'(,01 ; I HD F GDC <03 ; @K. 70..,HE*L ?.* .(?>*1( 803V (0 >,.-?.. *'-) 'EF*H3',- .(3'(*FG@

A-#.1B CDB&8( )8-''E9 !$ % !&

S'J* .(?>*1(. (?31 ()*,3 '((*1(,01 (0 .0EJ,1F <03 70,1(. 8)*3* (80 0< ()* O2*( :,((*3X -01.(3',1(.,1(*3.*-(@ P),E* ()*3* '3* +'1G 70,1(. 0< ,1(*3.*-(,01 H*(8**1 J'3,0?. -01.(3',1(.L 8* 8,EE-01.,>*3 (80 .?-) 70,1(. ,1 (),. ('.V@

5F%8&$( C+G-88 H$&#%E9

I)* -003>,1'(*. 0< ()* 70,1( 0< ,1(*3.*-(,01 <03 ()* .80>, '1> ./0-/?48 >$./ -01.(3',1(. '3* 10( 8)0E*1?+H*3.@ I),. .)0?E> +0(,J'(* '1 'EF*H3',- .0E?(,01 .(3'(*FG@ :(?>*1(. 8)0 '3* 803V,1F 8,() ()*-01.(3',1(. 83,((*1 ,1 .('1>'3> <03+ +'G <,1> ' .?H.(,(?(,01 .(3'(*FG +03* *<<,-,*1( ()'( .0EJ,1FH0() *Y?'(,01. <03 ' J'3,'HE* '1> .*((,1F ()*+ *Y?'E (0 *'-) 0()*3@ P'(-) <03 H0() 'EF*H3',-.(3'(*F,*.@

>I'1#'' CDB&8( )8-''E9

&'V* .?3* ()'( H0() 'EF*H3',- .(3'(*F,*. <03 .0EJ,1F .G.(*+. 0< (80 E,1*'3 *Y?'(,01. R,@*@L.?H.(,(?(,01 '1> .*((,1F *`73*..,01. *Y?'ET F*( 73*.*1(*> '1> >,.-?..*>@ i,.-?.. ()* ,..?* 0< ()*.0E?(,01 (0 ()* .80>, '1> ./0-/?48 >$./. .G.(*+ 0< *Y?'(,01. 10( )'J,1F 8)0E* 1?+H*3 -003>,1'(*.@P),E* 8* -'1b( )'J* %\ %j\ >0F. '1> A #j\ -'(. '. ' 3*'.01'HE* .0E?(,01 (0 ()* O2*( :,((*3.X



! #$%# &'()*+'(,-. /,.,01 2304*-( 5 &!2

61 7'3(1*3.),7 8,() ()* 9(') :('(* ;<<,-* 0< =>?-'(,01!"#$%&$' )%'$* +,$ -*$.+"/$ -0110%& 2++*"3)+"0%450%-011$*#".647,.*$26"8$ 9:; <%=0*+$' 6"#$%&$:

.-*1'3,0L (),. ,. ()* .0E?(,01 (0 ()* .G.(*+ 0< *Y?'(,01.@ K. 8* )'J* 10(,-*>L ()* .0E?(,01 (0 ()*-01(*`(?'E,k*> .,(?'(,01 +'G H* ' 70,1( ,1 ()* ,1(*3,03 0< ()* 3*F,01@ S08*J*3L ,( ,. ,+703('1( (010(* ()'( (),. ,. '1 *`'+7E* 0< ()* +0>*E,1F .('1>'3>W8* )'J* >*-01(*`(?'E,k*> ()* .,(?'(,01 (0<,1> ' +'()*+'(,-'E +0>*E ()'( 8,EE )*E7 ?. 3*'.01 'H0?( ()* O2*( :,((*3.X -01(*`(@ =J*1(?'EEGL '1G-01-E?.,01. 8* +'V* ?.,1F ()* +'()*+'(,-'E +0>*E 8,EE )'J* (0 H* ,1(*373*(*> ,1 (*3+. 0< ()*03,F,1'E -01(*`(@

28IJ.(/ 4(-/K= +(,= H&9 +K',(G' <37



Name: Systems 2.6




Ready, Set, Go! Ready

Topic: Determine patterns

Find the next two values in the pattern. Describe how you determined these values.

1. 3, 6, 9, 12, ____, ____ Description:

2. 3, 6, 12, 24, ____, ___ Description:

3. 24, 20, 16, 12, ___, ___ Description:

4. 24, 12, 6, 3, ___, ___ Description:

Set

Topic: Solve systems of equations using substitution

For questions 5-8 solve the system of equations using substitution. Check your work by

graphing.

5. { 6. {

7. { 8. {

9. Tickets to a show cost $10 in advance and $15 at the door. If 120 tickets are sold for a total of$1390, how many of the tickets were bought in advance?

© 2012 www.flickr.com/photos/photosteve101

24

Systems 24

http://www.flickr.com/photos/photosteve

http://www.flickr.com/photos/photosteve



Name: Systems 2.6




Go

Topic: Graph two variable inequalities

Graph the following inequalities.

10. 11. 12.

13. 14. 15.


http://www.khanacademy.org/math/algebra/systems-of-eq-and-ineq/v/solving-systems-by-substitution-3

http://www.khanacademy.org/math/algebra/linear-equations-and-inequalitie/v/solving-and-graphing-linear-inequalities-in-two-

variables-1

http://www.khanacademy.org/math/algebra/linear-equations-and-inequalitie/v/graphing-inequalities-2

25

Systems 25



http://www.khanacademy.org/math/algebra/linear-equations-and-inequalitie/v/solving-and-graphing-linear-inequalities-in-two-variables-1











! #$%# &'()*+'(,-. /,.,01 2304*-( 5 &! 2

61 7'3(1*3.),7 8,() ()* 9(') :('(* ;<<,-* 0< =>?-'(,01!"#$%&$' )%'$* +,$ -*$.+"/$ -0110%& 2++*"3)+"0%450%-011$*#".647,.*$26"8$ 9:; <%=0*+$' 6"#$%&$:

#@A :)077,1B <03 C'(. '1> D0B. ! #$%$&'( )*+$,-./*+0*1 2/-3

CE'3,(' ,. ?7.*( 8,() C'3E0. F*-'?.* )* )'. F**1 F?G,1B

-'( '1> >0B <00> 8,()0?( 3*-03>,1B ()* 73,-* 0< *'-) (G7*0< <00> ,1 ()*,3 '--0?1(,1B 3*-03>.@ 61.(*'>H C'3E0. )'. 4?.( 3*-03>*> ()* (0('E 73,-* 0< *'-)

7?3-)'.*H *I*1 ()0?B) ()* (0('E -0.( ,1-E?>*. +03* ()'( 01* (G7* 0< <00>@ C'3E0. ,. 108 (3G,1B (0

<,B?3* 0?( ()* 73,-* 0< *'-) (G7* 0< <00> FG 3*I,*8,1B .0+* 3*-*1( 7?3-)'.*.@ :** ,< G0? -'1 )*E7

),+ <,B?3* 0?( ()* -0.( 0< 7'3(,-?E'3 ,(*+. <03 *'-) 7?3-)'.*H '1> F* 73*7'3*> (0 *J7E',1 G0?3

3*'.01,1B (0 C'3E0.@

%@ ;1* 8**K C'3E0. F0?B)( L F'B. 0< 2/40.5/ 20+40.- '1> M F'B. 0< 601/,' 6&/3$- <03 NML@$$@O)* 1*J( 8**K )* F0?B)( L F'B. 0< 2/40.5/ 20+40.- '1> P F'B. 0< 601/,' 6&/3$- <03 NQM@$$@R'.*> 01 (),. ,1<03+'(,01H <,B?3* 0?( ()* 73,-* 0< 01* F'B 0< *'-) (G7* 0< -'( <00>@ =J7E',1G0?3 3*'.01,1B@

#@ ;1* 8**K C'3E0. F0?B)( # F'B. 0< 7,8.8- 70.$- '1> L F'B. 0< 98:3; 90:3- <03 NM#@Q$@ O)*1*J( 8**K )* F0?B)( Q F'B. 0< 7,8.8- 70.$- '1> P F'B. 0< 98:3; 90:3- <03 NSM@#Q@ R'.*> 01(),. ,1<03+'(,01H <,B?3* 0?( ()* 73,-* 0< 01* F'B 0< *'-) (G7* 0< >0B <00>@ =J7E',1 G0?33*'.01,1B@

L@ C'3E0. 7?3-)'.*> P >0B E*'.)*. '1> P -'( F3?.)*. <03 NMQ@$$ <03 CE'3,(' (0 ?.* 8),E*7'+7*3,1B ()* 7*(.@ T'(*3 ,1 ()* .?++*3 )* 7?3-)'.*> L '>>,(,01'E >0B E*'.)*. '1> # -'(F3?.)*. <03 N%S@$$@ R'.*> 01 (),. ,1<03+'(,01H <,B?3* 0?( ()* 73,-* 0< *'-) ,(*+@ =J7E',1G0?3 3*'.01,1B@

M@ ;1* 8**K C'3E0. F0?B)( # 7'-K'B*. 0< >0B F01*. '1> M 7'-K'B*. 0< -'( (3*'(. <03 N%U@Q$@R*-'?.* ()* <,1,-KG -'(. >,>1V( E,K* ()* -'( (3*'(.H ()* 1*J( 8**K C'3E0. 3*(?31*> L ?107*1*>7'-K'B*. 0< -'( (3*'(. '1> F0?B)( # +03* 7'-K'B*. 0< >0B F01*.@ W<(*3 F*,1B 3*<?1>*> <03()* -'( (3*'(.H C'3E0. 01EG )'> (0 7'G N%@$$ <03 ),. 7?3-)'.*@ R'.*> 01 (),. ,1<03+'(,01H<,B?3* 0?( ()* 73,-* 0< *'-) ,(*+@ =J7E',1 G0?3 3*'.01,1B@

Q@ C'3E0. )'. 10(,-*> ()'( F*-'?.* *'-) 0< ),. 7?3-)'.*. )'I* F**1 .0+*8)'( .,+,E'3H ,( )'.F**1 *'.G (0 <,B?3* 0?( ()* -0.( 0< *'-) ,(*+@ X08*I*3H ),. E'.( .*( 0< 3*-*,7(. )'. ),+7?YYE*>@ ;1* 8**K )* (3,*> 0?( -)*'7*3 F3'1>. 0< -'( '1> >0B <00>@ ;1 &01>'G )*7?3-)'.*> L .+'EE F'B. 0< -'( <00> '1> Q .+'EE F'B. 0< >0B <00> <03 N##@AQ@ R*-'?.* )*8*1( ()30?B) ()* .+'EE F'B. Z?,(* Z?,-KEGH )* )'> (0 3*(?31 (0 ()* .(03* 01 O)?3.>'G (0 F?G# +03* .+'EE F'B. 0< -'( <00> '1> L +03* .+'EE F'B. 0< >0B <00>H 8),-) -0.( ),+ N%M@#Q@

R'.*> 01 (),. ,1<03+'(,01H <,B?3* 0?( ()* 73,-* 0< *'-) F'B 0< ()* -)*'7*3 -'( '1> >0B <00>@=J7E',1 G0?3 3*'.01,1B@

:?++'3,Y* ()* .(3'(*B,*. G0? )'I* ?.*> (0 3*'.01 'F0?( ()* 73,-* 0< ,1>,I,>?'E ,(*+. ,1 ()*

730FE*+. B,I*1 'F0I*@ [)'( '3* .0+* K*G ,>*'. ()'( .**+ )*E7<?E\

! >

; ? > @ @ @ : A 6 " # 8 * : # 0 1 B = , 0 + 0 & B + ) ' 0 *

26

Systems 26



! #$%# &'()*+'(,-. /,.,01 2304*-( 5 &!2

61 7'3(1*3.),7 8,() ()* 9(') :('(* ;<<,-* 0< =>?-'(,01

!"#$%&$' )%'$* +,$ -*$.+"/$ -0110%& 2++*"3)+"0%450%-011$*#".647,.*$26"8$ 9:; <%=0*+$' 6"#$%&$:

#@A :)077,1B <03 C'(. '1> D0B. ] O*'-)*3 ^0(*. ! #$%$&'( )*+$,-./*+0*1 2/-3

"#$%&'(_ 9.,1B .)077,1B 3*-*,7(. '. ' -01(*J(H .(?>*1(. 8,EE >*I*E07 ' .(3'(*BG <03 .0EI,1B '

.G.(*+ 0< (80 E,1*'3 *Z?'(,01. ?.,1B *E,+,1'(,01 0< I'3,'FE*.@ :(?>*1(. 8,EE 3*-0B1,Y* ()'( 8* -'10F(',1 '1 *Z?,I'E*1( .G.(*+ 0< *Z?'(,01. FG 3*7E'-,1B 01* 03 F0() *Z?'(,01. ,1 ()* .G.(*+ ?.,1B

01* 0< ()* <0EE08,1B .(*7._

• `*7E'-* '1 *Z?'(,01 ,1 ()* .G.(*+ 8,() ' -01.('1( +?E(,7E* 0< ()'( *Z?'(,01

• `*7E'-* '1 *Z?'(,01 ,1 ()* .G.(*+ 8,() ()* .?+ 03 >,<<*3*1-* 0< ()* (80 *Z?'(,01.

• `*7E'-* '1 *Z?'(,01 8,() ()* .?+ 0< ()'( *Z?'(,01 '1> ' +?E(,7E* 0< ()* 0()*3

='-) 0< ()*.* .(3'(*B,*. -'1 F* 4?.(,<,*> ,1 (*3+. 0< ()* 7?3-)'.* 73,-* 0< ' -0+F,1'(,01 0< (80

,(*+.@

O)* B0'E 0< ()*.* (80 730-*>?3*. ,. (0 0F(',1 ' .G.(*+ 0< *Z?'(,01. ,1 8),-) ()* -0*<<,-,*1( 0< 01*

0< ()* I'3,'FE*. ,. ()* .'+* ,1 F0() *Z?'(,01.@ O)*1 ()* >,<<*3*1-* ,1 ()* 3*.?E(. >*7*1>. .0E*EG 01()* >,<<*3*1-* ,1 ()* '+0?1(. 0< 01* Z?'1(,(GH 3'()*3 ()'1 (80@ 61 (*3+. 0< ()* .)077,1B -01(*J(H ,<

8* F?G ()* .'+* '+0?1( 0< 01* ,(*+H F?( >,<<*3*1( '+0?1(. 0< ()* 0()*3H ()*1 ()* >,<<*3*1-* ,1 ()*

7?3-)'.* 73,-* ,. >?* *1(,3*EG (0 ()* ,(*+ 0< 8),-) 8* 7?3-)'.*> >,<<*3*1( '+0?1(.@

)&$( +,-./-$/' 0&1#'_

2345637 230I* ()'(H B,I*1 ' .G.(*+ 0< (80 *Z?'(,01. ,1 (80 I'3,'FE*.H 3*7E'-,1B 01* *Z?'(,01 FG ()*

.?+ 0< ()'( *Z?'(,01 '1> ' +?E(,7E* 0< ()* 0()*3 730>?-*. ' .G.(*+ 8,() ()* .'+* .0E?(,01.@

2345638 :0EI* .G.(*+. 0< E,1*'3 *Z?'(,01. *J'-(EG '1> '7730J,+'(*EG a*@B@H 8,() B3'7).bH <0-?.,1B

01 7',3. 0< E,1*'3 *Z?'(,01. ,1 (80 I'3,'FE*.@

4(9-,(/ +,-./-$/': ;3<3=> 23++53=-> 23)5?3@> 23)5?3A

B-#.1C DEC&9( )9-''F:

`*'> ()30?B) ()* -01(*J( 'F0?( ?.,1B .)077,1B 3*-*,7(. (0 <,1> ()* 7?3-)'.* 73,-* 0< ,1>,I,>?'E

,(*+. 8)*1 01EG ()* 7?3-)'.* 73,-* 0< ' -0+F,1'(,01 0< ()* ,(*+. ,. K1081@ =1-0?3'B* .(?>*1(. (0

-'3*<?EEG 3*-03> ()*,3 '3B?+*1(. <03 )08 ()*G K108 ()* 73,-* 0< ' .,1BE* ,(*+ 8)*1 *J'+,1,1B (80

.,+,E'3H F?( >,<<*3*1(H 7?3-)'.*. 0< -0+F,1'(,01. 0< ()* .'+* ,(*+.@

5G%9&$( D+H-99 I$&#%F:

T,.(*1 <03 .(?>*1(. +'K,1B .*1.* 0< ()* .-*1'3,0. ()30?B) '3B?+*1(. .?-) '. ()* <0EE08,1B_

• c03 730FE*+ d%H .,1-* C'3E0. F0?B)( ()* .'+* 1?+F*3 0< F'B. 0< 2/40.5/ 20+40.-H ()*>,<<*3*1-* ,1 ()* (0('E -0.( +?.( F* >?* (0 ()* (80 *J(3' F'B. 0< 601/,' 6&/3$-<

• c03 230FE*+ d#H ,< 8* >0?FE* ()* <,3.( 7?3-)'.* 8* B*( 'E+0.( ()* .*-01> 7?3-)'.*H *J-*7(

8* 1**> (0 ,1-E?>* 01* +03* F'B 0< 7,8.8- 70.$-= 8),-) 80?E> '--0?1( <03 ()* >,<<*3*1-* ,1

(0('E -0.(@



! #$%# &'()*+'(,-. /,.,01 2304*-( 5 &!2

61 7'3(1*3.),7 8,() ()* 9(') :('(* ;<<,-* 0< =>?-'(,01

!"#$%&$' )%'$* +,$ -*$.+"/$ -0110%& 2++*"3)+"0%450%-011$*#".647,.*$26"8$ 9:; <%=0*+$' 6"#$%&$:

• c03 230FE*+ dLH ,< 8* >0?FE* ()* .*-01> 7?3-)'.* 8* B*( 'E+0.( ()* <,3.( 7?3-)'.*H *J-*7(

<03 # '>>,(,01'E -'( F3?.)*.H 8),-) 80?E> '--0?1( <03 ()* >,<<*3*1-* ,1 (0('E -0.(@

• c03 730FE*+ dMH 3*(?31,1B ,(*+. ,. E,K* +'K,1B ' 1*B'(,I* 7?3-)'.*H .0 8* F0?B)( A <*8*3

7'-K'B*. 0< -'( (3*'(. ,1 ()* .*-01> 7?3-)'.* ()'1 8* F0?B)( ,1 ()* <,3.( 7?3-)'.*H .0 ()0.*

A 7'-K'B*. +?.( '--0?1( <03 ()* >,<<*3*1-* ,1 ()* (0('E -0.(@

• c03 730FE*+ dQH 8* 1**> (0 (),1K 0< 70..,FE* 3*E'(*> 7?3-)'.*. .0 ()'( ()* '+0?1(

7?3-)'.*> 0< 01* 0< ()* ,(*+. ,. ()* .'+* <03 (80 >,<<*3*1( -0+F,1'(,01. 0< -'( <00> '1> >0B

<00>@ [* -0?E>H <03 *J'+7E*H -01.,>*3 ()* (0('E -0.( ,< C'3E0. )'> F0?B)( (8,-* '. +?-) -'(

'1> >0B <00> 01 &01>'G '1> ()3** (,+*. '. +?-) 0< *'-) 01 O)?3.>'G@ O)*1 )* 80?E>

)'I* 7?3-)'.*> P .+'EE F'B. 0< -'( <00> 01 *'-) >'GH '1> ()* >,<<*3*1-* ,1 ()* (0('E -0.(

80?E> F* >?* (0 ()* *J(3' F'B 0< >0B <00> 7?3-)'.*> 01 &01>'G@

&'K* .?3* ()'( .(?>*1(. <,B?3* 0?( ()* 7?3-)'.* 73,-* <03 *'-) ,(*+ ,1 *'-) .-*1'3,0@ O),. 8,EE

3*Z?,3* ()*+ (0 .?F.(,(?(* ()* 73,-* 0< ()* <,3.( ,(*+ ()*G <,B?3* 0?( F'-K ,1(0 ()* -01(*J( (0 <,1> ()*

73,-* 0< ()* .*-01> ,(*+@

?J'1#'' DEC&9( )9-''F:

61.(*'> 0< >,.-?..,1B *'-) ,1>,I,>?'E 730FE*+H '.K .(?>*1(. (0 B*1*3'(* ' E,.( 0< K*G ,>*'. ()'(

)*E7*> ()*+ (0 .0EI* *'-) .-*1'3,0@ O)* E,.( 0< K*G ,>*'. .)0?E> ,1-E?>*_

• 6< ()* 1?+F*3 0< 01* (G7* 0< ,(*+ ,. ()* .'+* ,1 F0() 7?3-)'.*.H ()*1 ()* >,<<*3*1-* ,1 (0('E

-0.( ,. >?* (0 ()* >,<<*3*1-* ,1 ()* 1?+F*3 0< ()* .*-01> ,(*+ 7?3-)'.*>@

• 6< 8* K108 ()* (0('E -0.( 0< ' -0+F,1'(,01 0< (80 ,(*+.H 8* K108 ()* (0('E -0.( 0< F?G,1B

(8,-* '. +?-)H 03 ()3** (,+*. '. +?-)H 03 '1G 0()*3 +?E(,7E* 0< ()* 03,B,1'E 7?3-)'.*@

• O80 >,<<*3*1( -0+F,1'(,01. 0< ()* .'+* ,(*+. -'1 F* '>>*> (0B*()*3 a,@*@H 7?3-)'.*>(0B*()*3b '1> ()* (0('E -0.( 8,EE F* ()* .?+ 0< ()* -0.(. 0< *'-) .*7'3'(* 7?3-)'.*@

61 ()* 1*J( ('.K .(?>*1(. 8,EE +'K* ()*.* ,>*'. +03* 'EB*F3',- FG 83,(,1B ()* >,<<*3*1( 7?3-)'.*. '.

.G.(*+. 0< *Z?'(,01. '1> ?.,1B (),. ,1(?,(,I* 3*'.01,1B (0 .0EI* ()* .G.(*+@

29JK.(/ 4(-/L> +(,> I&: +L',(H' @3M



Name: Systems 2.7



Ready, Set, Go!

ReadyTopic: Exponents

Write the following in exponential notation.

1. 4 × 4 × 4 × 4 × 4 2.

Find each value.

3. 4. 5. 6. 7.

Set

Topic: Solving systems

8. Nadia and Peter visit the candy store. Nadia buys three candy bars and four fruit roll-ups for$2.84. Peter also buys three candy bars, but can only afford one additional fruit roll-up. Hispurchase costs $1.79. What is the cost of a candy bar and a fruit roll-up individually?

9. A farmer noticed that his chickens were loose and were running around with the cows in the cowpen. He quickly counted 100 heads and 270 legs. How many chickens did he have and how manycows?

© 2012 www.flickr.com/photos/tudor

27

Systems 27



Name: Systems 2.7



Go

Topic: Solve one variable inequalities.

Solve the following inequalities. Write the solution set in interval notation and graph the

solution set on a number line.

10. 4 x + 10 < 2 x + 14

11. 2 x + 6 > 55 – 5 x

12.

13.

Solve each inequality. Give the solution in inequality notation and set notation.

14.

15.

16.

17.

18. 19.

Need help? Check out these related videos?

Exponential notation:

http://www.khanacademy.org/math/algebra/exponents-radicals/v/understanding-exponents

Solving inequalities:http://www.khanacademy.org/math/algebra/solving-linear-inequalities/v/solving-inequalities

http://www.khanacademy.org/math/algebra/solving-linear-inequalities/v/multi-step-inequalities-2

Set notation and interval notation:

http://patrickjmt.com/using-interval-notation-to-express-inequalities-ex1/

28

Systems 28



http://www.khanacademy.org/math/algebra/solving-linear-inequalities/v/solving-inequalities












! #$%# &'()*+'(,-. /,.,01 2304*-( 5 &!2

61 7'3(1*3.),7 8,() ()* 9(') :('(* ;<<,-* 0< =>?-'(,01!"#$%&$' )%'$* +,$ -*$.+"/$ -0110%& 2++*"3)+"0%450%-011$*#".647,.*$26"8$ 9:; <%=0*+$' 6"#$%&$:

> ?;@? AAA:B6"#8*:#01C=,0+0&CDA"610*$

#@A B'1 C0? D*( (0 ()* 20,1(E F00G ! #$%&'&() *+',-./0+'&+1 20.3

"#$% '

61 H:)077,1I <03 B'(. '1> J0I.EK B'3L0. <0?1> ' 8'M (0 <,1> ()* -0.( 0<,1>,N,>?'L ,(*+. 8)*1 I,N*1 ()* 7?3-)'.* 73,-* 0< (80 >,<<*3*1(

-0+O,1'(,01. 0< ()0.* ,(*+.@ P* 80?L> L,Q* (0 +'Q* ),. .(3'(*IM +03* *<<,-,*1( OM 83,(,1I ,( 0?(

?.,1I .M+O0L. '1> 'LI*O3'@ P*L7 ),+ <03+'L,R* ),. .(3'(*IM OM >0,1I ()* <0LL08,1IS

• T03 *'-) .-*1'3,0 ,1 H:)077,1I <03 B'(. '1> J0I.K 83,(* ' ()(%*+ ,- *./#%0,1( (0

3*73*.*1( ()* (80 7?3-)'.*.@

• :)08 )08 M0?3 .(3'(*I,*. <03 <,1>,1I ()* -0.( 0< ,1>,N,>?'L ,(*+. -0?L> O* 3*73*.*1(*> OM

+'1,7?L'(,1I ()* *U?'(,01. ,1 ()* .M.(*+@ V3,(* 0?( ,1(*3+*>,'(* .(*7. .M+O0L,-'LLME .0

()'( .0+*01* *L.* -0?L> <0LL08 M0?3 803Q@

• ;1-* M0? <,1> ()* 73,-* 0< 01* 0< ()* ,(*+. ,1 ()* -0+O,1'(,01E .)08 )08 M0? 80?L> <,1> ()*73,-* 0< ()* 0()*3 ,(*+@

"#$% 2

V3,(,1I 0?( *'-) .M.(*+ 0< *U?'(,01. 3*+,1>*> B'3L0. 0< ),. 803Q 8,() .0LN,1I .M.(*+. 0<

*U?'(,01. I3'7),-'LLM@ :)08 )08 *'-) .-*1'3,0 ,1 H:)077,1I <03 B'(. '1> J0I.K -'1 O* 3*73*.*1(*>

I3'7),-'LLME '1> )08 ()* -0.( 0< *'-) ,(*+ .)08. ?7 ,1 ()* I3'7).@

"#$% 3

B'3L0. 'L.0 3*'L,R*> ()'( ()* 'LI*O3',- .(3'(*IM )* -3*'(*> ,1 7'3( % -0?L> O* ?.*> (0 <,1> ()* 70,1(.

0< ,1(*3.*-(,01 <03 ()* H2*( :,((*3.K -01.(3',1(.@ 9.* ()* *40+01#%0,1 ,- 5#$0#64*( +*()0>

>*N*L07*> ,1 7'3( % (0 <,1> ()* 70,1( 0< ,1(*3.*-(,01 <03 *'-) 0< ()* <0LL08,1I 7',3. 0< H2*( :,((*3K

-01.(3',1(.@

• #/0-/456 7$./. '1> .607, -01.(3',1(.

• 8096,-&+1 /&9, '1> (,,'&+1 /&9, -01.(3',1(.

• W1M 0()*3 7',3 0< H2*( :,((*3K -01.(3',1(. 0< M0?3 -)0,-*

29

Systems 29



! #$%# &'()*+'(,-. /,.,01 2304*-( 5 &!2

61 7'3(1*3.),7 8,() ()* 9(') :('(* ;<<,-* 0< =>?-'(,01

!"#$%&$' )%'$* +,$ -*$.+"/$ -0110%& 2++*"3)+"0%450%-011$*#".647,.*$26"8$ 9:; <%=0*+$' 6"#$%&$:

#@A B'1 C0? D*( (0 ()* 20,1(E F00G X F*'-)*3 Y0(*. ! #$%&'&() *+',-./0+'&+1 20.3

"/$7,(*S F),. ('.Q .0L,>,<,*. ()* .(3'(*I,*. <03 .0LN,1I .M.(*+. 0< *U?'(,01. ()'( .?3<'-*> >?3,1I

()* 73*N,0?. ('.Q@ :(?>*1(. 8,LL O*I,1 OM 83,(,1I ' .M.(*+ 0< *U?'(,01. (0 3*73*.*1( ()* .)077,1I.-*1'3,0.@ :(?>*1(. 8,LL 3*-0I1,R* ()'( 8* -'1 0O(',1 '1 *U?,N'L*1( .M.(*+ 0< *U?'(,01. OM

3*7L'-,1I 01* 03 O0() *U?'(,01. ,1 ()* .M.(*+ ?.,1I 01* 0< ()* <0LL08,1I .(*7.S

• Z*7L'-* '1 *U?'(,01 ,1 ()* .M.(*+ 8,() ' -01.('1( +?L(,7L* 0< ()'( *U?'(,01

• Z*7L'-* '1 *U?'(,01 ,1 ()* .M.(*+ 8,() ()* .?+ 03 >,<<*3*1-* 0< ()* (80 *U?'(,01.

• Z*7L'-* '1 *U?'(,01 8,() ()* .?+ 0< ()'( *U?'(,01 '1> ' +?L(,7L* 0< ()* 0()*3

F)* I0'L 0< ()*.* .(*7. ,. (0 0O(',1 ' .M.(*+ 0< *U?'(,01. ,1 8),-) ()* -0*<<,-,*1( 0< 01* 0< ()*

N'3,'OL*. ,. ()* .'+* ,1 O0() *U?'(,01.@ F)*1E 8)*1 8* .?O(3'-( 01* 0< ()* *U?'(,01. <30+ ()*

0()*3E 8* 8,LL 0O(',1 '1 *U?'(,01 ()'( -01(',1. 01LM 01* N'3,'OL*@ F),. *U?'(,01 -'1 O* .0LN*> <03

,(. N'3,'OL* '1> ()* 3*.?L( -'1 O* .?O.(,(?(*> O'-Q ,1(0 01* 0< ()* 03,I,1'L *U?'(,01. (0 0O(',1 '1

*U?'(,01 ()'( -'1 O* .0LN*> <03 ()* 0()*3 N'3,'OL*@

8,$* 9%#1:#$:( ;,</(S

=>?@A>B 230N* ()'(E I,N*1 ' .M.(*+ 0< (80 *U?'(,01. ,1 (80 N'3,'OL*.E 3*7L'-,1I 01* *U?'(,01 OM ()*

.?+ 0< ()'( *U?'(,01 '1> ' +?L(,7L* 0< ()* 0()*3 730>?-*. ' .M.(*+ 8,() ()* .'+* .0L?(,01.@

=>?@A>C :0LN* .M.(*+. 0< L,1*'3 *U?'(,01. *['-(LM '1> '7730[,+'(*LM \*@I@E 8,() I3'7).]E <0-?.,1I

01 7',3. 0< L,1*'3 *U?'(,01. ,1 (80 N'3,'OL*.@

?*4#%*: 9%#1:#$:(D E>F>'G =>99@>'#G =>8@H>2G =>8@H>3

I#/1<J KLJ,4* 84#((MD

230N,>* ' +0>*L 0< )08 .(?>*1(. +,I)( 803Q 8,() .M.(*+. 0< *U?'(,01. ?.,1I ()* ,1(?,(,N*

3*'.01,1I ()*M >*N*L07*> ,1 ()* 73*N,0?. ('.Q OM 803Q,1I ()30?I) .-*1'3,0 % <30+ H:)077,1I <03

B'(. '1> J0I.K (0I*()*3@ V3,(* 0?( ()* .M.(*+ ?.,1I *U?'(,01. ,1 .('1>'3> <03+S

:,1-* ()* -0*<<,-,*1(. 0< 2 '3* ()* .'+* ,1 O0() *U?'(,01.E 8* 8,LL .?O(3'-( *U?'(,01 ^% <30+

*U?'(,01 ^# (0 I*( #: _ %%@$$@ V* -'1 .0LN* (),. *U?'(,01 <03 : OM >,N,>,1I O0() .,>*. 0< ()*

*U?'(,01 OM # (0 I*( : _ `@`$E 8),-) +?.( O* ()* 73,-* 0< ' O'I :&10-$ :%03,.; V* -'1 .?O.(,(?(* (),.'+0?1( ,1(0 *,()*3 *U?'(,01 (0 .0LN* <03 ()* 73,-* 0< 20<&/=0 2&'<&/.; T03 *['+7L*E .?O.(,(?(,1I @`$

,1(0 ()* <,3.( *U?'(,01 <03 : M,*L>. a2 b ##@$$ _ ca@$$@ F)*3*<03*E a2 _ #%@$$E 03 2 _ d@$$@



! #$%# &'()*+'(,-. /,.,01 2304*-( 5 &!2

61 7'3(1*3.),7 8,() ()* 9(') :('(* ;<<,-* 0< =>?-'(,01

!"#$%&$' )%'$* +,$ -*$.+"/$ -0110%& 2++*"3)+"0%450%-011$*#".647,.*$26"8$ 9:; <%=0*+$' 6"#$%&$:

@N74,$* K9+#44 O$,/7MD

V'(-) '1> L,.(*1 <03 ()* 8'M. .(?>*1(. 83,(* '1> .0LN* ()* .M.(*+. 0< *U?'(,01. 3*73*.*1(*> ,1

*'-) 0< ()* 0()*3 .-*1'3,0.@ =1-0?3'I* ()*+ (0 -011*-( ()*,3 ,1(?,(,N* 3*'.01,1I 8,() ()* .)077,1I

.-*1'3,0. (0 ()* .M+O0L,- 3*'.01,1I 8,() N'3,'OL*.@ 2'3( # 0< ()* ('.Q I,N*. .(?>*1(. '1 07703(?1,(M

(0 -011*-( (),. 803Q (0 .0LN,1I ' .M.(*+ 0< L,1*'3 *U?'(,01. I3'7),-'LLME '1> 7'3( a *[(*1>. ()*

*L,+,1'(,01 +*()0> (0 +03* -0+7L,-'(*> .M.(*+. 8)*3* I*((,1I ()* .'+* -0*<<,-,*1( <03 01* 0< ()*N'3,'OL*. ,1 O0() *U?'(,01. ,. +03* >,<<,-?L(@

H0(</(( KLJ,4* 84#((MD

61N,(* .(?>*1(. (0 '3(,-?L'(* ' I*1*3'L .(3'(*IM <03 .0LN,1I .M.(*+. 0< *U?'(,01. OM *L,+,1'(,1I '

N'3,'OL*@ P'N* .(?>*1(. >*+01.(3'(* (),. .(3'(*IM 8,() 01* 0< ()* +03* -)'LL*1I,1I .M.(*+. <30+

()* H2*( :,((*3.K -01(*[(E .?-) '. ()* <0LL08,1I .M.(*+ ()'( ,1N0LN*. ()* .607, -01.(3',1( '1> ()*

6096,-&+1 /&9, -01.(3',1(@

24 x + 6 y = 360

1

3 x+

4

15 y= 8

!

" #

$ #

;1* 70..,OL* .(3'(*IM <03 .0LN,1I (),. .M.(*+ 80?L> O* (0 +?L(,7LM ()* O0((0+ *U?'(,01 OM %` (0

0O(',1 8)0L* 1?+O*3 -0*<<,-,*1(.@

24 x + 6 y = 360

5 x + 4 y = 120

! " #

F)*1 +?L(,7LM ()* (07 *U?'(,01 OM c '1> ()* O0((0+ *U?'(,01 OM e (0 I*( ()* ) f-0*<<,-,*1( ()* .'+*

,1 O0() *U?'(,01.@

96 x + 24 y = 1440

30 x + 24 y = 720

! " #

:?O(3'-(,1I ()* O0((0+ *U?'(,01 <30+ ()* (07 M,*L>. ()* .,1IL* N'3,'OL* *U?'(,01 @

:0LN,1I (),. *U?'(,01 <03 > I,N*. @ F)* -0+7L*(* .0L?(,01 ,. @

T03(?1'(*LME (),. ,. 10( 01* 0< ()* ,+703('1( 70,1(. 0< ,1(*3.*-(,01 ,1 ()* H2*( :,((*3.K -01(*[(E .,1-*

,( L,*. 0?(.,>* ()* <*'.,OL* 3*I,01@

=40P1*: ?*#:)G 9*%G O,D 9)(%*+( 2>Q



Name: Systems 2.8




Ready, Set, Go!

Ready

Topic: Evaluate exponents

Simplify and evaluate the following.

1. 3-2 2. (0.5)-2 3. 24 4. 4-2

Write the following expression three different ways (one way can include the simplified value).

5. 6.

Set

Topic: Solve systems of equations

Solve the following systems of equations using elimination of variables, then justify

graphically.

7. { 8. {

9.

10. A 150-yard pipe is cut to provide drainagefor two fields. If the length of one piece (a) isthree yards less than twice the length of thesecond piece (b), what are the lengths of the twopieces?

-5

-5

5

5

-5

-5

5

5

© 2012 www.flickr.com/photos/gwilmore

-5

-5

5

5

30

Systems 30



Name: Systems 2.8




Go

Topic: Graph two variable linear inequalities

Graph the following linear inequalities on the graphs below. Include constraints.

11. Ben has enough money to buy up to eight yogurts. If hisfavorite flavors are blueberry and strawberry, what are all

the possible combinations he can buy? Graph the inequality

that shows all possible combinations of his favorite flavors.

12. Peggy is buying a balloon bouquet. Her favorite colors

are silver and purple. The silver balloons are $1 and the

purple balloons are $0.80. Graph an inequality that shows

how many of each color balloon she can put in her bouquet if

she doesn’t spend more than $20.


Negative exponentshttp://patrickjmt.com/negative-exponents/ http://www.khanacademy.org/math/algebra/ck12-algebra-1/v/zero--negative--and-fractional-exponents

Solving systems by elimination

http://www.khanacademy.org/math/algebra/systems-of-eq-and-ineq/v/solving-systems-by-elimination-2

Solving systems by graphing

http://www.khanacademy.org/math/algebra/systems-of-eq-and-ineq/v/solving-linear-systems-by-graphing

Blueberry

S t r a w b e r r y

Silver

P u r p l e

31

Systems 31

http://patrickjmt.com/negative-exponents/


http://www.khanacademy.org/math/algebra/ck12-algebra-1/v/zero--negative--and-fractional-exponents












! #$%# &'()*+'(,-. /,.,01 2304*-( 5 &! 2

61 7'3(1*3.),7 8,() ()* 9(') :('(* ;<<,-* 0< =>?-'(,01

!"#$%&$' )%'$* +,$ -*$.+"/$ -0110%& 2++*"3)+"0%450%-011$*#".647,.*$26"8$ 9:; <%=0*+$' 6"#$%&$:

#@A B00> <03 B,>0 '1> BC?<<D ! #$%&'&() *+',-./0+'&+1 20.3

E'3C0. '1> EC'3,(' )'F* <0?1> (80 >,<<*3*1( -'( <00>. ()'( .**+ (0 '77*'C (0

*F*1 ()* +0.( <,1,-GD 0< -'(.H 204&/50 2&'4&/. '1> 6&10-$ 6%03,.7 ='-) 0?1-*0< 204&/50 2&'4&/. -01(',1. # I3'+. 0< 730(*,1J K I3'+. 0< -'3L0)D>3'(*. '1> K I3'+. 0< <'(@

='-) 0?1-* 0< 6&10-$ 6%03,. -01(',1. M I3'+. 0< 730(*,1J K I3'+. 0< -'3L0)D>3'(*. '1> # I3'+. 0<

<'(@ :,1-* 204&/50 2&'4&/. ,. <',3CD *N7*1.,F*J 8),C* 6&10-$ 6%03,. ,. F*3D -)*'7J ()* (8,1. )'F*

>*-,>*> (0 -3*'(* ' 1*8 -'( <00> LD +,N,1I ()* (80@ O<(*3 .(?>D,1I .0+* 1?(3,(,01'C I?,>*C,1*. <03

-'(.J E'3C0. '1> EC'3,(' )'F* >*-,>*> (0 -3*'(* ' +,N(?3* L'.*> 01 ()* <0CC08,1I -01.(3',1(.@

• !8$9+/ $( :-$/,&+H ='-) +*'C .)0?C> -01(',1 '( C*'.( %# I3'+. 0< 730(*,1@

• !8$9+/ $( ;0-4$5)'-0/,.< ='-) +*'C .)0?C> -01(',1 +03* ()'1 %P I3'+. 0< -'3L0)D>3'(*.@

• !8$9+/ $( 60/.< ='-) +*'C .)0?C> -01(',1 10 +03* ()'1 %Q I3'+. 0< <'(@

• #&=, $( 0 6,,'&+1< ='-) +*'C .)0?C> -01.,.( 0< C*.. ()'1 %$ 0?1-*. 0< <00>@

B03 ()* 803G ()'( <0CC08.J C*( 2 3*73*.*1( ()* 1?+L*3 0< 0?1-*. 0< 204&/50 2&'4&/. ,1 ' +*'C '1> C*(6 3*73*.*1( ()* 1?+L*3 0< 0?1-*. 0< 6&10-$ 6%03,.7

%@ R3,(* '1 ,1*S?'C,(D <03 *'-) 0< ()* -01.(3',1(.@

#@ ;1 .*7'3'(* -003>,1'(* I3,>.J I3'7) ()* .0C?(,01 .*( <03 *'-) 0< ()* ,1*S?'C,(,*. D0? 830(* ,1

T%@ U08 >0 D0? G108 01 8),-) .,>* 0< ()* L0?1>'3D C,1* D0? .)0?C> .)'>* ()* )'C<V7C'1*

()'( 3*73*.*1(. ()* .0C?(,01 .*(W

M@ X*-,>* ,< ()* L0?1>'3D C,1* <03 *'-) ,1*S?'C,(D 3*73*.*1(*> ,1 T# .)0?C> L* ' .0C,> C,1* 03 '

>0((*> C,1*@ R),-) 803>. 03 7)3'.*. ,1 ()* -01.(3',1(. .?II*.(*> ' .0C,> C,1*W O >0((*>

C,1*W

K@ B,1> '( C*'.( Y -0+L,1'(,01. 0< 204&/50 2&'4&/. '1> 6&10-$ 6%03,. E'3C0. '1> EC'3,(' -'1 +,N

(0I*()*3 (0 -3*'(* ' 1?(3,(,0?. -'( +*'C@ :)08 ()'( ()*.* 70,1(. C,* 8,(),1 ' <*'.,LC* 3*I,01

<03 ()*.* -01.(3',1(.@

Y@ >-9/9. >&/,. ,. ' L3'1> 0< >0I <00> ()'( -01(',1. K I3'+. 0< 730(*,1 '1> P I3'+. 0< <'( 7*3

0?1-*@ ?9@3) ?&@3. ,. '10()*3 L3'1> 0< >0I <00> ()'( -01(',1. %# I3'+. 0< 730(*,1 '1> K

I3'+. 0< <'( 7*3 0?1-*@ E'3C0. 8'1(. (0 +'G* ' +*'C <03 >0I. ()'( -01(',1. '( C*'.( Q I3'+.

0< 730(*,1 '1> 10 +03* ()'1 P I3'+. 0< <'(@ R3,(* '1> .0CF* ' .D.(*+ 0< ,1*S?'C,(,*. ()'(E'3C0. -'1 ?.* (0 >*(*3+,1* ' -0+L,1'(,01 0< >-9/9. >&/,. '1> ?9@3) ?&@3. ()'( 8,CC .'(,.<D

()*.* -01.(3',1(.@

32

Systems 32



! #$%# &'()*+'(,-. /,.,01 2304*-( 5 &!2

61 7'3(1*3.),7 8,() ()* 9(') :('(* ;<<,-* 0< =>?-'(,01

!"#$%&$' )%'$* +,$ -*$.+"/$ -0110%& 2++*"3)+"0%450%-011$*#".647,.*$26"8$ 9:; <%=0*+$' 6"#$%&$:

#@A B00> <03 B,>0 '1> BC?<<D Z [*'-)*3 \0(*. ! #$%&'&() *+',-./0+'&+1 20.3

"#$%&'(H [)* ,1(*1( 0< (),. ('.G ,. (0 .0C,>,<D ' 730-*>?3* <03 .0CF,1I ' .D.(*+ 0< C,1*'3 ,1*S?'C,(,*.

?.,1I ()* <0CC08,1I .(*7.H

• X*<,1* F'3,'LC*. <03 ()* +0>*C,1I -01(*N( ]<03 *N'+7C*J 2 '1> 6 ,1 ()* 204&/50 2&'4&/. '1>

6&10-$ 6%03,. -'( <00> .-*1'3,0^

• R3,(* ,1*S?'C,(,*. ,1 (*3+. 0< ()*.* F'3,'LC*. <03 *'-) 0< ()* -01.(3',1(.

• _3'7) *'-) 0< ()* L0?1>'3D C,1*. 3*73*.*1(*> LD ()* -01.(3',1(.J ?.,1I ' .0C,> C,1* <03

-01.(3',1(. ()'( ,1-C?>* ()* 70,1(. 01 ()* L0?1>'3DJ '1> ' >0((*> C,1* <03 .(3,-( ,1*S?'C,(,*.

• :)'>* ()* '773073,'(* )'C<V7C'1* 3*73*.*1(*> LD *'-) -01.(3',1(J *,()*3 LD ?.,1I ' `(*.(

70,1(a (0 >*(*3+,1* 01 8),-) .,>* 0< ()* L0?1>'3D C,1* (0 .)'>*J 03 LD ?.,1I ()* C0I,-'C

,+7C,-'(,01 0< ()* ,1*S?'C,(D .,I1

• B,1> ()* 3*I,01 0< ,1(*3.*-(,01 <03 'CC 0< ()* -01.(3',1(.J ],1-C?>,1I -01.(3',1(. ,+7C,*> LD

()* -01(*N(J ,< ()* .D.(*+ 0< ,1*S?'C,(,*. ,. +0>*C,1I ' 3*'CV803C> .-*1'3,0^

• [),. 3*I,01 0< ,1(*3.*-(,01b70,1(. -0++01 (0 'CC 0< ()* )'C<V7C'1*.b,. ()* .0C?(,01 .*( (0()* .D.(*+ 0< ,1*S?'C,(,*.

)&$( +,-./-$/' 0&1#'H

23)4536 c*73*.*1( -01.(3',1(. LD *S?'(,01. 03 ,1*S?'C,(,*.J '1> LD .D.(*+. 0< *S?'(,01. '1>d03

,1*S?'C,(,*.J '1> ,1(*373*( .0C?(,01. '. F,'LC* 03 101VF,'LC* 07(,01. ,1 ' +0>*C,1I -01(*N(@

7(8-,(/ +,-./-$/'9 :3;3<= 23++43>= 23)453<

?-#.1@ AB@&8( )8-''C9

c*'> ()* 204&/50 2&'4&/. '1> 6&10-$ 6%03,. .-*1'3,0 (0I*()*3@ &0>*CJ 8,() .(?>*1( ,17?(J )08 (083,(* ()* -01.(3',1( <03 ()* '+0?1( 0< 730(*,1 ,1 ()* +,N(?3*@ U*C7 .(?>*1(. ,>*1(,<D ()* C'1I?'I*

,1 ()* -01.(3',1( ()'( ,+7C,*. ,( ,. '1 ,1*S?'C,(D ]`'( C*'.(a^ '1> >*(*3+,1* (0I*()*3 8)'( ,1*S?'C,(D

10('(,01 ()*D 80?C> ?.* (0 3*73*.*1( (),. C'1I?'I* ]`!a^@ O..,I1 .(?>*1(. (0 803G 01 S?*.(,01. %VY

8,() ' 7'3(1*3@

4D%8&$( A+E-88 F$&#%C9

R'(-) <03 )08 .(?>*1(. '3* ,1(*373*(,1I ()* ,1*S?'C,(D C'1I?'I* ,1 *'-) -01.(3',1( ]`'( C*'.(aJ !e

`+03* ()'1aJ fe `10 +03* ()'1aJ "e `C*.. ()'1aJ g^@ h,.(*1 <03 )08 .(?>*1(. '3* >*(*3+,1,1I ()* )'C<V

7C'1* ()*D .)0?C> .)'>* <03 *'-) ,1*S?'C,(DH ?.,1I ' (*.( 70,1(J 03 ?.,1I ()* ,1*S?'C,(D 3*C'(,01.),7

(0 >*(*3+,1* ,1 8),-) )'C<V7C'1* ()* .0C?(,01 .*( C,*.@ \0(* ()'( (),. ,. ()* <,3.( ('.G ,1 (),. +0>?C*

()'( ,1-C?>*. L0() `I3*'(*3 ()'1a '1> `C*.. ()'1a ,1*S?'C,(,*.@ OC.0 C,.(*1 <03 )08 .(?>*1(. '3*>*(*3+,1,1I 8)*()*3 ()* 70,1(. 01 ()* L0?1>'3D C,1* .)0?C> L* ,1-C?>*> ],1>,-'(*> LD ' .0C,> C,1*^

03 10( ],1>,-'(*> LD ' >0((*> C,1*^@ _00> S?*.(,01. (0 '.G 80?C> L* 0< ()* <03+J `E'1 ()* +,N(?3*

-01(',1 *N'-(CD %Y I3'+. 0< <'(Wa



! #$%# &'()*+'(,-. /,.,01 2304*-( 5 &!2

61 7'3(1*3.),7 8,() ()* 9(') :('(* ;<<,-* 0< =>?-'(,01

!"#$%&$' )%'$* +,$ -*$.+"/$ -0110%& 2++*"3)+"0%450%-011$*#".647,.*$26"8$ 9:; <%=0*+$' 6"#$%&$:

O. .(?>*1(. <,1,.) ()* 204&/50 2&'4&/. '1> 6&10-$ 6%03,. .-*1'3,0J ()*D .)0?C> +0F* 01 (0 ()* >-9/9.

>&/,. '1> ?9@3) ?&@3. .-*1'3,0@ [),. C'.( -01(*N( 8,CC L* >,.-?..*> '. ' 8)0C* I30?7J .0 'CC .(?>*1(.

1**> 10( <,1,.) ,( L*<03* +0F,1I (0 ()* 8)0C* -C'.. >,.-?..,01@

5G'1#'' AB@&8( )8-''C9

:('3( LD )'F,1I .(?>*1(. >*.-3,L*J ,1 ()*,3 081 803>.J ' .(*7VLDV.(*7 730-*>?3* <03 .0CF,1I '

.D.(*+ 0< C,1*'3 *S?'(,01.@ 23*.. <03 -C*'3 *N7C'1'(,01. '1> >*(',C.J .?-) '. ()0.* I,F*1 ,1 ()*7?370.* .('(*+*1( 'L0F*@ ;1-* ' C,.( 0< .(*7. )'. L**1 '3(,-?C'(*>J 803G ()30?I) ()* >-9/9. >&/,.

'1> ?9@3) ?&@3. .-*1'3,0 (0I*()*3J ?.,1I ()* .(?>*1(VI*1*3'(*> .*( 0< .(*7.@ U'F* .(?>*1(. -C'3,<D

()*,3 C,.( 0< .(*7. ,< ()*D <,1> ()*+.*CF*. >0,1I .0+*(),1I >,<<*3*1( ()'1 ()* .(*7. C,.(*>@ B03

*N'+7C*J >,> ()*D >*<,1* F'3,'LC*. ]> <03 0?1-*. 0< >-9/9. >&/,. '1> ? <03 0?1-*. 0< ?9@3) ?&@3.^

L*<03* 83,(,1I ()* -01.(3',1(.W

U*3* '3* ()* C,1*'3 .D.(*+. '1> <*'.,LC* 3*I,01. <03 ()* (80 >,<<*3*1( .-*1'3,0.@

204&/50 2&'4&/. '1> 6&10-$ 6%03,. >-9/9. >&/,. '1> ?9@3) ?&@3.

2T + 3F ! 12

4T + 4F >16

4T + 2F " 18

T + F <10

#

$

% %

&

% %

4 B +12 L ! 86 B + 4 L " 6# $

%

28GH.(/ 7(-/I= +(,= F&9 +I',(E' <3J



Name: Systems 2.9




Ready, Set, Go!

Ready

Topic: Solving two variable inequalities

1. A theater wants to take in at least $2000 for a certain matinee. Children’s tickets cost $5 eachand adult tickets cost $10 each.

a. Write an inequality describing the number of tickets that will allow the theater to meet theirgoal of $2000.

b. If the theater has a maximum of 350 seats, write an inequality describing the number of both

types of tickets the theater can sell.

c. Find the number of children and adult tickets that can be sold so that all seats are sold and the$2000 goal is reached.

Set

Topic: Writing equations of two variable inequalities

Given the graph with the regions that are shaded write the inequality or system of

inequalities.

2. 3. 4.

-8

-8 8

8

-8

-8 8

8

-8

-8 8

8

© 2012 www.flickr.com/photos/msciba

33

Systems 33



Name: Systems 2.9




Go

Topic: Graph two variable inequalities

Graph each set of inequalities below. Include the shaded region of both, plus indicate the

region that is true for all inequalities.

5. { 6. {

Solve the following systems of equations.

7. Nadia and Peter visit the candy store. Nadia buys three candy bars and four fruit roll-ups for$2.84. Peter also buys three candy bars, but can only afford one additional fruit roll-up. Hispurchase costs $1.79. What is the cost of a candy bar and a fruit roll-up individually?

8. { 9. {


http://www.khanacademy.org/math/algebra/systems-of-eq-and-ineq/v/graphing-systems-of-inequalities-2

-10

-10

10

10

-10

-10

10

10

34

Systems 34






! #$%# &'()*+'(,-. /,.,01 2304*-( 5 &! 2

61 7'3(1*3.),7 8,() ()* 9(') :('(* ;<<,-* 0< =>?-'(,01!"#$%&$' )%'$* +,$ -*$.+"/$ -0110%& 2++*"3)+"0%450%-011$*#".647,.*$26"8$ 9:; <%=0*+$' 6"#$%&$:

#@%$ A'B*1 ;?( 0< C01(*D( ! #$%&'(&) *+,)$-'%+,(+. /%-0

E3,(* ' .)077,1F .-*1'3,0 .,+,G'3 (0 ()0.* ,1 H:)077,1F <03 C'(. '1>

I0F.J (0 <,( *'-) 0< ()* <0GG08,1F .K.(*+. 0< *L?'(,01.@ A)*1 ?.* ()**G,+,1'(,01 0< M'3,'NG*. +*()0> K0? ,1M*1(*> ,1 HC'1 O0? P*( (0 ()*

20,1(Q A00J (0 .0GM* ()* .K.(*+@ :0+* 0< ()* .K.(*+. +'K )'M* ,1(*3*.(,1F 03 ?1?.?'G .0G?(,01.@

:** ,< K0? -'1 *D7G',1 ()*+ ,1 (*3+. 0< ()* .)077,1F .-*1'3,0. K0? 830(*@

%@ #@

R@ S@

A)3** 0< C'3G0.T '1> CG'3,('T. <3,*1>. '3* 7?3-)'.,1F .-)00G .?77G,*. '( ()* N00B.(03*@ :('1 N?K. '

10(*N00BQ ()3** 7'-B'F*. 0< 7*1-,G. '1> (80 +'3B*3. <03 UV@W$@ X'1 N?K. (80 10(*N00B.Q .,D

7'-B'F*. 0< 7*1-,G. '1> <,M* +'3B*3. <03 U%W@W$@ Y3'1 N?K. ' 10(*N00BQ (80 7'-B'F*. 0< 7*1-,G. '1>

(80 +'3B*3. <03 UZ@#W@ [08 +?-) >0 *'-) 0< ()*.* ()3** ,(*+. -0.(\

=D7G',1 ,1 803>. 03 8,() .K+N0G. )08 K0? -'1 ?.* K0?3 ,1(?,(,M* 3*'.01,1F 'N0?( ()*.* 7?3-)'.*.

(0 <,1> ()* 73,-* 0< *'-) ,(*+@

> ?

; @ ? A A A : B

6 " # 8 * : # 0 1 C = , 0 + 0 & C 1 0 1 1 . / $ %

35

Systems 35



! #$%# &'()*+'(,-. /,.,01 2304*-( 5 &!2

61 7'3(1*3.),7 8,() ()* 9(') :('(* ;<<,-* 0< =>?-'(,01!"#$%&$' )%'$* +,$ -*$.+"/$ -0110%& 2++*"3)+"0%450%-011$*#".647,.*$26"8$ 9:; <%=0*+$' 6"#$%&$:

#@%$ A'B*1 ;?( 0< C01(*D( ] A*'-)*3 ^0(*. ! #$%&'(&) *+,)$-'%+,(+. /%-0

"#$%&'(_ A),. ('.B F,M*. .(?>*1(. 73'-(,-* ,1 .0GM,1F .K.(*+. 0< (80 G,1*'3 *L?'(,01. ,1 (80

M'3,'NG*.@ 61 '>>,(,01Q ' -0?7G* 0< ,>*'. ()'( )'M* 10( N**1 *D'+,1*> ,1 (),. +0>?G* 8,GG '3,.*@ ;1*0< ()* .K.(*+. )'. 10 .0G?(,01. `'1 ,1-01.,.(*1( .K.(*+a .,1-* ()* G,1*. '3* 7'3'GG*G@ ;1* 0< ()*.K.(*+. )'. '1 ,1<,1,(* .*( 0< .0G?(,01.b'GG ()* 70,1(. 01 ()* G,1* `' >*7*1>*1( .K.(*+ab.,1-* N0()*L?'(,01. '-(?'GGK >*.-3,N* ()* .'+* G,1*@ :(?>*1(. .)0?G> 10(* )08 (0 ,>*1(,<K *L?,M'G*1( <03+. 0<()* .'+* G,1* 8)*1 83,((*1 ,1 .('1>'3> <03+Q '. 8*GG '. )08 (0 ,>*1(,<K 7'3'GG*G G,1*. 8)*1 83,((*1,1 .('1>'3> <03+@

61 ()* <,1'G 7'3( 0< ()* ('.B .(?>*1(. *D(*1> ()* .(3'(*FK <03 .0GM,1F ' .K.(*+ 0< G,1*'3 *L?'(,01. (0 '.,(?'(,01 ()'( -01(',1. ()3** *L?'(,01. '1> ()3** ?1B1081.@ A),. 803B 730M,>*. '>>,(,01'G73'-(,-* 8,() ()* -01-*7(?'G ,>*'. 0< .0GM,1F ' G,1*'3 .K.(*+ 0< *L?'(,01.@

)&$( +,-./-$/' 0&1#'_

2345637 :0GM* .K.(*+. 0< G,1*'3 *L?'(,01. *D'-(GK '1> '7730D,+'(*GK `*@F@Q 8,() F3'7).aQ <0-?.,1F01 7',3. 0< G,1*'3 *L?'(,01. ,1 (80 M'3,'NG*.@

4(8-,(/ +,-./-$/'9

:-#.1; <=;&8( )8-''>9

20,1( 0?( ()'( 8),G* ()* .K.(*+. 730M,>*> ,1 L?*.(,01. %cS >0 10( -0+* <30+ ' -01(*D(Q 8* -'1,+70.* ' -01(*D(Q .?-) '. ()* .)077,1F .-*1'3,0 .?FF*.(*> ,1 ()* ,1.(3?-(,01.Q (0 -3*'(* ' .(03K ()'(-'1 )*G7 ?. 3*'.01 ()30?F) ()* +'()*+'(,-. 0< .0GM,1F ()* .K.(*+Q 7'3(,-?G'3GK '. ?1?.?'G.,(?'(,01. '3,.* ()'( +'K .**+ -01<?.,1F 8)*1 803B,1F 01GK 8,() ()* *L?'(,01.@

20,1( 0?( 'G.0 ()'( L?*.(,01 W 8,GG '.B .(?>*1(. (0 *D(*1> ()* ,1(?,(,M* 3*'.01,1F 0< ()* H.)077,1F.-*1'3,0J (0 7?3-)'.*. ,1M0GM,1F -0+N,1'(,01. 0< ()3** ,(*+.@

5?%8&$( <+@-88 A$&#%>9

230NG*+. % '1> S '3* .(3',F)(<038'3>d ()*K -'1 N* .0GM*> ?.,1F ()* .(3'(*F,*. >*M*G07*> '1>3*M,*8*> ,1 (),. +0>?G*@ :0+* .(?>*1(. +'K -)00.* (0 ?.* 01GK ' F3'7),-'G 03 .?N.(,(?(,01 +*()0>@eGG08 ()*.* +*()0>. (0 N* ?.*> '. ' -)*-B 01 ()*,3 803BQ N?( *1-0?3'F* 'GG .(?>*1(. (0 73'-(,-* ()**G,+,1'(,01 +*()0>@

e.B .(?>*1(. (0 *D7G',1 ()* >,G*++'. ()'( 0--?3 8)*1 .0GM,1F 730NG*+. # '1> R 'GF*N3',-'GGK ?.,1F()* .)077,1F .-*1'3,0@ `20..,NG* *D7G'1'(,01 <03 f#_ :,1-* 8* N0?F)( *D'-(GK (8,-* '. +?-)Q ()*73,-*. 0< ()* ,1>,M,>?'G ,(*+. -'110( N* >*(*3+,1*>@ e1K -0+N,1'(,01 0< 73,-*. ()'( 8,GG +'B* ()*

<,3.( *L?'(,01 (3?* 8,GG 'G.0 +'B* ()* .*-01> *L?'(,01 (3?*Q .0 ()* .0G?(,01 (0 ()* .K.(*+ ,. ()*8)0G* -0GG*-(,01 0< 03>*3*>c7',3. ()'( .'(,.<K *,()*3 *L?'(,01@ 20..,NG* *D7G'1'(,01 <03 fR_ E*7?3-)'.*> ()3** (,+*. '. +?-) ,1 ()* .*-01> 7?3-)'.* ()'1 ,1 ()* <,3.(Q N?( ()* (0('G -0.( 8'. 10(()3** (,+*. '. +?-)@ A),. .?FF*.(. ()'( ()* 8'K ()* (0('G -0.( 8'. >*(*3+,1*> ,1 ()* .*-01>7?3-)'.* 8'. ,1-01.,.(*1( 8,() ()* 8'K ()* (0('G 73,-* 8'. >*(*3+,1*> ,1 ()* <,3.( 7?3-)'.*b



! #$%# &'()*+'(,-. /,.,01 2304*-( 5 &!2

61 7'3(1*3.),7 8,() ()* 9(') :('(* ;<<,-* 0< =>?-'(,01!"#$%&$' )%'$* +,$ -*$.+"/$ -0110%& 2++*"3)+"0%450%-011$*#".647,.*$26"8$ 9:; <%=0*+$' 6"#$%&$:

7*3)'7. ' G'3F* >,.-0?1( ,1 ()* 73,-* 0< 01* 0< N0() 0< ()* ,(*+.@ E* -'110( >*(*3+,1* ()* 73,-* 0<*,()*3 ,(*+Q .,1-* ()* 73,-*. >,> 10( 3*+',1 -01.,.(*1( ()30?F)0?( ()* .-*1'3,0@a

E),G* ' F3'7),-'G +*()0> -'1 N* ?.*> (0 -)*-B ()* 3*'.01'NG*1*.. 0< ()* .0G?(,01. (0 ()* .K.(*+. ,1730NG*+. % '1> SQ ,( 730M,>*. '>>,(,01'G ,1.,F)( ,1(0 8)'( ,. )'77*1,1F ,1 730NG*+. # 03 R@=1-0?3'F* H<'.( <,1,.)*3.J (0 F3'7) f# '1> fR (0 73*.*1( >?3,1F ()* 8)0G* -G'.. >,.-?..,01@

BC'1#'' <=;&8( )8-''>9

91G*.. .(?>*1(. )'M* *1-0?1(*3*> >,<<,-?G(K 8,() .0GM,1F L?*.(,01. % '1> SQ K0? -'1 <0-?. ()*>,.-?..,01 01 ()* 1*8 ,..?*. ()'( '3,.* ,1 L?*.(,01. #Q R '1> W@ Y03 *'-) 0< 730NG*+. # '1> R )'M*.(?>*1(. *D7G',1 8)'( ,. )'77*1,1F ?.,1F ' F3'7),-'G +*()0> '1> ' M*3N'G >*.-3,7(,01 0< ()*'..0-,'(*> .)077,1F .-*1'3,0@ 61 f# .(?>*1(. .)0?G> 10(* ()'( 8),G* ()* F3'7). 0< ()* G,1*. '3* ()*.'+*Q ()* .)077,1F .-*1'3,0. 3*73*.*1(*> NK ()* *L?'(,01. '3* 10(Q 'G()0?F) ()* 01* 7?3-)'.* ,.4?.( ' -01.('1( +?G(,7G* 0< ()* 0()*3@ e((*+7(,1F (0 ?.* ()* *G,+,1'(,01 +*()0> 80?G> 3*.?G( ,1N0() M'3,'NG*. '1> ()* (0('G -0.( N*,1F *G,+,1'(*>Q 3*.?G(,1F ,1 ()* ,>*1(,(K $ g $@ 61 fR .(?>*1(..)0?G> 10(* ()'( ()* G,1*. '3* 7'3'GG*G NK *D'+,1,1F ()* .G07*c,1(*3-*7( <03+ 0< ()* (80 G,1*.Q '1>3*-0F1,h* ()'( (),. 3*.?G(. N*-'?.* ()* -0*<<,-,*1(. 0< 1 '1> 2 ,1 ()* (80 *L?'(,01. '3* 730703(,01'GQ

N?( ()* (0('G -0.(. '3* 10(@

['M* .(?>*1(. .)'3* ()*,3 ,1(?,(,M* 3*'.01,1F 01 fWQ '1> ()*1 -011*-( (),. (0 ()* 'GF*N3',- 803B 0<

.0GM,1F ()* .K.(*+ 0< ()3** *L?'(,01. ,1 ()3** M'3,'NG*. NK *G,+,1'(,01@

:('1

X'1

Y3'1

:,1-* :('1 N0?F)( 'G+0.( ()* .'+* ,(*+. '. Y3'1Q ()* >,<<*3*1-* ,1 ()* (0('G -0.( ,. >?* *1(,3*GK (0()* '>>,(,01'G 7'-B'F* 0< 7*1-,G.Q 8),-) +?.( -0.( U%@#W@ 6< K0? >0?NG* :('1T. 7?3-)'.* ,( 8,GG N*'G+0.( ()* .'+* '. X'1T.Q *D-*7( X'1 80?G> )'M* 7?3-)'.*> 01* +03* +'3B*3 <03 U$@W$@ ^08 ()'(

8* B108 ()* 73,-* 0< (80 0< ()* ,(*+. 8* -'1 <,1> ()* 73,-* 0< ()* (),3>@ :('1 7?3-)'.*> R 7'-B'F*.0< 7*1-,G. <03 UR@VW '1> # +'3B*3. <03 U%@$$Q .0 ' 10(*N00B +?.( -0.( U#@VW@ A)*.* 73,-*. 'G.0 -)*-B,1 ()* 0()*3 (80 7?3-)'.*.@

eGF*N3',-'GGKQ ,< 8* .?N(3'-( Y3'1T. *L?'(,01 <30+ :('1T. 8* F*( 3 g %@#W@ 6< 8* >0?NG* :('1T.*L?'(,01 '1> .?N(3'-( ,( <30+ X'1T. 8* F*( 4 g $@W$@ E* -'1 .?N.(,(?(* ()*.* M'G?*. ,1(0 '1K 01* 0<()* ()3** *L?'(,01. (0 .0GM* <03 +5 A)*3* '3* 0()*3 'GF*N3',- 8'K. 8* -0?G> '7730'-) (),. .K.(*+Q

.?-) '. *G,+,1'(,1F + ,1 (80 0< ()* *L?'(,01. (0 <03+ ' 1*8 # ! # .K.(*+@ 6( ,. 10( 1*-*..'3K (0

<03+'G,h* (),. .(3'(*FK ,1 (),. -0?3.*@

28CD.(/ 4(-/EF +(,F A&9 +E',(@' G3HI

n + 3 p + 2m = 7.50

2n + 6 p + 5m = 15.50

n + 2 p + 2m = 6.25

!

" #

$ #



!"#$% '()*$#) !"#$

! #$%# &'()*+'(,-. /,.,01 2304*-( 5 & 2

61 7'3(1*3.),7 8,() ()* 9(') :('(* ;<<,-* 0< =>?-'(,01"#$%&'%( )&(%* +,% -*%.+#/% -0110&' 2++*#3)+#0&450&-011%*$#.647,.*%26#8% 9:; <&=0*+%( 6#$%&'%

&'()*+ ,'-+ ./0

&'()*

+,-./% '()*$#) ,0 12$34"5.*.$)

1/2 '(34 /5 -4' 6*6-'76 /5 89':;(<8-8'6+ )'-'2789' 85 -4' =8>'9 3//2)89(-'6 (2' 6/<;-8/96 -/

-4' 6*6-'7"

67 87 y > !2 x+9

y " 5 x !6

#$%

&'(

97 y < !

1

2 x+ 9

y > 6 x !10

"

#$

%$

&

'$

($

"7 : ; < 6= > "7 : ?8 < ?@ > "7 : ?8 < ?@ >

A7 : 6 < B > A7 : ?6 < 68 > A7 : C < 9 >

/7 : D < 6@ > /7 : @ < = > /7 : ?D < 6= >

,'-

+,-./% E$*$F#.2$ *G$ 24#A$F ,0 ),54*.,2) .2 " )()*$# ,0 $34"*.,2)

?@A2'66 '(34 ':;(-8/9 89 6</A'B89-'23'A- 5/27" !"#$%&# ()*+$",(+ 6-(-' C4'-4'2 -4' 6*6-'7

/5 ':;(-8/96 4(6 D'2/+ /9' /2 895898-' 6/<;-8/96" E/C )/ */; F9/CG

B7!! ! !! ! !"

! ! !!! ! !

@7!! ! !! ! !

! ! ! ! !

;7!!!! ! ! ! !"

!!!! ! ! ! !!"

C7!! ! !! ! !!

!! ! !! ! !!"

y ! 3 x " 5

y # x +2

$%&

'()

> ?;@? AAA:B6#$8*:$01C=,0+0'C1011./%&

36

Systems 36



!"#$% '()*$#) !"#$

! #$%# &'()*+'(,-. /,.,01 2304*-( 5 & 2

61 7'3(1*3.),7 8,() ()* 9(') :('(* ;<<,-* 0< =>?-'(,01"#$%&'%( )&(%* +,% -*%.+#/% -0110&' 2++*#3)+#0&450&-011%*$#.647,.*%26#8% 9:; <&=0*+%( 6#$%&'%

./

+,-./% HF"-G *I, J"F."A5$ .2$34"5.*.$)

.2(A4 -4' 5/<</C89= 89':;(<8-8'6" H' 6;2' -/ <(I'< */;2 (@'6 (9) 63(<'" J;6-85* -4' 2'=8/9 */;

64()' I* 64/C89= -42'' A/89-6 89 -4' 2'=8/9 (6 I'89= 6/<;-8/96 -/ -4' A2/I<'7" ,4/C ( A/89-

*/; 4(>' -'6-') -/ A2/>' */;2 64()') 2'=8/9 86 (33;2(-'"

D7 !! ! !! ! !" K7 ! ! !! ! !

6=7 !! ! !! ! ! 667 ! !!

!! ! !

687 L2 *G$ )"#$ )$* ,0 "M$)< NF"-G ! O " P 8 "2Q ( R " P @ 7

SG"* J"54$) Q, *G$)$ *I, G"J$ .2 /,##,2T

!$$Q G$5-T UG$/V ,4* *G$)$ F$5"*$Q J.Q$,)

+$)*.2N " ),54*.,2 *, "2 $34"*.,2

G**-%WWIII7VG"2"/"Q$#(7,FNW#"*GW"5N$AF"W)()*$#)?,0?$3?"2Q?.2$3WJW*$)*.2N?"?),54*.,2?0,F?"?)()*$#?,0?$34"*.,2)

!4#A$F ,0 ),54*.,2)

G**-%WWIII7VG"2"/"Q$#(7,FNW#"*GW"5N$AF"W)()*$#)?,0?$3?"2Q?.2$3WJW)-$/."5?*(-$)?,0?5.2$"F?)()*$#)

',5J.2N .2$34"5.*.$)

G**-%WWIII7VG"2"/"Q$#(7,FNW#"*GW"5N$AF"W),5J.2N?5.2$"F?.2$34"5.*.$)WJW),5J.2N?.2$34"5.*.$)

-5

-5

5

5

-5

-5

5

5

-5

-5

5

5

-5

-5

5

5

-5

-5

5

5

37

Systems 37



! #$%# &'()*+'(,-. /,.,01 2304*-( 5 &! 2

61 7'3(1*3.),7 8,() ()* 9(') :('(* ;<<,-* 0< =>?-'(,01

!"#$%&$' )%'$* +,$ -*$.+"/$ -0110%& 2++*"3)+"0%450%-011$*#".647,.*$26"8$ 9:; <%=0*+$' 6"#$%&$:

>

? ; @ ? A A A

B 6 " # 8 * # 0 1 C = , 0 + 0 & C ' 0 6 1 . % & . D 6 " 6

#@%% &03* A),1B. A'C*1 ;?( 0< D01(*E( ! #$%&'(&) *+,)$-'%+,(+. /%-0

:0FG* ()* <0FF08,1B .H.(*+. 0< ,1*I?'F,(,*.J

%@!5 x + 3 y " 45

2 x + 3 y > 24

# $ %

#@!10 x + 6 y " 90

6 x + 9 y > 36

# $ %

K@ 6. ()* 70,1( LMKN %$O ' .0F?(,01 (0 ()* .H.(*+ ,1 730PF*+ Q%R S)H 03 8)H 10(R

T@ U08 '3* ()* ,1*I?'F,(,*. 3*73*.*1(,1B ()* P0?1>'3,*. 0< ()* .0F?(,01 .*(. ,1 730PF*+. Q% '1> Q#

.,+,F'3 (0 *'-) 0()*3R S)'( '--0?1(. <03 ()*.* .,+,F'3,(,*.R

V@ S3,(* ()* .H.(*+ 0< ,1*I?'F,(,*. 8)0.* .0F?(,01 .*( ,. .)081 P*F08J

W@ X+'1>' ,. *E'+,1,1B Y3'1CZ. 803C 01 QVN 8)*1 .)* *E-F',+.N

[\0? )'G* 83,((*1 'FF 0< H0?3 ,1*I?'F,(,*. P'-C8'3>.@ A)*

.0F?(,01 .*( (0 H0?3 .H.(*+ 80?F> F00C F,C* (),.@]

S)'( >0 H0? (),1C 'P0?( X+'1>'Z. .('(*+*1(R

38

Systems 38



! #$%# &'()*+'(,-. /,.,01 2304*-( 5 &!2

61 7'3(1*3.),7 8,() ()* 9(') :('(* ;<<,-* 0< =>?-'(,01

!"#$%&$' )%'$* +,$ -*$.+"/$ -0110%& 2++*"3)+"0%450%-011$*#".647,.*$26"8$ 9:; <%=0*+$' 6"#$%&$:

#@%% &03* A),1B. A'C*1 ;?( 0< D01(*E( ^

A*'-)*3 _0(*. ! #$%&'(&) *+,)$-'%+,(+. /%-0

"#$%&'(J A)* 7?370.* 0< (),. ('.C ,. (0 73'-(,-* .0FG,1B .H.(*+. 0< F,1*'3 ,1*I?'F,(,*. PH ,>*1(,<H,1B

()* 0G*3F'77,1B L03 ,1(*3.*-(,1BO 3*B,01 0< ()* )'F<M7F'1*. ()'( <03+ ()* .0F?(,01 .*(. 0< *'-) 0< ()*

(80MG'3,'PF* ,1*I?'F,(,*. ,1 ()* .H.(*+@ :(?>*1(. 73'-(,-* (),. ,1 730PF*+. % '1> # PH <,1>,1B ()*

0G*3F'77,1B 3*B,01 '1> ,1 K PH -3*'(,1B ' .H.(*+ 0< ,1*I?'F,(,*. 8)0.* .0F?(,01 ,. B,G*1 '. ' .)'>*>

3*B,01 ,1 ()* -003>,1'(* 7F'1*@ :(?>*1(. 'F.0 +?.( 3*-0B1,`* ()* >,<<*3*1-* P*(8**1 ' .(3,-(

,1*I?'F,(H '1> 01* ()'( ,1-F?>*. ()* 70,1(. 01 ()* P0?1>'3H F,1* '. 7'3( 0< ()* .0F?(,01 .*(@ A)'( ,.N

,1 730PF*+ TN ()*H +?.( >,.(,1B?,.) ()* >,<<*3*1-* P*(8**1 a '1> !N '1> P*(8**1 b '1> " '.

3*F'(,01.),7.@ :(?>*1(. 'F.0 B*( '>>,(,01'F 73'-(,-* ,1 3*-0B1, ,1B 7'3'FF*F 03 *I?,G'F*1( F,1*. <03

F,1*'3 *I?'(,01. 83,((*1 ,1 .('1>'3> <03+@

)&$( +,-./-$/' 0&1#'J

23456378 c3'7) ()* .0F?(,01. (0 ' F,1*'3 ,1*I?'F,(H ,1 (80 G'3,'PF*. '. ' )'F<M7F'1* L*E-F?>,1B ()*

P0?1>'3H ,1 ()* -'.* 0< ' .(3,-( ,1*I?'F,(HON '1> B3'7) ()* .0F?(,01 .*( (0 ' .H.(*+ 0< F,1*'3

,1*I?'F,(,*. ,1 (80 G'3,'PF*. '. ()* ,1(*3.*-(,01 0< ()* -033*.701>,1B )'F<M7F'1*.@

4(9-,(/ +,-./-$/':

;-#.1< =><&9( )9-''?:

:(?>*1(. .)0?F> 1**> G*3H F,((F* ,1(30>?-(,01 (0 (),. ('.C@ 20,1( 0?( ()* I?*.(,01. %N # '1> V '3*

73'-(,-* 730PF*+.N '1> ()'( I?*.(,01. KN T '1> W 8,FF *E(*1> ()*,3 (),1C,1B@

5@%9&$( =+A-99 B$&#%?:

A)* +',1 ,..?* (0 8'(-) <03 ,. 8)*()*3 .(?>*1(. -'1 <,1> '1> .)'>* ()* -033*-( 3*B,01. ,1 ()*

-003>,1'(* 7F'1* <03 % '1> #N '1> ,< ()*H -'1 83,(* ' .H.(*+ 0< ()3** F,1*'3 ,1*I?'F,(,*. ,1 (80

G'3,'PF*. <03 V .0 ()'( ()* 0G*3F'77,1B 3*B,01 0< ()* ()3** )'F<M7F'1*. ,. ()* (3,'1BF* B,G*1@ A),. 8,FF

3*I?,3* .(?>*1(. (0 'F.0 P* 'PF* (0 83,(* ()* *I?'(,01. 0< ()* P0?1>'3H F,1*. .)081@ S'(-) <03 )08

.(?>*1(. >0 (),.J ?.,1B ()* .F07* '1> HM,1(*3-*7( 0< *'-) F,1* (0 83,(* ()* *I?'(,01 ,1 .F07*M,1(*3-*7(

<03+N 03 ?.,1B ()* (80 70,1(. 0< ,1(*3.*-(,01 ()'( F,* 01 *'-) F,1*@ XF.0 10(* ,< .(?>*1(. '3* 8,FF,1B (0

F*'G* ()*,3 *I?'(,01. ,1 .F07*M,1(*3-*7( <03+ 8)*1 ()*H 83,(* ()*,3 .H.(*+ 0< ,1*I?'F,(,*. L8),-) ,.

<,1*O 03 ,< ()*H <**F -0+7*FF*> (0 (3'1.<03+ ()*,3 *I?'(,01. ,1(0 .('1>'3> <03+ L8),-) ,. 10(

1*-*..'3HN P?( +'H <**F +03* [-033*-(] (0 .(?>*1(. .,1-* (),. ,. ()* <03+ ,1 8),-) +0.( .H.(*+.

)'G* '77*'3*> ()30?B)0?( ()* +0>?F*O@



! #$%# &'()*+'(,-. /,.,01 2304*-( 5 &!2

61 7'3(1*3.),7 8,() ()* 9(') :('(* ;<<,-* 0< =>?-'(,01

!"#$%&$' )%'$* +,$ -*$.+"/$ -0110%& 2++*"3)+"0%450%-011$*#".647,.*$26"8$ 9:; <%=0*+$' 6"#$%&$:

:0F?(,01 .*( <03 730PF*+ %J :0F?(,01 .*( <03 730PF*+ #J

:H.(*+ <03 730PF*+ VJ

y ! 3

2 x +10

y ! " 5

2 x +10

y # 1

6 x + 2

$

%

&

'

& 03

CD'1#'' =><&9( )9-''?:

6< ()*3* '3* 10 ,..?*. 8,() .)'>,1B ()* '773073,'(* .0F?(,01 .*( <03 I?*.(,01. % '1> #N P*B,1 ()*

>,.-?..,01 8,() I?*.(,01 V@ U'G* ' .(?>*1( 8)0 830(* ()* *I?'(,01. ,1 .F07*M,1(*3-*7( <03+

73*.*1( ()*,3 .H.(*+ 0< ,1*I?'F,(,*.N <0FF08*> PH ' .(?>*1( 8)0 830(* ()* *I?'(,01. ,1 .('1>'3>

<03+@ 6< P0() +*()0>. '3* 10( 73*.*1( ,1 H0?3 -F'..N P3,1B ?7 ()* I?*.(,01 0< 8),-) ,. [-033*-(]@ 6( ,.

,+703('1( <03 .(?>*1(. (0 3*-0B1,`* ()'( ()* <03+ ,1 8),-) 8* 83,(* ()* *I?'(,01 0< ()* P0?1>'3H

F,1* ,. 10( ,+703('1(@

A?31 ()* >,.-?..,01 (0 ()* ,>*'. 3*73*.*1(*> ,1 I?*.(,01. KN T '1> W@

Y03 I?*.(,01 KN .(?>*1(. .)0?F> P* 'PF* (0 -01-F?>* ()'( ()* 70,1( LMKN %$O ,. 10( ' .0F?(,01 (0 ()*

.H.(*+ .,1-* ,( >0*.1Z( .'(,.<H ()* .*-01> L.(3,-(O ,1*I?'F,(H ,1 ()* .H.(*+ ,1 Q%@

Y03 I?*.(,01 TN .(?>*1(. .)0?F> 3*-0B1,`* ()* <,3.( ,1*I?'F,(,*. ,1 *'-) .H.(*+ -01(',1 ()* .'+*

P0?1>'3H F,1* .,1-* 01* ,1*I?'F,(H ,. 4?.( ' -01.('1( +?F(,7F* 0< ()* 0()*3@ A)* 0()*3 P0?1>'3H F,1*.

,1 ()* (80 .H.(*+. '3* 7'3'FF*F (0 *'-) 0()*3 .,1-* ()* -0*<<,-,*1(. 0< ()* 1 M(*3+. '1> ()* 2 M(*3+.

'3* 730703(,01'FN P?( ()* -01.('1( (*3+. '3* 10(@

!3 x + 2 y " 20

5 x + 2 y " 20

! x + 6 y # 12

$

%

&

' &



! #$%# &'()*+'(,-. /,.,01 2304*-( 5 &!2

61 7'3(1*3.),7 8,() ()* 9(') :('(* ;<<,-* 0< =>?-'(,01

!"#$%&$' )%'$* +,$ -*$.+"/$ -0110%& 2++*"3)+"0%450%-011$*#".647,.*$26"8$ 9:; <%=0*+$' 6"#$%&$:

Y03 I?*.(,01 WN .(?>*1(. .)0?F> 3*-0B1,`* ()'( ,< Y3'1C >,> ,1>**> (?31 'FF 0< ),. ,1*I?'F,(H

.('(*+*1(. '30?1> L8* '..?+* X+'1>' +*'1. )* .8,(-)*> '30?1> ),. [B3*'(*3 ()'1] '1> [F*..

()'1] .,B1.ON ()* .0F?(,01 (0 Y3'1CZ. .H.(*+ 80?F> P* '1 *+7(H .*( 3'()*3 ()'1 ()* .0F?(,01

73070.*> PH X+'1>'N .,1-* 'FF ()3** )'F<M7F'1*. 80?F> 10( .)'3* '1H 70,1(. ,1 -0++01@

29DE.(/ 4(-/FG +(,G B&: +F',(A' 8377



!"#$% '()$ *+,-./ *"0$- (12 (3 4(-2$52 !"##

! #$%# &'()*+'(,-. /,.,01 2304*-( 5 & 2

61 7'3(1*3.),7 8,() ()* 9(') :('(* ;<<,-* 0< =>?-'(,01"#$%&'%( )&(%* +,% -*%.+#/% -0110&' 2++*#3)+#0&450&-011%*$#.647,.*%26#8% 9:; <&=0*+%( 6#$%&'%

%&'()* +&,* -./

%&'()

*(6,7% 8$2$)#,-$ " .((9 :,$;,-. ;,-9(; 3() .)"6+/

"#$% &'$()#*%+ , +-,.# /0 ,

01%)(*/%2 *( *& *3./-(,%( (#,(

4$ &$$ *3./-(,%( ./*%(&5 6/-7*%$,- 01%)(*/%&2 4$ 4,%( ,

4*%8/4 (#,( &#/4& *3./-(,%(

*%0/-3,(*/% -$7,($8 (/ (#$

&(/-95 :0($%2 (#*& 3$,%&

*%)718*%+ ;/(# (#$ <= ,%8 9=*%($-)$.(&5

0.1 ,2& 3.44.5678 &9:',6.7;* ;,',& ' 567(.5 ,2', 5.:4( <& ;',6;3'=,.1) 3.1 ,2& 86>&7

&9:',6.7" ?2&7 ;@&,=2 ' 81'A2 67 ,2& <.B&; A1.>6(&("

<= !!!! ! !! @ !"" >= !! ! !! ! !"

5% ? @ A BC C% ? @ A 5% ? @ A BC C% ? @ A

5D/7"E$% CD/7"E$% 5D/7"E$% CD/7"E$%

F5"#6E$% . G5H I3

15 J K

L,-9(;% ? D<M@ <MA BC ? D<M@<MA5D /7"E$% < CD/7"E$% <

!N* " .((9 ;,-9(;

L,-9(;% ?D<M@ >OA BC ? D<M@ OA5D/7"E$% O CD/7"E$% O

P((9 ;,-9(;

> ?;@? AAA:B6#$8*:$01C=,0+0'C(061.&'.D6#6

39

Systems 39



!"#$% '()$ *+,-./ *"0$- (12 (3 4(-2$52 !"##

! #$%# &'()*+'(,-. /,.,01 2304*-( 5 & 2

61 7'3(1*3.),7 8,() ()* 9(') :('(* ;<<,-* 0< =>?-'(,01"#$%&'%( )&(%* +,% -*%.+#/% -0110&' 2++*#3)+#0&450&-011%*$#.647,.*%26#8% 9:; <&=0*+%( 6#$%&'%

+&,

*(6,7% 4)$"2,-. "-9 /(E:,-. 2;( :"),"BE$ ,-$Q1"E,2,$/

R= S"22C #"0$/ TU 6$) +(1) #(;,-. E";-/ "-9 T<> 6$) +(1) B"BC/,22,-.= V+$ ;"-2/ 2( #"0$ "2

E$"/2 T<MM 6$) ;$$0 B12 7"- ;()0 -( #()$ 2+"- <> +(1)/ " ;$$0= L),2$ "-9 .)"6+ " /C/2$# (3

E,-$") ,-$Q1"E,2,$/= W,-"EEC@ E,/2 > 6(//,BE$ 7(#B,-"2,(-/ (3 +(1)/ 2+"2 S"22C 7(1E9 ;()0 "2 $"7+ X(B=

-.

*(6,7% V(E:$ /C/2$#/ (3 $Q1"2,(-/

+.4>& &'=2 ;);,&C .3 &9:',6.7; :;678 '7) C&,2.( ).: A1&3&1

Y=!! ! !! ! !!

! ! !! ! !

!

!

O=! ! ! ! !

!"

!

!! ! !! ! !!

E0)*' 3.3F'#++#&G

E 0 ) * ' 1 0 A # & G 6 . A & '

5 10 15 20

20

18

16

14

12

10

8

6

4

2

40

Systems 40



!"#$% '()$ *+,-./ *"0$- (12 (3 4(-2$52 !"##

! #$%# &'()*+'(,-. /,.,01 2304*-( 5 & 2

61 7'3(1*3.),7 8,() ()* 9(') :('(* ;<<,-* 0< =>?-'(,01"#$%&'%( )&(%* +,% -*%.+#/% -0110&' 2++*#3)+#0&450&-011%*$#.647,.*%26#8% 9:; <&=0*+%( 6#$%&'%

D1&',& ' ;);,&C .3 &9:',6.7; '7( :;& 6, ,. ;.4>& ,2& 3.44.5678 9:&;,6.7;"

K= N3 2+$ 2;( -(-D),.+2 "-.E$/ ,- " ),.+2 2),"-.E$@ (-$ #$"/1)$/ 2;,7$ "/ #"-C 9$.)$$/ "/ 2+$ (2+$)=L+"2 ")$ 2+$ "-.E$/Z

[= *+$ /1# (3 2;( -1#B$)/ ,/ [M "-9 2+$ 9,33$)$-7$ ,/ <<= L+"2 ")$ 2+$ -1#B$)/Z

U= \ )$72"-.1E") 3,$E9 ,/ $-7E(/$9 BC " 3$-7$ (- 2+)$$ /,9$/ "-9 " ;"EE (- 2+$ 3(1)2+ /,9$= *+$ 2(2"EE$-.2+ (3 2+$ 3$-7$ ,/ R>M C")9/= ]3 2+$ 3,$E9 +"/ " 2(2"E 6$),#$2$) (3 YMM C")9/@ ;+"2 ")$ 2+$

9,#$-/,(-/ (3 2+$ 3,$E9Z

^= \ )"C 712/ " E,-$ 3()#,-. 2;( "-.E$/= *+$ 9,33$)$-7$ B$2;$$- 2+$ 2;( "-.E$/ ,/ = L+"2 9($/

$"7+ "-.E$ #$"/1)$Z

!$$9 _$E6Z 4+$70 (12 2+$/$ )$E"2$9 :,9$(/%

+226%``;;;=0+"-"7"9$#C=().`#"2+`"E.$B)"`/C/2$#/D(3D$QD"-9D,-$Q`:`/C/2$#D(3D,-$Q1"E,2,$/D"66E,7"2,(-

41

Systems 41



! #$%# &'()*+'(,-. /,.,01 2304*-( 5 &! 2

61 7'3(1*3.),7 8,() ()* 9(') :('(* ;<<,-* 0< =>?-'(,01!"#$%&$' )%'$* +,$ -*$.+"/$ -0110%& 2++*"3)+"0%450%-011$*#".647,.*$26"8$ 9:; <%=0*+$' 6"#$%&$:

>

? ; @ ? A A A B

6 " # 8 * # 0 1 C = , 0 + 0 & C ' ) D & = *

#@%# 2*( :,((*3. A*B,.,(*> ! #$%$&'( )*+$,-./*+0*1 2/-3

C'3D0. '1> CD'3,(' )'B* .?--*..<?DDE <0?1> ' 8'E (0 3*73*.*1( /&& 0<

()* -0+F,1'(,01. 0< -'(. '1> >0G. ()'( ()*E -'1 F0'3> F'.*> 01 /&& 0< ()* <0DD08,1G -01.(3',1(.@

• 4(/5$H C'( 7*1. 8,DD 3*I?,3* J <( # 0< .7'-*K 8),D* >0G 3?1. 3*I?,3* #L <( #@ C'3D0. '1> CD'3,(')'B* ?7 (0 MJ$ <( # 'B',D'FD* ,1 ()* .(03'G* .)*> <03 7*1. '1> 3?1.K 8),D* .(,DD D*'B,1G *10?G)300+ (0 +0B* '30?1> ()* -'G*.@

• 6$$+0*1 207$H C'3D0. '1> CD'3,(' *.(,+'(* ()'( -'(. 8,DD 3*I?,3* J +,1?(*. (8,-* ' >'EN+031,1G '1> *B*1,1GN(0 <**> '1> -D*'1 ()*,3 D,((*3 F0O*.K <03 ' (0('D 0< %# +,1?(*. 7*3 >'E<03 *'-) -'(@ P0G. 8,DD 3*I?,3* %$ +,1?(*. (8,-* ' >'E (0 <**> '1> 8'DQK <03 ' (0('D 0< #$+,1?(*. 7*3 >'E <03 *'-) >0G@ C'3D0. -'1 .7*1> ?7 (0 R )0?3. *'-) >'E <03 ()* +031,1G '1>*B*1,1G <**>,1G.K F?( 1**>. ()* +,>>D* 0< ()* >'E 0<< <03 F'.*F'DD 73'-(,-* '1> G'+*.@

• 8/7($,0*1 207$H S)* (8,1. 7D'1 (0 .7*1> %J +,1?(*. *'-) >'E F3?.),1G '1> 7*((,1G *'-)-'(K '1> #$ +,1?(*. *'-) >'E F'(),1G 03 7D'E,1G 8,() *'-) >0G@ CD'3,(' 1**>. (,+* 0<< ,1 ()*+031,1G <03 .8,+ (*'+ '1> *B*1,1G <03 )*3 '3( -D'..K F?( .)* -'1 .7*1> ?7 (0 R )0?3.>?3,1G ()* +,>>D* 0< ()* >'E (0 7'+7*3 '1> 7D'E 8,() ()* 7*(.@

• 4./,.9:( ;'-.-H C'3D0. '1> CD'3,(' 7D'1 (0 ,1B*.( +?-) 0< ()* T%#R$ ()*E *'31*> <30+ ()*,3D'.( F?.,1*.. B*1(?3* (0 7?3-)'.* -'( 7*1. '1> >0G 3?1.@ 6( 8,DD -0.( TM# <03 *'-) -'( 7*1'1> TR$ <03 *'-) >0G 3?1@

U08 ()*E '3* (3E,1G (0 >*(*3+,1* )08 +'1E 0< *'-) (E7* 0< 7*( ()*E .)0?D> 7D'1 (0 '--0++0>'(*@;< -0?3.*K C'3D0. '1> CD'3,(' 8'1( (0 +'Q* '. +?-) +01*E '. 70..,FD* <30+ ()*,3 F?.,1*..K .0 ()*E

1**> (0 7'E '((*1(,01 (0 F0() ()*,3 >',DE ,1-0+* '. 8*DD '. ()*,3 >',DE -0.(.@ S)*E 7D'1 (0 -)'3G* TR7*3 >'E <03 F0'3>,1G *'-) -'( '1> T#$ 7*3 >'E <03 *'-) >0G@ S)*E *.(,+'(* ()'( *'-) -'( 8,DD 3*I?,3*T#@$$ 7*3 >'E ,1 <00> '1> .?77D,*.K '1> ()'( *'-) >0G 8,DD 3*I?,3* TL@$$ 7*3 >'E ,1 -0.(.@

V<(*3 .?3B*E,1G ()* -0++?1,(E 3*G'3>,1G ()* 7*( F0'3>,1G 1**>.K C'3D0. '1> CD'3,(' '3* -01<,>*1(()'( ()*E -'1 Q**7 'DD 0< ()*,3 F0'3>,1G .7'-*. <,DD*> <03 ()* .?++*3@

:0 ()* I?*.(,01 ,.K )08 +'1E 0< *'-) (E7* 0< 7*( .)0?D> ()*E 73*7'3* <03 ,1 03>*3 (0 +'Q* '. +?-)+01*E '. 70..,FD*W

X)'( -0+F,1'(,01 0< -'(. '1> >0G. >0 E0? (),1Q 8,DD +'Q* ()* +0.( +01*EW X)'(3*-0++*1>'(,01. 80?D> E0? G,B* (0 C'3D0. '1> CD'3,('K '1> 8)'( '3G?+*1( 80?D> E0? ?.* (0-01B,1-* ()*+ ()'( E0?3 3*-0++*1>'(,01 ,. 3*'.01'FD*W

S0 G*( .('3(*> 01 (),. ('.QK E0? +,G)( 8'1( (0 D00Q <03 -0DD*-(,01. 0< 70,1(. 8)*3* ()* >',DE 730<,( ,.()* .'+*@ Y03 *O'+7D*K -'1 E0? <,1> ' -0DD*-(,01 0< 70,1(. 8)*3* <03 *'-) 70,1( ()* >',DE 730<,( ,.T%#$W X)'( 'F0?( T%R$W

42

Systems 42



! #$%# &'()*+'(,-. /,.,01 2304*-( 5 &!2

61 7'3(1*3.),7 8,() ()* 9(') :('(* ;<<,-* 0< =>?-'(,01!"#$%&$' )%'$* +,$ -*$.+"/$ -0110%& 2++*"3)+"0%450%-011$*#".647,.*$26"8$ 9:; <%=0*+$' 6"#$%&$:

?:@? 2*( :,((*3. A*B,.,(*> Z S*'-)*3 U0(*.

! #$%$&'( )*+$,-./*+0*1 2/-3

"#$%&'(H X),D* (),. +0>?D* )'. ?.*> ()* [2*( :,((*3\ -01(*O( (0 *D,-,( ,+703('1( +'()*+'(,-.

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

• 6>*1(,<E,1G B'3,'FD*. ,1 ()* .,(?'(,01 '1> .*D*-(,1G ()0.* ()'( 3*73*.*1( *..*1(,'D <*'(?3*.

• Y03+?D'(,1G ' +0>*D FE -3*'(,1G '1> .*D*-(,1G G*0+*(3,-K G3'7),-'DK ('F?D'3K 'DG*F3',- 03.('(,.(,-'D 3*73*.*1('(,01. ()'( >*.-3,F* 3*D'(,01.),7. F*(8**1 ()* B'3,'FD*.

• V1'DE],1G '1> 7*3<03+,1G 07*3'(,01. 01 ()*.* 3*D'(,01.),7. (0 >3'8 -01-D?.,01.• 61(*373*(,1G ()* 3*.?D(. 0< ()* +'()*+'(,-. ,1 (*3+. 0< ()* 03,G,1'D .,(?'(,01

• /'D,>'(,1G ()* -01-D?.,01. FE -0+7'3,1G ()*+ 8,() ()* .,(?'(,01K '1> ()*1 *,()*3 ,+730B,1G()* +0>*DK 03

• A*703(,1G 01 ()* -01-D?.,01. '1> ()* 3*'.01,1G F*),1> ()*+

S),. B*3.,01 0< [2*( :,((*3.\ ,1(30>?-*. ()* I?'1(,(E (0 F* +'O,+,]*>N (,'<0. @ 230<,( ,. >*(*3+,1*>FE 3*B*1?* +,1?. -0.(.K '1> ,1 (),. -'.* ,. ' <?1-(,01 0< ()* 1?+F*3 0< -'(. '1> >0G. F*,1G F0'3>*>@X),D* 'DD 70,1(. ,1 ()* <*'.,FD* 3*G,01 3*73*.*1( -0+F,1'(,01. 0< -'(. '1> >0G. ()'( .'(,.<E 'DD 0< ()*-01.(3',1(.K ()* 730<,( B'3,*. <30+ 70,1( (0 70,1(@ 61(?,(,B*DEK 70,1(. <'3()*3 <30+ ()* 03,G,1 80?D>3*73*.*1( ),G)*3 730<,(.K .?GG*.(,1G ()'( 70,1(. 'D01G ()* F0?1>'3E D,1*. 80?D> F* '+01G ()* +0.(730<,('FD* -0+F,1'(,01.@ X)*1 .(?>*1(. (?31 ()*,3 '((*1(,01 (08'3>. <,1>,1G ' .*( 0< 70,1(. <03

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`@ 6( ,. 10( 1*-*..'3E <03 .(?>*1(. (0 <?DDE >*B*D07 (),.()*03EN8)'(*B*3 '3G?+*1(. ()*E ?.* (0 4?.(,<E ()*,3 3*-0++*1>'(,01 <03 ()* 1?+F*3 0< -'(. '1>>0G. C'3D0. '1> CD'3,(' -'1 F0'3> -'1 '1> .)0?D> F* -3,(,I?*> FE ()*,3 7**3. <03 '773073,'(*1*..@

)&$( +,-./-$/' 0&1#'H

2-,3(4-,51-6 "$-1,51( +,-./-$/ 78 2&/(6 95,3 4-,3(4-,51': &'()*+'(,-'DDE 730<,-,*1(.(?>*1(. -'1 '77DE ()* +'()*+'(,-. ()*E Q108 (0 .0DB* 730FD*+. '3,.,1G ,1 *B*3E>'E D,<*K .0-,*(EK'1> ()* 803Q7D'-*@ &'()*+'(,-'DDE 730<,-,*1( .(?>*1(. 8)0 -'1 '77DE 8)'( ()*E Q108 '3*-0+<03('FD* +'Q,1G '..?+7(,01. '1> '7730O,+'(,01. (0 .,+7D,<E ' -0+7D,-'(*> .,(?'(,01K 3*'D,],1G



! #$%# &'()*+'(,-. /,.,01 2304*-( 5 &!2

61 7'3(1*3.),7 8,() ()* 9(') :('(* ;<<,-* 0< =>?-'(,01!"#$%&$' )%'$* +,$ -*$.+"/$ -0110%& 2++*"3)+"0%450%-011$*#".647,.*$26"8$ 9:; <%=0*+$' 6"#$%&$:

()'( ()*.* +'E 1**> 3*B,.,01 D'(*3@ S)*E '3* 'FD* (0 ,>*1(,<E ,+703('1( I?'1(,(,*. ,1 ' 73'-(,-'D.,(?'(,01 '1> +'7 ()*,3 3*D'(,01.),7. ?.,1G .?-) (00D. '. >,'G3'+.K G3'7).K '1> <03+?D'.@ S)*E -'1'1'DE]* ()0.* 3*D'(,01.),7. +'()*+'(,-'DDE (0 >3'8 -01-D?.,01.@ S)*E 30?(,1*DE ,1(*373*( ()*,3+'()*+'(,-'D 3*.?D(. ,1 ()* -01(*O( 0< ()* .,(?'(,01 '1> 3*<D*-( 01 8)*()*3 ()* 3*.?D(. +'Q* .*1.*K70..,FDE ,+730B,1G ()* +0>*D ,< ,( )'. 10( .*3B*> ()* 7?370.*@

;(6-,(/ +,-./-$/'8 <:=:>? <:=:@? A:+BB:>-? A:)BC:@? A:)BC:D? A:;BE:>F? A:;BE:>@? 0:E0:7? 0:E0:G?0:EB:>H? 0:EB:G? A:;BE:G? A:;BE:I

J-#.13 KL3&6( )6-''M8

A*B,*8 ()* [2*( :,((*3.\ .-*1'3,0 >081 (0 ()* I?*.(,01 [X)'( -0+F,1'(,01 0< -'(. '1> >0G. >0 E0?(),1Q 8,DD +'Q* ()* +0.( +01*EW\ a,G)D,G)( ()* 1*8 ,..?*. ,1 (),. B*3.,01 0< [2*( :,((*3.\N'D()0?G) ()*E -)'3G* ' >',DE F0'3>,1G <**K ()* 730<,( ()* (8,1. *'31 *'-) >'E ,. >*(*3+,1*> 10( 01DEFE ()* >',DE 3*B*1?*K F?( 'D.0 FE ()* '>>,(,01'D -0.(. ,1-?33*> *'-) >'E <03 <00>K D,((*3K '1> 0()*3.?77D,*.@ b,B* .(?>*1(. ' <*8 +,1?(*. (0 803Q ,1>*7*1>*1(DE (0 >*(*3+,1* 70..,FD* .(3'(*G,*. <03+'O,+,],1G ()* 730<,(@ V<(*3 ' <*8 +,1?(*.K )'B* .(?>*1(. -0+7'3* ()*,3 .(3'(*G,*. 8,() 0()*3. '1>()*1 803Q 8,() ' 7'3(1*3 (0 -01.,>*3 )08 (0 ,+730B* ()*,3 .(3'(*G,*.@ V( (),. 70,1(K ,1(30>?-* ()*

[S0 G*( .('3(*> @ @ @\ ,>*' <30+ ()* D'.( 7'3'G3'7) 0< ()* )'1>0?(K '1> 70,1( 0?( ()* D'1G?'G* ,1 ()*.*-01> (0 D'.( 7'3'G3'7) 'F0?( +'Q,1G ' -01B,1-,1G '3G?+*1(@

BN%6&$( K+4-66 O$&#%M8

6( ,. '1(,-,7'(*> ()'( +'1E .(?>*1(. 8,DD F*G,1 8,() ' G?*.. '1> -)*-Q .(3'(*GEH 7,-Q ' 7'3(,-?D'3-0+F,1'(,01 0< >0G. '1> -'(. '. 3*73*.*1(*> FE ()* 70,1(. ,1 ()* <*'.,FD* 3*G,01 '1> ()*1 -'D-?D'(*()* 730<,(@ :(?>*1(. 8,DD 730F'FDE -0+* (0 3*'D,]* ()'( 70,1(. 01 ()* F0?1>'3E D,1*. 730>?-* ),G)*3730<,(. ()'1 70,1(. ,1 ()* ,1(*3,03@ V. .(?>*1(. <,1> .*(. 0< 70,1(. 8,() ()* .'+* 730<,( _'. .?GG*.(*>FE ()* [),1(\ '( ()* *1> 0< ()* ('.Q` ()*E 8,DD 10(,-* ()'( ()*.* 70,1(. <03+ ' D,1* '1> ()'( .*(. 0<70,1(. <03 >,<<*3*1( 730<,(. <03+ >,<<*3*1( D,1*. ()'( '3* 'DD 7'3'DD*D (0 *'-) 0()*3@ S)*E +'E.7*-?D'(* '. (0 8'E. (0 ?.* (),. .*( 0< 7'3'DD*D D,1*. (0 +'O,+,]* ()* 730<,(@

X'(-) <03 .(?>*1(. 8)0 -'D-?D'(* >',DE 3*B*1?* ,1.(*'> 0< >',DE 730<,(K 7'3(,-?D'3DE 8)*1 83,(,1G[730<,( *I?'(,01.\ <03 ' 730<,( 0< T%#$ 03 T#L$@ :?GG*.( ()'( ()*E 7D0( .*B*3'D [730<,( D,1*.\ '1>-01.,>*3 8)'( ,. )'77*1,1G (0 ()* D,1*. '. ()* 730<,( ,1-3*'.*.@ c00Q <03 .(?>*1(. 8)0 -'1 '3(,-?D'(* 8)E 70,1(. 01 ()* F0?1>'3E D,1*. E,*D> ),G)*3 730<,(.@ VD.0D00Q <03 .(?>*1(. 8)0 )'B* .?--*..<?DDE 7D0((*> .0+* 730<,( D,1*.K '1> 7*3)'7. )'B* '1 '3G?+*1(F'.*> 01 ()* 7'3'DD*D 730<,( D,1*. '. (0 8)E ()* +'O,+?+ 730<,( 8,DD 0--?3 '( ' B*3(*O 70,1(@

C5'1#'' KL3&6( )6-''M8

S)* ,1(*1( 0< (),. D*..01 ,. (0 .?3<'-* ,1(?,(,B* 8'E. (0 (),1Q 'F0?( ()* +'O,+?+ 730<,(K '1> 10(1*-*..'3,DE (0 <03+'D,]* ()* ,>*' ()'( ()* +'O,+?+ 730<,( 0--?3. '( 01* 0< ()* B*3(*O 70,1(.@ :)'3*

.(3'(*G,*. ()'( )'B* .?3<'-*> 8),-) 70,1( .(?>*1(. (08'3>. ()* +'O,+?+ 730<,( 0--?33,1G 'D01G01* 0< ()* F0?1>'3E D,1*. 03 '( ' B*3(*O@ VD.0 >,.-?.. )08 ()* [730<,( *I?'(,01.\ '3* D,1*. ()'( D,*8,(),1 ()* <*'.,FD* 3*G,01 '1> ()'( *B*3E 70,1( 01 ()* [730<,( D,1*\ 3*73*.*1(. ' -0+F,1'(,01 0< -'(.'1> >0G. <03 8),-) ()* 730<,( ,. ()* .'+* '. '1E 0()*3 -0+F,1'(,01 3*73*.*1(*> FE 0()*3 70,1(. 01()* .'+* D,1*@





Name: Systems 2.12




Ready, Set, Go!

Ready

Topic: Solve exponential equations

Find the value of x for each situation.

1. 2. 3. 4.

Set

Topic: Create and solve two variable inequalities

5. Jane is buying fruit salad and potato salad for a picnic. Fruit salad costs $2.00 per pound and

potato salad costs $4.00 per pound. Jane needs to buy at least 6 pounds of salads and she doesn’t

want to spend more than $20. Write and graph a system of linear inequalities. Also, list 2 possible

combinations of salad Jane could buy.

Let x = pounds of fruit salad and

y = pounds of potato salad.

Go

Topic: Find the solution region of the following systems of

inequalities.

6. Write the system of inequalities that is represented in thegraph to the right.

t

t

l

Pounds of Fruit Salad

10

10


-5

-5

5

5

43

Systems 43



Name: Systems 2.12




Graph each set of inequalities and determine the solution region.

7. { 8. {

9. 10.


Exponents http://patrickjmt.com/exponents-intro-to-evaluating-a-few-truefalse-questions/

Rules for exponents http://patrickjmt.com/basic-exponent-properties/

Solving a system of inequalities http://www.khanacademy.org/math/algebra/ck12-algebra-1/v/systems-of-linear-inequalities

44

Systems 44

http://patrickjmt.com/exponents-intro-to-evaluating-a-few-truefalse-questions/



http://patrickjmt.com/basic-exponent-properties/



http://www.khanacademy.org/math/algebra/ck12-algebra-1/v/systems-of-linear-inequalities








! #$%# &'()*+'(,-. /,.,01 2304*-( 5 &! 2

61 7'3(1*3.),7 8,() ()* 9(') :('(* ;<<,-* 0< =>?-'(,01

!"#$%&$' )%'$* +,$ -*$.+"/$ -0110%& 2++*"3)+"0%450%-011$*#".647,.*$26"8$ 9:; <%=0*+$' 6"#$%&$:

#@%AB C0 &'3D*( 8,() &'(3,-*. ! #$%&'&() *+',-./0+'&+1 20.3

E'3F0. F*'31*> 'G0?( +'(3,-*. 8)*1 =FH,3'I ()* +'1'J*3 0< ()*

.-)00F -'<*(*3,'I 8'. '.D*> (0 .?G.(,(?(* (*'-) >?3,1J 01* 0< ()*F'.( >'K. 0< .-)00F G*<03* .?++*3 H'-'(,01@ L08 ()'( )* )'.

803D*> 0?( ' .(3'(*JK <03 .0FH,1J .K.(*+. 0< *M?'(,01. GK

*F,+,1'(,01 0< H'3,'GF*.I )* ,. 801>*3,1J ,< +'(3,-*. -'1 )*F7

),+ D**7 (3'-D 0< ),. 803D@

E'3F0. ,. 3*-01.,>*3,1J ()* <0FF08,1J .-*1'3,0 <30+ N:)077,1J

<03 E'(. '1> O0J.PI 8),F* (3K,1J (0 3*-03> ),. (),1D,1J ?.,1J

+'(3,-*.@

4+, 5,,3 60-%$. 78-9:0.,' ; '$1 %,0.:,. 0+' ; 90/ <-8.:,. ($- =>?@AA ($- 6%0-&/0 /$ 8., 5:&%,

70B7,-&+1 /:, 7,/.@ C0/,- &+ /:, .8BB,- :, 78-9:0.,' D 0''&/&$+0% '$1 %,0.:,. 0+' E 90/<-8.:,. ($- =FG@AA@ H:0/ &. /:, 7-&9, $( ,09: &/,BI

E'3F0. 3*'F,Q*. ()'( )* -'1 3*73*.*1( (),. .-*1'3,0 ?.,1J ()* <0FF08,1J +'(3,RS

leashes brushes total

purchase 1

purchase 2

6 6 $45.00

3 2 $19.00

! " # $

% & B* 'F.0 3*'F,Q*. ()'( )* -'1 3*73*.*1( ()* -0.( 0< *'-) ,(*+ 8,() ' +'(3,R ()'( F00D. F,D* (),.S

leashes brushes total

purchase 1

purchase 2

1 0 $4.00

0 1 $3.50

! " # $

% & :0I 108 )* ,. (3K,1J (0 <,1> ' .*M?*1-* 0< +'(3,-*. ()'( -'1 <,FF ,1 ()* J'7. G*(8**1 ()* <,3.( +'(3,R

'1> ()* F'.(@ B* D108. <30+ ),. 73*H,0?. 803D 8,() .0FH,1J .K.(*+. 0< *M?'(,01. ()'( )* -'1 >0

'1K 0< ()* <0FF08,1J +'1,7?F'(,01. 8,() *M?'(,01.T'1> )* 3*'F,Q*. ()'( *'-) 0< ()*.*

+'1,7?F'(,01. 80?F> J,H* ),+ ' 1*8 308 0< 1?+G*3. ,1 ' -033*.701>,1J +'(3,R@

• U*7F'-* '1 *M?'(,01 ,1 ()* .K.(*+ 8,() ' -01.('1( +?F(,7F* 0< ()'( *M?'(,01

• U*7F'-* '1 *M?'(,01 ,1 ()* .K.(*+ 8,() ()* .?+ 03 >,<<*3*1-* 0< ()* (80 *M?'(,01.• U*7F'-* '1 *M?'(,01 8,() ()* .?+ 0< ()'( *M?'(,01 '1> ' +?F(,7F* 0< ()* 0()*3

! > ; ? > @ @ @ : A 6 " #

8 * : # 0 1 B = , 0 + 0 & B + 0 1 1 C , D B

45

Systems 45



! #$%# &'()*+'(,-. /,.,01 2304*-( 5 &! 2

61 7'3(1*3.),7 8,() ()* 9(') :('(* ;<<,-* 0< =>?-'(,01

!"#$%&$' )%'$* +,$ -*$.+"/$ -0110%& 2++*"3)+"0%450%-011$*#".647,.*$26"8$ 9:; <%=0*+$' 6"#$%&$:

%@ B*F7 E'3F0. <,1> ' .*M?*1-* 0< +'(3,-*. ()'( .('3(. 8,() ()* +'(3,R ()'( 3*73*.*1(. ()* 03,J,1'F

7?3-)'.*.I '1> *1>. 8,() ()* +'(3,R ()'( 3*73*.*1(. 7?3-)'.,1J 01* F*'.) 03 7?3-)'.,1J 01*

G3?.)@ V03 *'-) +'(3,R ,1 K0?3 .*M?*1-*I 83,(* 0?( ()* 4?.(,<,-'(,01 ()'( 'FF08. K0? (0 83,(* ()'(

+'(3,R G'.*> 01 ()* ()3** +'1,7?F'(,01. 8* -'1 7*3<03+ 01 ()* *M?'(,01. ,1 ' .K.(*+@ V03

*R'+7F*I ()* <0FF08,1J +'(3,R (3'1.<03+'(,01 -'1 G* 4?.(,<,*> GK 83,(,1J N6 3*7F'-*> ()* <,3.(

308 0< ()* +'(3,R GK +?F(,7FK,1J ()* <,3.( 308 GK 16@P

6 6 45.00

3 2 19.00

! " # $

% & ' 1 1 7.50

3 2 19.00

! " # $

% &

#@ V,1> '1> 4?.(,<K ' .*M?*1-* 0< +'(3,-*. ()'( -0?F> G* ?.*> (0 .0FH* ()* <0FF08,1J .-*1'3,0@

4+, 5,,3 60-%$. /-&,' $8/ 9:,07,- <-0+'. $( 90/ 0+' '$1 ($$'@ 4+ J$+'0) :, 78-9:0.,' D

.B0%% <01. $( 90/ ($$' 0+' ? .B0%% <01. $( '$1 ($$' ($- =EE@K?@ L,908., :, 5,+/ /:-$81:

/:, .B0%% <01. M8&/, M8&93%)N :, :0' /$ -,/8-+ /$ /:, ./$-, $+ 2:8-.'0) /$ <8) E B$-, .B0%%

<01. $( 90/ ($$' 0+' D B$-, .B0%% <01. $( '$1 ($$'N 5:&9: 9$./ :&B =F>@E?@ L0.,' $+ /:&.

&+($-B0/&$+N 90+ )$8 (&18-, $8/ /:, 7-&9, $( ,09: <01 $( /:, 9:,07,- 90/ 0+' '$1 ($$'I

46

Systems 46



! #$%# &'()*+'(,-. /,.,01 2304*-( 5 &!2

61 7'3(1*3.),7 8,() ()* 9(') :('(* ;<<,-* 0< =>?-'(,01!"#$%&$' )%'$* +,$ -*$.+"/$ -0110%& 2++*"3)+"0%450%-011$*#".647,.*$26"8$ 9:; <%=0*+$' 6"#$%&$:

#@%AB C0 &'3D*( 8,() &'(3,-*. W C*'-)*3 L0(*. ! #$%&'&() *+',-./0+'&+1 20.3

"#$%&'(S C)* 7?370.* 0< (),. ('.D ,. (0 ,1(30>?-* +'(3,-*. '. ' 8'K 0< D**7,1J (3'-D 0< ()* .(*7.

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

)&$( +,-./-$/' 0&1#'S

23 4&.&$'5 :0FH* .K.(*+. 0< F,1*'3 *M?'(,01. ?.,1J +'(3,-*.@

6(7-,(/ +,-./-$/'5 896:;9<

=-#.1> ?@>&7( )7-''A5

\*<03* >,.(3,G?(,1J ()* ('.D )'1>0?( (0 .(?>*1(.I 3*+,1> ()*+ 0< ()*,3 803D 01 ()* ('.D. N:)077,1J<03 E'(. '1> O0J.P '1> NE'1 ]0? ^*( (0 ()* 20,1(I C00_P 8)*3* ()*K >*H*F07*> ' .(3'(*JK <03 .0FH,1J' .K.(*+ 0< F,1*'3 *M?'(,01. GK *F,+,1'(,01 0< H'3,'GF*.@ C)*,3 .(3'(*JK ,1H0FH*> 83,(,1J *M?,H'F*1(.K.(*+. 0< *M?'(,01. GK +0>,<K,1J 01* 03 +03* 0< ()* *M?'(,01. ,1 ()* .K.(*+@

U*H,*8 ()*,3 803D 01 ()*.* 73*H,0?. (80 ('.D. GK '.D,1J .(?>*1(. (0 3*J*1*3'(* ()* F,.( 0< (),1J.

()*K -'1 >0 (0 ()* *M?'(,01. ,1 ' .K.(*+ (0 0G(',1 '1 *M?,H'F*1( .K.(*+@ V03 *R'+7F*I N]0? -'13*7F'-* '1 *M?'(,01 ,1 ' .K.(*+ 8,() ' -01.('1( +?F(,7F* 0< ()'( *M?'(,01@P [)'( *F.* -'1 K0? >0_

`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a. F0-'(,01 ,1 ()* +'(3,R 8,FF (*FF ()* '+0?1( 0< ' 7'3(,-?F'3,(*+ '-M?,3*> >?3,1J ' 7'3(,-?F'3 7?3-)'.*@ `F.0 70,1( 0?( ()'( ()* +'(3,-*. ?.*> (0 .0FH* .K.(*+.0< *M?'(,01. GK 308 3*>?-(,01 '3* 081B,+/,' 8,() ' -0F?+1 ()'( 3*73*.*1(. ()* .?+ 0< ()* F,1*'3-0+G,1'(,01 0< (*3+. ,1 ()* *M?'(,01.@ 61 (),. -'.*I ()* <,1'F -0F?+1 ,1 ()* +'(3,R 8,FF 3*73*.*1( ()*

<,1'F 7?3-)'.* 73,-*@ U*'> '1> >,.-?.. ()* ('.D ?1(,F .(?>*1(. '3* 3*'>K (0 803D 01 M?*.(,01 %@

bL0(*S B'H* .(?>*1(. 803D 01 M?*.(,01 %I ()*1 >,.-?.. ,(I G*<03* )'H,1J ()*+ 803D 01 M?*.(,01 #@c



! #$%# &'()*+'(,-. /,.,01 2304*-( 5 &!2

61 7'3(1*3.),7 8,() ()* 9(') :('(* ;<<,-* 0< =>?-'(,01!"#$%&$' )%'$* +,$ -*$.+"/$ -0110%& 2++*"3)+"0%450%-011$*#".647,.*$26"8$ 9:; <%=0*+$' 6"#$%&$:

:B%7&$( ?+C-77 D$&#%A5 E#(',F&. G

[),F* ()*3* ,. '1 *<<,-,*1( 8'K (0 '7730'-) 308 3*>?-(,01 0< +'(3,-*. X.** ()* 10(*. ,1 ()* O,.-?...*-(,01ZI ,( ,. 10( 1*-*..'3K (0 73*.. <03 '1 *<<,-,*1( .(3'(*JK '( (),. 70,1( ,1 (,+*@ U'()*3I ,( ,.,+703('1( <03 .(?>*1(. (0 3*-0J1,Q* ()'( ()*K '3* >0,1J ()* .'+* '3,()+*(,- 803D ()*K )'H* >01*73*H,0?.FK 8)*1 .0FH,1J .K.(*+. 0< *M?'(,01.I G?( 108 ()*K '3* 3*-03>,1J ()* 3*.?F(. 0< ()'( 803D,1 ()* <03+ 0< ' +'(3,R@ C)* *R'+7F* ,1 730GF*+ % ,. ,1(*1>*> (0 G* ' ),1( ()'( ,( ,. 70..,GF*I '1>

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

6 6 45.00

3 2 19.00

" #

% & ' 6 6 45.00

6 4 38.00

" #

% & ' 6 6 45.00

0 2 7.00

" #

% & ' 6 6 45.00

0 1 3.50

" #

% & 61 ()*,3 73*H,0?. 803D 8,() .0FH,1J .K.(*+.I 01-* .(?>*1(. <0?1> ()* H'F?* 0< 01* 0< ()* H'3,'GF*.,1 ()* .K.(*+I ()*K 80?F> .?G.(,(?(* ()'( H'F?* ,1(0 ()* 0()*3 *M?'(,01 (0 <,1> ()* H'F?* 0< ()* 0()*3H'3,'GF*@ C)*K +'K 801>*3 )08 (0 3*73*.*1( (),. 803D 8,() +'(3,-*.@ 61 ()* 'G0H* *R'+7F*I 8*108 D108 ()* H'F?* 0< % G3?.) ,. dA@e$@ C0 <,1> ()* H'F?* 0< % F*'.)I 8* -0?F> +?F(,7FK ()* G0((0+308 0< ()* +'(3,R GK fY '1> '>> ()* 3*.?F( (0 ()* (07 308 (0 *F,+,1'(* G3?.)*. <30+ ()* <,3.(7?3-)'.*@ [* -'1 ()*1 >,H,>*> ()* (07 308 GK YT()* 1?+G*3 0< G3?.)*. 7?3-)'.*> <03 d#g@$$T(0J*( ()* -0.( 0< ' .,1JF* G3?.)@

6 6 45.00

0 1 3.50

" #

% & ' 6 0 24.00

0 1 3.50

" #

% & ' 1 0 4.00

0 1 3.50

" #

% &

\*-'?.* *'-) +'(3,R 1**>. (0 G* ' # " A +'(3,RI .(?>*1(. 8,FF 1**> (0 3*-03> (80 7?3-)'.*. X*@J@I(80 308.Z ,1 *'-) +'(3,RI '1> ()*K .)0?F> G* 'GF* (0 4?.(,<K ()* '+0?1(. 7?3-)'.*> '1> -0.( 0< *'-)7?3-)'.* 3*73*.*1(*> GK *'-) 308@ h**7 (),. -011*-(,01 '( ()* <03*<301( 0< .(?>*1(.a 803D GK'.D,1J ()*+ (0 >*.-3,G* 8)'( *'-) 308 0< ()*,3 +'(3,-*. 3*73*.*1(. ,1 (*3+. 0< 7?3-)'.*.@

:(?>*1(. +,J)( '7730'-) (),. 803D ,1 ' H'3,*(K 0< 8'K.I '1> ()*K +'K 10( 3*-03> ()*,3 803D'773073,'(*FK ?.,1J +'(3,-*.@ [03D (08'3>. -01.*1.?. 01 8)'( ()*,3 803D .)0?F> F00D F,D*T,1-F?>,1J ' 83,((*1 >*.-3,7(,01 0< 8)'( ()*K >,> (0 (3'1.<03+ 01* +'(3,R ,1(0 '10()*3@

['(-) <03 J30?7. ()'( ?.* ()* .(*7 ),1(*> '( ,1 M?*.(,01 % '. ()*,3 <,3.( .(*7 ,1 ()*,3 .*M?*1-* 0<+'(3,-*.@ L0(* )08 ()*K ?.* ()* %. ,1 ()* <,3.( 308T3*73*.*1(,1J % F*'.) '1> % G3?.)@ O0 ()*K ?.*()*.* %. (0 *F,+,1'(* *,()*3 ()* 1?+G*3 0< F*'.)*. 03 1?+G*3 0< G3?.)*. ,1 ()* .*-01> 7?3-)'.*

X,@*@I J*( $. ,1 *,()*3 ()* <,3.( 03 .*-01> -0F?+1 0< ()* .*-01> 308 0< ()* +'(3,RZ_ [)*1 .(?>*1(.)'H* N7F'K*> '30?1>P 8,() +'(3,R 10('(,01 <03 ' .?<<,-,*1( '+0?1( 0< (,+* (0 )'H* '--*.. (0 ' 8)0F*-F'.. >,.-?..,01T'1> 7*3)'7. )'H* *1-0?1(*3*> ' 1?+G*3 0< ,..?*. ()'( -'1 G* 3*.0FH*> >?3,1J '8)0F* -F'.. >,.-?..,01T<**F <3** (0 G*J,1 >,.-?..,1J M?*.(,01 % '. ' 8)0F* -F'..@



! #$%# &'()*+'(,-. /,.,01 2304*-( 5 &!2

61 7'3(1*3.),7 8,() ()* 9(') :('(* ;<<,-* 0< =>?-'(,01!"#$%&$' )%'$* +,$ -*$.+"/$ -0110%& 2++*"3)+"0%450%-011$*#".647,.*$26"8$ 9:; <%=0*+$' 6"#$%&$:

HF'1#'' ?@>&7( )7-''A5 E#(',F&. G

:)'3* .?--*..<?F +'(3,R (3'1.<03+'(,01. '1> '773073,'(* +'(3,R 10('(,01 ()'( .(?>*1(. +'K )'H*J*1*3'(*>I *H*1 ,< ()*K >0 10( F*'> (0 ' -0+7F*(* .(3'(*JK@ &'D* .?3* .(?>*1(. >*.-3,G* 8)'( ()*K'3* >0,1J ,1 (*3+. 0< ()* .)077,1J -01(*R(@ 6< 70..,GF*I )'H* ' .(?>*1( .)'3* )08 ()*K -'1 ?.* ' % ,1' 7'3(,-?F'3 -0F?+1 (0 J*( $ ,1 ()'( .'+* -0F?+1 ,1 ()* 0()*3 308@

B*F7 .(?>*1(. (3'1.,(,01 <30+ ()* F'1J?'J* 0< .0FH,1J .K.(*+. GK *F,+,1'(,01 (0 ()* F'1J?'J* 0<.0FH,1J .K.(*+. 8,() +'(3,-*.@ C)'( ,.I ()* <0FF08,1J 3?F*. <03 .0FH,1J .K.(*+. 0< *M?'(,01.S

• U*7F'-* '1 *M?'(,01 ,1 ()* .K.(*+ 8,() ' -01.('1( +?F(,7F* 0< ()'( *M?'(,01

• U*7F'-* '1 *M?'(,01 ,1 ()* .K.(*+ 8,() ()* .?+ 03 >,<<*3*1-* 0< ()* (80 *M?'(,01.

• U*7F'-* '1 *M?'(,01 8,() ()* .?+ 0< ()'( *M?'(,01 '1> ' +?F(,7F* 0< ()* 0()*3

-'1 G* 3*H,.*> (0S

• U*7F'-* ' 308 ,1 ' +'(3,R 8,() ' -01.('1( +?F(,7F* 0< ()'( 308

• U*7F'-* ' 308 ,1 ' +'(3,R 8,() ()* .?+ 03 >,<<*3*1-* 0< ()'( 308 '1> '10()*3 308 0< ()*+'(3,R

• U*7F'-* ' 308 ,1 ' +'(3,R 8,() ()* .?+ 0< ()'( 308 '1> ' -01.('1( +?F(,7F* 0< '10()*3 3080< ()* +'(3,R

XL0(* ()'( ()* +,>>F* .('(*+*1( ,. ' .7*-,'F -'.* 0< ()* G0((0+ .('(*+*1(I '1> ()*3*<03* ,.3*>?1>'1(I G?( .(,FF +'K G* )*F7<?F (0 .('(* *R7F,-,(FK@Z

`10()*3 *F*+*1('3K 308 07*3'(,01 ()'( +'K 10( )'H* -0+* ?7 *R7F,-,(FK ,1 (),. ('.D ,. ()* ,>*' ()'(K0? -'1 .8,(-) '1K (80 308. ,1 ' +'(3,R@ C),. ,. 4?.(,<,*> GK ()* <'-( ()'( ()* 03>*3 ,1 8),-) K0?83,(* ()* *M?'(,01. ,1 ' F,1*'3 .K.(*+ >0*. 10( +'((*3@ ]0? -'1 ,1(30>?-* (),. ,>*' 108I 03 'FF08 ,((0 '3,.* +03* 1'(?3'FFK ,1 ()* 1*R( ('.D@

B*F7 .(?>*1(. 3*-0J1,Q* ()'( ()* J0'F 0< 803D,1J 8,() ()* +'(3,-*. ,. (0 J*( %. ,1 ()* >,'J01'F 0< ()*703(,01 0< ()* +'(3,R 3*73*.*1(,1J ()* -0*<<,-,*1(. 0< ()* H'3,'GF* (*3+.I '1> $. ,1 ()* 3*+',1,1J

70.,(,01. 0< ()'( 703(,01 0< ()* +'(3,RI .0 ()'( ()* F'.( -0F?+1 0< ()* +'(3,R 8,FF J,H* ()* H'F?* 0< '>,<<*3*1( H'3,'GF* <03 *'-) 308 0< ()* +'(3,R@

:B%7&$( ?+C-77 D$&#%A5 E#(',F&. I

`FF08 .(?>*1(. '>>,(,01'F (,+* (0 803D 01 ()* .)077,1J .-*1'3,0 ,. M?*.(,01 #@ ['(-) <03 +03**<<,-,*1( .(3'(*J,*. (0 *+*3J*I G'.*> 01 ()* 73*H,0?. >,.-?..,01@

HF'1#'' ?@>&7( )7-''A5 E#(',F&. I

:)'3* '1> >,.-?.. .(3'(*J,*. ()'( ,FF?.(3'(* ()* <0FF08,1J .(*7. <03 '1 *<<,-,*1( '1> -01.,.(*1( 8'K (0308 3*>?-* ' +'(3,R@ B'H* .(?>*1(. .?++'3,Q* ()* >,.-?..,01 GK '3(,-?F'(,1J ,>*'. .,+,F'3 (0 ()*<0FF08,1J@ XL0(* ()'( ()*.* ,>*'. 8,FF G* <03+'F,Q*> ,1 ()* 1*R( ('.D@Z

C0 308 3*>?-* ' +'(3,RS

• 2*3<03+ *F*+*1('3K 308 07*3'(,01. (0 K,*F> ' i%i ,1 ()* <,3.( 308I <,3.( -0F?+1@

• E3*'(* Q*30. ,1 'FF 0< ()* 0()*3 308. 0< ()* <,3.( -0F?+1 GK '>>,1J ()* <,3.( 308 (,+*. '-01.('1( (0 *'-) 0()*3 308@



! #$%# &'()*+'(,-. /,.,01 2304*-( 5 &!2

61 7'3(1*3.),7 8,() ()* 9(') :('(* ;<<,-* 0< =>?-'(,01!"#$%&$' )%'$* +,$ -*$.+"/$ -0110%& 2++*"3)+"0%450%-011$*#".647,.*$26"8$ 9:; <%=0*+$' 6"#$%&$:

• 2*3<03+ *F*+*1('3K 308 07*3'(,01. (0 K,*F> ' i%i ,1 ()* .*-01> 308I .*-01> -0F?+1@

• E3*'(* Q*30. ,1 'FF 0< ()* 0()*3 308. 0< ()* .*-01> -0F?+1 GK '>>,1J ()* .*-01> 308 (,+*. '-01.('1( (0 *'-) 0()*3 308@

• 2*3<03+ *F*+*1('3K 308 07*3'(,01. (0 K,*F> ' i%i ,1 ()* (),3> 308I (),3> -0F?+1@

• E3*'(* Q*30. ,1 'FF 0< ()* 0()*3 308. 0< ()* (),3> -0F?+1 GK '>>,1J ()* (),3> 308 (,+*. '-01.('1( (0 *'-) 0()*3 308@

• E01(,1?* (),. 730-*.. ?1(,F ()* <,3.( + j + *1(3,*. <03+ ' .M?'3* +'(3,R 8,() %. ,1 ()*>,'J01'F '1> $. *H*3K8)*3* *F.*@

87FJ.(/ 6(-/KL +(,L D&5 +K',(C' I9GM4



Name: To Market with Matrices 2.13H



Ready, Set, Go!

Ready

Topic: Solving Systems by Substitution and Elimination

Solve each system of equations using any algebraic method.

1.3 x y 1

3 x 2 y 16

2. x 2 y 5

3 x 5 y 14

3.4 x 2 y 8

x 2 y 7

4.2 x 3 y 2

3 x

4 y

14

5. x 2 y 11

x 4

y 2

6.2 x y 0

5 x

3 y

1

2 0 1 2

. f l i c k r

. c o

/ p h o t o s / t o

y h j /

47

Systems 47






Set

Topic: Row reductions in Matrices

7. Create a matrix to match each step in the solving of the system of equations given. Also, write a

description of what happened to the equation and the matrix between steps.

System of Equations Description Matrix

Given System { [

|]

Step 1 { [

| ]

Step 2 { [ | ]

Step 3 { [ | ]

Step 4 { [ | ]

Step 5 { [ | ]

48

Systems 48






Go

Topic: Solving Systems of Equations by Graphing

Solve each system of equations by graphing.

8. y 3 x 3

y 3 x 3

9. y 4 x 1

y x 4

10. y 2 x 7

3 x y 8

11.4 x y 7

3 x 2 y 8


http://www.khanacademy.org/math/algebra/ck12-algebra-1/v/solving-linear-systems-by-substitution

http://patrickjmt.com/row-reducing-a-linear-system-of-equations/

http://www.khanacademy.org/math/algebra/systems-of-eq-and-ineq/v/graphings-systems-of-equations

49

Systems 49












! #$%# &'()*+'(,-. /,.,01 2304*-( 5 &! 2

61 7'3(1*3.),7 8,() ()* 9(') :('(* ;<<,-* 0< =>?-'(,01

!"#$%&$' )%'$* +,$ -*$.+"/$ -0110%& 2++*"3)+"0%450%-011$*#".647,.*$26"8$ 9:; <%=0*+$' 6"#$%&$:

#@%AB :0CD,1E :F.(*+. 8,() &'(3,-*. ! #$%&'(&) *+,)$-'%+,(+. /%-0

61 ()* ('.G HI0 &'3G*( 8,() &'(3,-*.J F0? >*D*C07*> ' .(3'(*EF <03

.0CD,1E .F.(*+. 0< C,1*'3 *K?'(,01. ?.,1E +'(3,-*.@ L1 *<<,-,*1( '1>-01.,.(*1( 8'F (0 -'33F 0?( (),. .(3'(*EF -'1 M* .?++'3,N*> '.

<0CC08.O

I0 308 3*>?-* ' +'(3,PO

• 2*3<03+ *C*+*1('3F 308 07*3'(,01. (0 F,*C> ' Q%Q ,1 ()* <,3.( 308R <,3.( -0C?+1@

• S3*'(* N*30. ,1 'CC 0< ()* 0()*3 308. 0< ()* <,3.( -0C?+1 MF '>>,1E ()* <,3.( 308 (,+*. '

-01.('1( (0 *'-) 0()*3 308@

• 2*3<03+ *C*+*1('3F 308 07*3'(,01. (0 F,*C> ' Q%Q ,1 ()* .*-01> 308R .*-01> -0C?+1@

• S3*'(* N*30. ,1 'CC 0< ()* 0()*3 308. 0< ()* .*-01> -0C?+1 MF '>>,1E ()* .*-01> 308 (,+*. '

-01.('1( (0 *'-) 0()*3 308@

• 2*3<03+ *C*+*1('3F 308 07*3'(,01. (0 F,*C> ' Q%Q ,1 ()* (),3> 308R (),3> -0C?+1@

• S3*'(* N*30. ,1 'CC 0< ()* 0()*3 308. 0< ()* (),3> -0C?+1 MF '>>,1E ()* (),3> 308 (,+*. '-01.('1( (0 *'-) 0()*3 308@

• S01(,1?* (),. 730-*.. ?1(,C ()* <,3.( + T + *1(3,*. <03+ ' .K?'3* +'(3,P 8,() %. ,1 ()*

>,'E01'C '1> $. *D*3F8)*3* *C.*@

23'-(,-* (),. .(3'(*EF MF -3*'(,1E ' .*K?*1-* 0< +'(3,-*. <03 *'-) 0< ()* <0CC08,1E ()'( M*E,1. 8,()

()* E,D*1 +'(3,P '1> *1>. 8,() ()* C*<( 703(,01 0< ()* +'(3,P U()* <,3.( + ! + *1(3,*.V ,1 308W

3*>?-*> <03+@ X3,(* ' >*.-3,7(,01 0< 8)'( F0? >,> (0 E*( <30+ 01* +'(3,P (0 '10()*3 ,1 *'-) .(*7 0<

F0?3 .*K?*1-* 0< +'(3,-*.@

%@ 2 4 0

3 5 !2" # $ %

& '

#@4 !2 2

1 3 11

" # $ %

& '

Y@

4 !2 1 3

2 1 !1 1

3 !1 2 7

" # $ $ $

% & ' ' '

50

Systems 50



! #$%# &'()*+'(,-. /,.,01 2304*-( 5 &! 2

61 7'3(1*3.),7 8,() ()* 9(') :('(* ;<<,-* 0< =>?-'(,01

!"#$%&$' )%'$* +,$ -*$.+"/$ -0110%& 2++*"3)+"0%450%-011$*#".647,.*$26"8$ 9:; <%=0*+$' 6"#$%&$:

A@ ='-) 0< ()* 'M0D* +'(3,-*. 3*73*.*1(. ' .F.(*+ 0< *K?'(,01.@ Z03 *'-) 730MC*+R 83,(* ()*

.F.(*+ 0< *K?'(,01. 3*73*.*1(*> MF ()* 03,E,1'C +'(3,P@ [*(*3+,1* ()* .0C?(,01 <03 *'-) .F.(*+

?.,1E ()* 308W3*>?-*> +'(3,P F0? 0M(',1*>R '1> ()*1 -)*-G ()* .0C?(,01. ,1 ()* 03,E,1'C .F.(*+@

\@ :0CD* ()* <0CC08,1E 730MC*+ MF ?.,1E ' +'(3,P (0 3*73*.*1( ()* .F.(*+ 0< *K?'(,01. >*.-3,M*> ,1

()* .-*1'3,0R '1> ()*1 -)'1E,1E ()* +'(3,P (0 308W3*>?-*> <03+ (0 0M(',1 ()* .0C?(,01@

/1$)) 23 4%$52-6 %+, 45%$('%6- 3$()+,- %$) 78$&1%-(+. -&1225 -8775()- %' '1) 9220-'2$): ;'%+98<- % +2')9220= '1$)) 7%&0%.)- 23 7)+&(5- %+, '>2 ?%$0)$- 32$ @A:BC: D%+ 98<- '>2+2')9220-= -(E 7%&0%.)- 23 7)+&(5- %+, 3(F) ?%$0)$- 32$ @GB:BC: H$%+ 98<- % +2')9220= '>2

7%&0%.)- 23 7)+&(5- %+, '>2 ?%$0)$- 32$ @I:JB: K2> ?8&1 ,2 )%&1 23 '1)-) '1$)) (')?- &2-'L

]@ S3*'(* ' C,1*'3 .F.(*+ ()'( ,. *,()*3 >*7*1>*1( UM0() *K?'(,01. ,1 ()* .F.(*+ 3*73*.*1( ()* .'+*

C,1*V 03 ,1-01.,.(*1( U()* *K?'(,01. ,1 ()* .F.(*+ 3*73*.*1( 101W,1(*3.*-(,1E C,1*.V@ X)'(

)'77*1. 8)*1 F0? (3F (0 308 3*>?-* ()* # ! Y +'(3,P ()'( 3*73*.*1(. (),. C,1*'3 .F.(*+ 0<

*K?'(,01.^

51

Systems 51



! #$%# &'()*+'(,-. /,.,01 2304*-( 5 &!2

61 7'3(1*3.),7 8,() ()* 9(') :('(* ;<<,-* 0< =>?-'(,01

!"#$%&$' )%'$* +,$ -*$.+"/$ -0110%& 2++*"3)+"0%450%-011$*#".647,.*$26"8$ 9:; <%=0*+$' 6"#$%&$:

#@%AB :0CD,1E :F.(*+. 8,() &'(3,-*._

I*'-)*3 `0(*. ! #$%&'(&) *+,)$-'%+,(+. /%-0

"#$%&'(O I)* 7?370.* 0< (),. ('.G ,. (0 73'-(,-* '1 *<<,-,*1(R 03E'1,N*> '1> -01.,.(*1( .(3'(*EF <03

(3'1.<03+,1E ' +'(3,P ,1(0 308W3*>?-*> <03+@ :(?>*1(. -01(,1?* (0 -011*-( ()* +'(3,P

3*73*.*1('(,01 (0 ()* .F.(*+ 0< C,1*'3 *K?'(,01. ,( 3*73*.*1(.R '1> (0 ()* .-*1'3,0. ()'( E,D* 3,.* (0

M0() 3*73*.*1('(,01.@ 61 K?*.(,01 ] .(?>*1(. -01.,>*3 )08 >*7*1>*1( .F.(*+. '1> ,1-01.,.(*1(

.F.(*+. -3*'(* ' 308 0< $. U10 %. ,1 ()* >,'E01'CV 8)*1 3*83,((*1 ,1 308W3*>?-*> <03+@

)&$( +,-./-$/' 0&1#'O

23 4&.&$'5 :0CD* .F.(*+. 0< C,1*'3 *K?'(,01. ?.,1E +'(3,-*.@

6(7-,(/ +,-./-$/'5 896:;9<

=-#.1> ?@>&7( )7-''A5

a*D,*8 ()* H*C*+*1('3F 308 07*3'(,01.J ()'( -'1 M* 7*3<03+*> 01 +'(3,-*. 8)*1 ?.*> (0

3*73*.*1( C,1*'3 .F.(*+. 0< *K?'(,01.O U()*.* *+*3E* <30+ ()* 803G 8,() +'(3,-*. ,1 ()* 73*D,0?.

('.G HI0 &'3G*( X,() &'(3,-*.JV

• a*7C'-* ' 308 ,1 ' +'(3,P 8,() ' -01.('1( +?C(,7C* 0< ()'( 308

• a*7C'-* ' 308 ,1 ' +'(3,P 8,() ()* .?+ 03 >,<<*3*1-* 0< ()'( 308 '1> '10()*3 308 0< ()*

+'(3,P

• a*7C'-* ' 308 ,1 ' +'(3,P 8,() ()* .?+ 0< ()'( 308 '1> ' -01.('1( +?C(,7C* 0< '10()*3 308

0< ()* +'(3,P

U`0(* ()'( ()* +,>>C* .('(*+*1( ,. ' .7*-,'C -'.* 0< ()* M0((0+ .('(*+*1(R '1> ()*3*<03* ,.3*>?1>'1(R M?( .(,CC +'F M* )*C7<?C (0 .('(* *P7C,-,(CF@V

L10()*3 *C*+*1('3F 308 07*3'(,01 ()'( +'F 10( )'D* -0+* ?7 *P7C,-,(CF ,1 ()* 73*D,0?. ('.G ,. ()*

,>*' ()'( F0? -'1 .8,(-) '1F (80 308. ,1 ' +'(3,P@ I),. ,. 4?.(,<,*> MF ()* <'-( ()'( ()* 03>*3 ,1

8),-) F0? 83,(* ()* *K?'(,01. ,1 ' C,1*'3 .F.(*+ >0*. 10( +'((*3@ I),. ,>*' M*-0+*. ?.*<?C 8)*1

.(?>*1(. '3* 803G,1E 01 ()* H*<<,-,*1( '1> -01.,.(*1(J .(*7. (0 308 3*>?-* ' +'(3,P <03+'CCF

,1(30>?-*> ,1 (),. ('.G@ I)'( ,.R .0+*(,+*. ()* *'.,*.( 8'F (0 E*( %. 01 ()* >,'E01'C 0< ()* +'(3,P ,.

(0 *P-)'1E* 308.@ b0? -'1 ,1(30>?-* (),. ,>*' >,3*-(CFR 03 C*( ,( '3,.* 1'(?3'CCF '. .(?>*1(. >,.-?..

>,<<*3*1( 8'F. (0 E*( % ,1 ()* (07 308R <,3.( -0C?+1 <03 730MC*+. % '1> #@

c*<03* .*((,1E .(?>*1(. (0 803G 01 ()* 730MC*+.R >,.-?.. ()* .(*7. 0< ()* H*<<,-,*1( '1> -01.,.(*1(J308 3*>?-(,01 .(3'(*EF E,D*1 ,1 ()* ('.G MF '.G,1E .(?>*1(. 8)F (),. .(3'(*EF 80?C> 'C8'F. 803G@

L.G .?-) K?*.(,01. '.R HX)F '3* ()* %. 01 ()* >,'E01'C )*C7<?C J '1> HX)F >0 F0? 8'1( (0 E*( $. ,1

01* -0C?+1 M*<03* F0? +0D* 01(0 ()* 1*P(^J



! #$%# &'()*+'(,-. /,.,01 2304*-( 5 &!2

61 7'3(1*3.),7 8,() ()* 9(') :('(* ;<<,-* 0< =>?-'(,01

!"#$%&$' )%'$* +,$ -*$.+"/$ -0110%& 2++*"3)+"0%450%-011$*#".647,.*$26"8$ 9:; <%=0*+$' 6"#$%&$:

:B%7&$( ?+C-77 D$&#%A5

:(?>*1(. .)0?C> .** (),. C,.( 0< .(*7. '. ' )*C7<?C .(3'(*EFR 3'()*3 ()'1 ()* 01CF 8'F (0 308 3*>?-* '

+'(3,P@ L. .(?>*1(. 803G 01 730MC*+. %WYR )'D* ()*+ -01.,>*3 8)F (),. .(3'(*EF 803G. ,1 +03*

>*(',CR '1> *P'+,1* 'C(*31'(,D* 8'F. <03 '--0+7C,.),1E ()* .(*7. C,.(*>R .?-) '. .8,(-),1E 308. 03

>,D,>,1E ' 308 MF ' -01.('1(@ B*C7 .(?>*1(. '3(,-?C'(* )08 ()* %. ,1 ()* >,'E01'C 70.,(,01. '3* ?.*>

(0 E*( ()* $. ,1 ()* 0()*3 308. 0< ()* .'+* -0C?+1 U*@E@R ?.,1E ()* 3?C* Ha*7C'-* ' 308 ,1 ' +'(3,P8,() ()* .?+ 0< ()'( 308 '1> ' -01.('1( +?C(,7C* 0< '10()*3 308 0< ()* +'(3,PJ 8)*3* ()* 308 M*,1E

3*7C'-*> ,. ()* 308 8)*3* F0? 8'1( (0 E*( ' $R ()* 0()*3 308 ,. ()* 308 ()'( )'. % ,1 ()* .'+*

-0C?+1R '1> ()* -01.('1( +?C(,7C* ,. ()* '>>,(,D* ,1D*3.* 0< ()* 1?+M*3 ,1 ()* 70.,(,01 8)*3* F0?

8'1( (0 E*( ()* $V@

d?*.(,01. Y '1> \ E,D* .(?>*1(. '1 07703(?1,(F (0 *P(*1> ()* 730-*.. <03 .0CD,1E C,1*'3 .F.(*+.

?.,1E +'(3,-*. (0 .,(?'(,01. ,1D0CD,1E Y *K?'(,01. ,1 Y ?1G1081.@ I),. 'CC08. .(?>*1(. (0

3*-01.,>*3 ()* ,>*'. M*),1> ()* 308 3*>?-(,01 .(3'(*EF '. ()*F *P7'1> ,( (0 +'(3,-*. 0< C'3E*3

>,+*1.,01.@

EF'1#'' ?@>&7( )7-''A5e,D* <**>M'-G (0 ,1>,D,>?'C. 03 E30?7.R '. 1**>*> >?3,1E ()*,3 803G 01 ()* ('.G@ L. F0? +01,(03

()*,3 803GR ,>*1(,<F '1F G*F ,..?*. 03 +,.-01-*7(,01. ()'( +'F M*1*<,( <30+ ' 8)0C* -C'..

>,.-?..,01@ f*F ,..?*. <03 ' 8)0C* -C'.. >,.-?..,01 +,E)( ,1-C?>* <,3+,1E ?7 8)F ()* .(*7. C,.(*>

803G 'CC 0< ()* (,+*R 8)F F0? 1**> (0 E*( $. ,1 'CC 0< ()* 101W>,'E01'C 70.,(,01. 0< ' -0C?+1 M*<03*

+0D,1E (0 803G,1E 01 '10()*3 -0C?+1R '1> 8)'( )'77*1. 8)*1 ()* +'(3,P 3*73*.*1(. ' >*7*1>*1(

03 ,1-01.,.(*1( .F.(*+@

87FG.(/ 6(-/HI +(,I D&5 +H',(C' J9KL4



Name: Solving Systems with Matrices 2.14H




Ready, Set, Go!

Ready

Topic: Solving systems of equations using matrices.

1. In an earlier assignment you worked the following

problem:

“ A theater wants to take in $2000 for a certain matinee. Children’s tickets cost $5 each and

adult tickets cost $10 each. If the theater has a maximum of 350 seats, write a system of

equations that can be solved to determine the number of both children and adult tickets the

theater can sell .”

Set up a matrix that goes with the situation described above.

Set

Assume that the matrices below represent linear systems of equations. Practice the strategy

you used for reducing a given matrix so that the left portion of the matrix (the 2 rows and

first 2 columns of entries) has ones on the diagonal. Write a description of what you did to

get from one matrix to another in each step of your sequence of matrices.

2. [ | ] [

|] [ |]

3. [ |] [

| ]

4. [ | ]

5.

[ |

]

2012 www.flickr.com/photos/dansmath

52

Systems 52



Name: Solving Systems with Matrices 2.14H

Go

Topic: Solving systems of equations

Solve the following systems of equations with a method of your choice.

6. x y 11

2 x y 19

7.8 x y 16

3 x y 5

8.4 x 9 y 9

x 3 y 6

9.7 x y 19

2 x 3 y 19


http://www.khanacademy.org/math/algebra/ck12-algebra-1/v/solving-linear-systems-by-graphing


53





http://www.khanacademy.org/math/algebra/ck12-algebra-1/v/solving-systems-of-equations-by-elimination

http://www.khanacademy.org/math/algebra/ck12-algebra-1/v/solving-systems-of-equations-by-elimination



Documents

Sec1 Mod2 Systems Tnh 71713