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QtXK!drs/slideMasters/slideMaster1.xml͎0n>+i!.ن>ժ*!Ǔ/ގ*ͤHǿ3)Փ簿&5WTB2Ӥ֠~܏ճ˲d9&mх9:ϔִף7᳧3šM'Q5I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!/drs/downrev.xmlDn0V8 *jVz[%16IWOXe=98IT+7|@#\jy{\oC͢4!t9jH؎LdG4xKC떧I2 vP=k(ms,OW!X1?~U!cK:e,rbaPK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!/drs/downrev.xmlPK @H$ >  |A system of linear equations consists of two or more linear equations. This section focuses on only two equations at a time. The solution of a system of linear equations in two variables is any ordered pair that solves both of the linear equations.DI }ep   -B0fBYt  ___PPT10 .+,ODH ' = @B D ' = @BA?%,( < +O%,( < +D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*G%(D' =-s6Bwipe(left)*<3<*GD' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*G}%(D' =-s6Bwipe(left)*<3<*G}D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*}%(D' =-s6Bwipe(left)*<3<*}+8+0+ +"9%   pq (      rP0e0eRectangle 2"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!ydrs/downrev.xmlDAO1F&fLIk1V 1**exnunƃ'ob;P;#\qM ;Rp 폮:5,K\,PAR_p,Ƒebghxrq)=h\^h*zvP}?%U"Sq'N`t1QP sogPK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!ydrs/downrev.xmlPK0$ > ___PPT9 Determine whether the given point is a solution of the following system. point: ( 3, 1) system: x  y =  4 and 2x + 10y = 4 Plug the values into the equations. First equation:  3  1 =  4 true Second equation: 2( 3) + 10(1) =  6 + 10 = 4 true Since the point ( 3, 1) produces a true statement in both equations, it is a solution.I = t$ " b tX " I $ "  &"&""&"  F$"$d  &   x0e0eRectangle 6 `   ZSolution of a System   Y  d0-B??Text Box 7=D<4___PPT9 yExample*( " d2   -B0fBYt( ___PPT10.+D' = @B D[' = @BA?%,( < +O%,( < +D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*I%(D' =-s6Bwipe(left)*<3<*ID' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*I[%(D' =-s6Bwipe(left)*<3<*I[D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*[%(D' =-s6Bwipe(left)*<3<*[D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*%(D' =-s6Bwipe(left)*<3<*D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*%(D' =-s6Bwipe(left)*<3<*D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<* %(D' =-s6Bwipe(left)*<3<* D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<* d%(D' =-s6Bwipe(left)*<3<* d+8+0+ +"%     (       r0e0eRectangle 2"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!\drs/downrev.xmlDN0HHܨC*u+*?46&؎lMy8q54`;w Gr5 VDL4vޑ#EϦXhp%헩,q@mJ}!e[G'lghx`dei8^hǖ*ؘ|?eٻMzR=DC~_Ujn<0}9n%DAr/sPK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!\drs/downrev.xmlPK`$ > ___PPT9 HDetermine whether the given point is a solution of the following system point: (4, 2) system: 2x  5y =  2 and 3x + 4y = 4 Plug the values into the equations First equation: 2(4)  5(2) = 8  10 =  2 true Second equation: 3(4) + 4(2) = 12 + 8 = 20 4 false Since the point (4, 2) produces a true statement in only one equation, it is NOT a solution. H : # ts t] tH#"&"  "& " "& &  "  &  "     & ]$"$u  &   x00e0eRectangle 6 `   ZSolution of a System   Y  d-B??Text Box 7=D<4___PPT9 yExample*( " d2   -B0fBYt( ___PPT10.+D' = @B D[' = @BA?%,( < +O%,( < +D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<* H%(D' =-s6Bwipe(left)*<3<* HD' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<* HX%(D' =-s6Bwipe(left)*<3<* HXD' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<* X%(D' =-s6Bwipe(left)*<3<* XD' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<* %(D' =-s6Bwipe(left)*<3<* D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<* %(D' =-s6Bwipe(left)*<3<* D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<* %(D' =-s6Bwipe(left)*<3<* D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<* u%(D' =-s6Bwipe(left)*<3<* u+8+0+  +" 7 / $ (  $ $  r0e0eRectangle 2"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!=drs/downrev.xmlDMO1@&fLIjB4Xxnu⯷Ǚ7y7gG Q;+a<6NiJ˛`1U;Ka>bVtܤe%JR%`9O6 SCUSBszz.]/Lm?R^_ G`WW[ ZU y{3>PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!=drs/downrev.xmlPK`$ > >6___PPT9 6Since our chances of guessing the right coordinates to try for a solution are not that high, we ll be more successful if we try a different technique. Since a solution of a system of equations is a solution common to both equations, it would also be a point common to the graphs of both equations. So to find the solution of a system of 2 linear equations, graph the equations and see where the lines intersect. " }  0 $  xP0e0eRectangle 3 `   dFinding a Solution by Graphing    -B0fBYt  ___PPT10 .+DH ' = @B D ' = @BA?%,( < +O%,( < +D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*$%(D' =-s6Bwipe(left)*<3<*$D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*$*%(D' =-s6Bwipe(left)*<3<*$*D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*$*%(D' =-s6Bwipe(left)*<3<*$*+8+0+$ +" ||GG(|(  (= (  r00e0eRectangle 2"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!cdrs/downrev.xmlDn0H}k+N\hAU[w /؎lO_r\j>b5|#[;m#a[s`!U9K`-PdK:nbÒĆ%1n`lb{ t+$7Y6M -R9 ;-SV/[:|>Iys=<4sٳ7ZTZ"y 0&| PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!cdrs/downrev.xmlPK0 @ H$ >  bSolve the following system of equations by graphing. 2x  y = 6 and x + 3y = 105  5R  v r f (C (Group 6#"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!1drs/downrev.xmlDOMK@ efSݖRP[A i 5Is<>r=V &3@5\YB (( hjJ??Line 7BBfB )( hjJ??Line 8r B *( b??Line 9 B +( d??Line 10 B ,( d??Line 11V VB -( d??Line 12 B .( d??Line 13 B /( d??Line 14 B 0( d??Line 15V VB 1( d??Line 16V V B 2( d??Line 17  B 3( d??Line 18 B 4( d??Line 19 B 5( d??Line 20V V B 6( d??Line 21  B 7( d??Line 22 B 8( d??Line 236B 9( d??Line 246B :( d??Line 256B ;( d??Line 26BB6B <( d??Line 276B =( d??Line 286B >( d??Line 296B ?( d??Line 306B @( d??Line 316B A( d??Line 32  6B B( d??Line 33B B fB C( d??Line 34  6B D( d??Line 35  6B E( d??Line 36  6Y F( f??Text Box 37  @ D<4___PPT9 ox& " d  Y G( fе??Text Box 38K&D<4___PPT9 oy& " d   ( `??Text Box 39"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!΅drs/downrev.xmlDOK1G!EVdmZE"snf$~{=ox?l10DIB9.!5^[j%wϣ`1)Ҫ.a1J3mM-`R+ccЩ8=RfGJ -Au\=wR^0'Iv֚uy[MWֹELD9-_.`FńAB.̽PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!΅drs/downrev.xmlPKp p$ ><4___PPT9 0First, graph 2x  y = 6.P " d2  3x P p (C *Group 40#"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!6drs/downrev.xmlDOMk@ MbHR0JoCvLِ]ﻂԢUēqnuŅkˤFV˗Sm{>RBv)*(oR)]^A7 qζ5l [CeEsiPbCKv5 v=i.4=ĤuX40?yOYC\PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!6drs/downrev.xmlPKP p,$D >L2 &( d0??Oval 41  D<4___PPT9 d$ " d2  W '( `??Text Box 42PP pD<4___PPT9 s(0, -6)$ " d2  2x    (C *Group 43#"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!Zdrs/downrev.xmlDOMk@M]ERP mȎI0;${PԢUēqnuŅ;5֖IW/K̴H"PA}I mwA`[HbM-(KږO?F~3?}\RѰY4{di zPK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!Zdrs/downrev.xmlPK  ,$D >L2 $( d??Oval 44 00 D<4___PPT9 d$ " d2  V %( `??Text Box 450  D<4___PPT9 r(3, 0)$ " d2  3y  (C *Group 46#"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!s[3drs/downrev.xmlDOKk@)V7I*" P 1 fgCvM =~|jt g$E\x[sitxy"3Sa?RI T1֡aXF]mFgI2k [TT\7gsan]JL2 "( dp??Oval 47@pD<4___PPT9 d$ " d2  V #( `??Text Box 48D<4___PPT9 r(6, 6)$ " d2   ( `P??Text Box 50"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!qϞdrs/downrev.xmlDQK0FC_ĥ cH]6Ɔ:Ad|ms$}8h:6Y il唶#*% W ^,1Wb aIbCPd0L]O6y1%ME-AmB=m[F8܋u}Aʻɸyi/CQn?ګ"bk*0DRaMPK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!qϞdrs/downrev.xmlPK gp$  ><4___PPT9 Second, graph x + 3y = 10.P " d2  5x   (C *Group 51#"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!Ydrs/downrev.xmlDOj@ I⃒:"4Jw5 f̘Ŀw ^SZWYVO"Ĺ ϟw#k-;9X^FKL"KQA}J mwA`[HbM-(ZHږ_QL]w^sIaA,S;PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!Ydrs/downrev.xmlPK`C ,$D  >L2  ( d??Oval 52D<4___PPT9 d$ " d2  Y !( `??Text Box 53D<4___PPT9 u (0, 10/3)$ " d2    3x `  (C *Group 54#"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!gdrs/downrev.xmlDOMk@& ]EĊ)T6d$ m}PԢUqnuŅuydeR$e=Qw!2TPzdR$njmkR؇pS$RiPbCے(of;>=Ƥd0?yK4C\PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!gdrs/downrev.xmlPK@,$D  >L2 ( d??Oval 55D<4___PPT9 d$ " d2  W ( `п??Text Box 56`D<4___PPT9 s(-2, 4)$ " d2  4y  p  (C *Group 57#"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!{ndrs/downrev.xmlDOj@ IV4:.D&̝ByϗDK+-+GsmydeR$e=\v)*(S)]VA75qn1lrBdEciP`M낲a|uحMsŤ_@x#S=PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!{ndrs/downrev.xmlPK V@,$D  >L2 ( d0??Oval 58 @ pD<4___PPT9 d$ " d2  W ( `0??Text Box 59 D<4___PPT9 s(-5, 5)$ " d2  B  ( jo??Line 60"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!Ddrs/downrev.xmlDOMK@ -EP{&ݰ;w^o'߫Rb\PX8Aeఏ,(v3[c5T4QrZ:-y̋8PG^%0f=vAZ赥rVE.b6}S~>^@1M/s/SqN03% rJ. z PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!Ddrs/downrev.xmlPK ,$D >  ( `p??Text Box 61"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!/Wdrs/downrev.xmlDOK1G!ElhYb[ot$Koo Ǜ:۰3h0U^WK8_bB$a6`mK5KSjKcb\f',|, E-4}[ bs'F1\-q더7X.?__R1ç`bL$ܛ9PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!/Wdrs/downrev.xmlPK p$ ><4___PPT9 (The lines APPEAR to intersect at (4, 2).$) " d2))  2x ` (C *Group 62#"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!WVdrs/downrev.xmlDOj@-'҅>I0'$}(ٲ8Ū7hqe0Ԩհഥ|۳L2 ( dP??Oval 63D<4___PPT9 d$ " d2  V ( `0??Text Box 64`D<4___PPT9 r(4, 2)$ " d2  2 (  z0e0eRectangle 65 `   dFinding a Solution by Graphing   [ ( f-B??Text Box 66=D<4___PPT9 yExample*( " d2  { ( hP??Rectangle 67#"@PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!SPdrs/downrev.xmlDN0EH5Hl*4V<4___PPT9 z Continued.( " d2    v W_p(c0Z (C (Group 2"`0B ( jo??Line 49W_p(c0ZF2 ( P?Oval 1"@3nkA"lWAD<4___PPT9 d$ " d2  H2 ( R0?Oval 65"@h{giI=/hjJD<4___PPT9 d$ " d2  H2 ( Rp?Oval 66"@by5%z%D<4___PPT9 d$ " d2  $x  (C *Group 51#"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!_!drs/downrev.xmlDOk@ޚMH-=BUo3 f߆6}Pq0v4seI .Tp>V GX&9nfEї"@إMtEE]d[lgٕRw8i"_B-+*o}aL~O%OH[4_C+xMKrPK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!_!drs/downrev.xmlPK#v,$D >L2 ( d??Oval 52D<4___PPT9 d$ " d2  H ( `??Text Box 53D<4___PPT9 d$ " d2   -B0fBYtJ5B5___PPT10"5.+D3' = @B D3' = @BA?%,( < +O%,( < +D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*(5%(D' =-s6Bwipe(left)*<3<*(5D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*(5F%(D' =-s6Bwipe(left)*<3<*(5FD' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*(FR%(D' =-s6Bwipe(left)*<3<*(FRD' = @B%(D' = @B%(D7' =4@BBBB%(D' =1:Bvisible*o3>+#.<*(%(D' =-s6Bwipe(left)*<3<*(D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*(%(D' =-s6Bwipe(left)*<3<*(D' = @B%(D' = @B%(D7' =4@BBBB%(D' =1:Bvisible*o3>+#.<*(%(D' =-s6Bwipe(left)*<3<*(D' = @B%(D' = @B%(D7' =4@BBBB%(D' =1:Bvisible*o3>+#.<*(%(D' =-s6Bwipe(left)*<3<*(D' = @B%(D' = @B%(D7' =4@BBBB%(D' =1:Bvisible*o3>+#.<*(%(D' =-s6Bwipe(left)*<3<*(D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*(%(D' =-s6Bwipe(left)*<3<*(D' = @B%(D' = @B%(D7' =4@BBBB%(D' =1:Bvisible*o3>+#.<* (%(D' =-s6Bwipe(left)*<3<* (D' = @B%(D' = @B%(D7' =4@BBBB%(D' =1:Bvisible*o3>+#.<* (%(D' =-s6Bwipe(left)*<3<* (D' = @B%(D' = @B%(D7' =4@BBBB%(D' =1:Bvisible*o3>+#.<* (%(D' =-s6Bwipe(left)*<3<* (D' = @B%(D' = @B%(D7' =4@BBBB%(D' =1:Bvisible*o3>+#.<* (%(D' =-s6Bwipe(left)*<3<* (D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<* (%(D' =-s6Bwipe(left)*<3<* (D' = @B%(D' = @B%(D7' =4@BBBB%(D' =1:Bvisible*o3>+#.<*(%(D' =-s6Bwipe(left)*<3<*(D' = @B%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*(%(D' = @B%(D' = @B%(D7' =4@BBBB%(D' =1:Bvisible*o3>+#.<*(%(D' =-s6Bwipe(left)*<3<*(++0+( ++0+( ++0+( ++0+ ( ++0+( +"f    , (  , ,  r0e0eRectangle 2"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK! drs/downrev.xmlDMO0 @HH\K?ʲ uݼk MR%8zֳ|9^ȇY$Avm"ZB9ڝmIMlKl(PAPH 2;8o0=Ynz%T,_hq(w-1Ucyܽ*u{3>?4]3-C.1D {\PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-! drs/downrev.xmlPK0@H$ >  Although the solution to the system of equations appears to be (4, 2), you still need to check the answer by substituting x = 4 and y = 2 into the two equations. First equation, 2(4)  2 = 8  2 = 6 true Second equation, 4 + 3(2) = 4 + 6 = 10 true The point (4, 2) checks, so it is the solution of the system.>I }    >I }z          >A  0 ,  x0e0eRectangle 6 `   dFinding a Solution by Graphing   c , d-B??Text Box 700 D<4___PPT9 Example continued*( " d2   -B0fBYt ___PPT10.+BLD' = @B DE' = @BA?%,( < +O%,( < +D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*,%(D' =-s6Bwipe(left)*<3<*,D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*,%(D' =-s6Bwipe(left)*<3<*,D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*,%(D' =-s6Bwipe(left)*<3<*,D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*,%(D' =-s6Bwipe(left)*<3<*,D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*,%(D' =-s6Bwipe(left)*<3<*,D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*,A%(D' =-s6Bwipe(left)*<3<*,A+8+0+, +" An9n==0!n(  0V 0  r0e0eRectangle 2"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!~drs/downrev.xmlDN0HHܨPUVT~nnxudMcqjVivH> qFZ_݀Yc(jy~BK:nb#C BPd1L\Oy1䶓yͥEiŞZ6`g-~Sfou>J]^w "]_ՋN-l6 ϧ7+H7qPK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!~drs/downrev.xmlPK0 H$ >  zSolve the following system of equations by graphing.  x + 3y = 6 and 3x  9y = 95  5T  v r f 0C (Group 6#"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!Q0Edrs/downrev.xmlDOj@ I|:.DK3wBfLwTƕģqfuɹi ydeRp'm=\v)*(S)]VA75q.1lrB8iP`M낲f|wح&]/iŤ_}ax:PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!Q0Edrs/downrev.xmlPKp ,$D >B 0 hjJ??Line 7BBfB 0 hjJ??Line 8r B  0 b??Line 9 B !0 d??Line 10 B "0 d??Line 11V VB #0 d??Line 12 B $0 d??Line 13 B %0 d??Line 14 B &0 d??Line 15V VB '0 d??Line 16V V B (0 d??Line 17  B )0 d??Line 18 B *0 d??Line 19 B +0 d??Line 20V V B ,0 d??Line 21  B -0 d??Line 22 B .0 d??Line 236B /0 d??Line 246B 00 d??Line 256B 10 d??Line 26BB6B 20 d??Line 276B 30 d??Line 286B 40 d??Line 296B 50 d??Line 306B 60 d??Line 316B 70 d??Line 32  6B 80 d??Line 33B B fB 90 d??Line 34  6B :0 d??Line 35  6B ;0 d??Line 36  6Y <0 fP€??Text Box 37  @ D<4___PPT9 ox& " d  Y =0 f€??Text Box 38K&D<4___PPT9 oy& " d   0 `0??Text Box 39"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!drs/downrev.xmlDOK1GBCٮصiYZDGަ&I$mAޏ7egA;+`<ʀԶ"Z$JK.vK]lXPcWrjEud;:o07\z$iyeܠiAaG Ed䛻XV:%q;g`|L-yQ'`bԛ8PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!drs/downrev.xmlPKp p$ ><4___PPT9 4First, graph  x + 3y = 6.Z " d2   6{    0C *Group 40#"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!ovdrs/downrev.xmlDj@afMYEĊP !;&lnBǏofdZ1P (A\Zpr~{ZpYckf={1vNA.C]&+k2"j{>`_I䦕8~jhWSy;}2??NW&?>p"M~ rPK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!ovdrs/downrev.xmlPK ,$D >L2 0 d€??Oval 41@ p  D<4___PPT9 d$ " d2  W 0 `0€??Text Box 42 D<4___PPT9 s(-6, 0)$ " d2  2x  f 0C *Group 43#"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!G'drs/downrev.xmlDOMk@ zMR0JoCvLِ]]ԢUēqnuŅkˤFV˗Sm{>RBv)*(oR)]^A7 qζ5l [CeE`šĆ6%jzx.4=ĤiO{d6K)<ryPK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!G'drs/downrev.xmlPK f,$D >L2 0 d0€??Oval 44D<4___PPT9 d$ " d2  V 0 `€??Text Box 45 fD<4___PPT9 r(0, 2)$ " d2  2x 0 0C *Group 46#"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!Wʄdrs/downrev.xmlDOj@ I⃒:.Dh,.k 1^SZWYVO"Ĺ ~N;5֖IW/%M] BإItyI6ā;֠-nᦖI-CC mK/(L]w{IaKlD  PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!Wʄdrs/downrev.xmlPK0,$D  >L2 0 d€??Oval 47@p0D<4___PPT9 d$ " d2  V 0 `€??Text Box 48D<4___PPT9 r(6, 4)$ " d2  B 0 jo??Line 49"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!u 4drs/downrev.xmlDOMK@ eovc(Ec m&ݰɿw^Gש"eW6{{ .x20zu}†PZI 4)\5g'/)DI`xp<a륡^3P.6~:?w73}~xs{3n@%ӿne~>#t"'\PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!u 4drs/downrev.xmlPKP ` ,$D  >  0 `p€??Text Box 50"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!*drs/downrev.xmlDMK1@!l"mZE Kofv$KwǙ7cK `렭o%|^V]( ۛte$RL}y :ơGO ѩLclBrqQ)Q=>N؎D:Xk:+nXNe1oR;_ZS(K1zVT%ІzPK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!*drs/downrev.xmlPK gp$  ><4___PPT9 4Second, graph 3x  9y = 9.P " d2  3x  0   0C *Group 51#"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!]0drs/downrev.xmlDO]k0}?;mNFg9 :.͵-67m@|ϗEGkPqLŅkǤN9f Cg ɤyI5đb-iᶖi̤ŊcC Kʯcn{UeX}4mt:MC\PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!]0drs/downrev.xmlPK 0 ,$D  >L2 0 d€??Oval 52  D<4___PPT9 d$ " d2  W 0 `€??Text Box 53 0 D<4___PPT9 s(0, -1)$ " d2  2x P X  0C *Group 54#"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!M)drs/downrev.xmlDOMk@&1-]EĊ)T6d$ m}PԢUqnuŅuydeR$e=Qw!2TPzdR$njmkR؇pS$ޥCC mKo`cŻoRY4$M8C\PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!M)drs/downrev.xmlPK PX,$D >L2 0 d€??Oval 55M(}XD<4___PPT9 d$ " d2  V 0 `€??Text Box 56P @D<4___PPT9 r(6, 1)$ " d2  2x f   0C *Group 57#"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!_drs/downrev.xmlDOj@-'҅>I0'$}(ٲ8Ū7hqe0Ԩհഥ|۳L2 0 dЮ€??Oval 58 = D<4___PPT9 d$ " d2  V 0 `p€??Text Box 59f D<4___PPT9 r(3, 0)$ " d2  B  0 jo??Line 60"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK! k-drs/downrev.xmlDAK@aJmEi6NnY;ޛy:5P6x *'P['1W7K,l- ex.@R_hUCyz!:L2ZۈW wγl^.4kCեqOnfp؝c3Dc?y.)v(Tz PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-! k-drs/downrev.xmlPK p ,$D >~ 0 `p€??Text Box 61"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!(drs/downrev.xmlDOK1G!ElaYڴXAw/`1SyG.a>bmM-kSkcb\f,|,EQq==kj+ADU/}-ݰxhHXMV*b z6 0f| PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!(drs/downrev.xmlPKp$ ><4___PPT9  The lines APPEAR to be parallel.$! " d2!!  2 0  zP€0e0eRectangle 62 `   dFinding a Solution by Graphing   [ 0 f0€-B??Text Box 63=D<4___PPT9 yExample*( " d2  { 0 h€??Rectangle 64#"@PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK! drs/downrev.xmlD]K0C87cK UF]6`lil[7x<`Zv .f0S9*l% #X-OOX(w;:c͒Ć%41vl`lbc:}͕c,M vtPl+_wKKy~6\4[UmTjy.Dz?V; TzPK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-! drs/downrev.xmlPKJp$><4___PPT9 z Continued.( " d2     -B0fBYtA292___PPT102.+ND0' = @B D0' = @BA?%,( < +O%,( < +D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*05%(D' =-s6Bwipe(left)*<3<*05D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*05H%(D' =-s6Bwipe(left)*<3<*05HD' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*0HT%(D' =-s6Bwipe(left)*<3<*0HTD' = @B%(D' = @B%(D7' =4@BBBB%(D' =1:Bvisible*o3>+#.<*0%(D' =-s6Bwipe(left)*<3<*0D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*0%(D' =-s6Bwipe(left)*<3<*0D' = @B%(D' = @B%(D7' =4@BBBB%(D' =1:Bvisible*o3>+#.<*0%(D' =-s6Bwipe(left)*<3<*0D' = @B%(D' = @B%(D7' =4@BBBB%(D' =1:Bvisible*o3>+#.<*0%(D' =-s6Bwipe(left)*<3<*0D' = @B%(D' = @B%(D7' =4@BBBB%(D' =1:Bvisible*o3>+#.<*0%(D' =-s6Bwipe(left)*<3<*0D' = @B%(D' = @B%(D7' =4@BBBB%(D' =1:Bvisible*o3>+#.<*0%(D' =-s6Bwipe(left)*<3<*0D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<* 0%(D' =-s6Bwipe(left)*<3<* 0D' = @B%(D' = @B%(D7' =4@BBBB%(D' =1:Bvisible*o3>+#.<* 0%(D' =-s6Bwipe(left)*<3<* 0D' = @B%(D' = @B%(D7' =4@BBBB%(D' =1:Bvisible*o3>+#.<* 0%(D' =-s6Bwipe(left)*<3<* 0D' = @B%(D' = @B%(D7' =4@BBBB%(D' =1:Bvisible*o3>+#.<* 0%(D' =-s6Bwipe(left)*<3<* 0D' = @B%(D' = @B%(D7' =4@BBBB%(D' =1:Bvisible*o3>+#.<* 0%(D' =-s6Bwipe(left)*<3<* 0D ' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*0%(D' =-s6Bwipe(left)*<3<*0D' = @B%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*0%(++0+0 ++0+0 ++0+ 0 ++0+0 ++0+0 +"nG s&k&4S&(  4 4  r0€0e0eRectangle 2"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!0ΰndrs/downrev.xmlDO0&3&C' 7Ǹx{K[௧Ǘ{ݰ#90$ȔV*S  0Hl!'0L1397bQb|ڌs_֤lK&uC<]ť.u$są[zJx΋e_w+@}^~WJƖi8vN} ' ȁ.PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!0ΰndrs/downrev.xmlPK" H$ >  Although the lines appear to be parallel, you still need to check that they have the same slope. You can do this by solving for y. First equation,  x + 3y = 6 3y = x + 6 (add x to both sides)I }Z  Z@  Z   w `   4C (Group 6#"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!l4drs/downrev.xmlDOMk@ &ZKH[  4 HA ??Object 7P`    4 ^€??Text Box 8w ^ D<4___PPT9 . y = x + 2 (divide both sides by 3)F/ " dZ  /  @ 4 ^p€??Text Box 9"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!LYdrs/downrev.xmlDOK1G!ElZtmZXAdm7ɒvhg06Xh%Β3ϮXHw:ncÒĆP)2#XNpyMAmӂŽ^U# /rj?XRۛ~ ,RK/W%?or! SogPK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!LYdrs/downrev.xmlPKx pQ ~$ >JB___PPT9$ Second equation, 3x  9y = 9  9y =  3x + 9 (subtract 3x from both sides) " dZD " dZ   U   @x   , 4C *Group 10#"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!7drs/downrev.xmlDOMk@&VEhA UA 1 fgCvM{L)]aYA<@V)85IW%&ڶCg"KPA}HҜ wA`I]cM)GQ4  9V)Fg#5m&ߗCLJ_/Ɠx>C\PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!7drs/downrev.xmlPK $,$D  >  4 JA ??Object 11 ,   4 `€??Text Box 12  D<4___PPT9 Xy = x  1 (divide both sides by  9)R- " dZ -  @x   4C *Group 18#"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!*y#drs/downrev.xmlDOMk@ &J"Ҋ)6d$ m}W<>b՛JԸҲx ά.9W{~<2)mTێ}.Bש.+Ƞۚ8pW6 v!T24Xrh(MAgl;֓߮49cRj4sz?;'$Jq( PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!*y#drs/downrev.xmlPK ,$D  >  4 JA ??Object 14 6 *   4 `€??Text Box 15 D<4___PPT9 uBoth lines have a slope of , so they are parallel and do not intersect. Hence, there is no solution to the system.4v " }Ztv  2 4  z€0e0eRectangle 16 `   dFinding a Solution by Graphing   e 4 f€-B??Text Box 1700 D<4___PPT9 Example continued*( " d2   -B0fBYt{ s ___PPT10S .+[7LD' = @B Dv' = @BA?%,( < +O%,( < +D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*4%(D' =-s6Bwipe(left)*<3<*4D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*4%(D' =-s6Bwipe(left)*<3<*4D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*4%(D' =-s6Bwipe(left)*<3<*4D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*4%(D' =-s6Bwipe(left)*<3<*4D' = @B%(D' = @B%(D7' =4@BBBB%(D' =1:Bvisible*o3>+#.<*4%(D' =-s6Bwipe(left)*<3<*4D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*4%(D' =-s6Bwipe(left)*<3<*4D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*4%(D' =-s6Bwipe(left)*<3<*4D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*4U%(D' =-s6Bwipe(left)*<3<*4UD' = @B%(D' = @B%(D7' =4@BBBB%(D' =1:Bvisible*o3>+#.<*4%(D' =-s6Bwipe(left)*<3<*4D' = @B%(D' = @B%(D7' =4@BBBB%(D' =1:Bvisible*o3>+#.<*4%(D' =-s6Bwipe(left)*<3<*4+p+0+4 ++0+4 +" TbLb9984b(  8J 8  rP€0e0eRectangle 2"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!3Dddrs/downrev.xmlDN1@&C3&ޤjb4mvXzxy7y8YQluxx"ZBۛ)ڝmIUlDP6ƾ2-1&s17R{<'9w2ϲd46}h &_OY}fxˡ7J o " [n T:O|zv_7+Hԛ8/PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!3Dddrs/downrev.xmlPK 0 H$ >  pSolve the following system of equations by graphing. x = 3y  1 and 2x  6y =  25  5T  v r f 8C (Group 6#"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!<drs/downrev.xmlDOj@ I⃒:.Dh,.k 1^SZWYVO"Ĺ ~N;5֖IW/%M] BإItyI6ā;֠-nᦖI-CC mK/(L]w{IaKl3 PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!<drs/downrev.xmlPKp ,$D >B 8 hjJ??Line 7BBfB 8 hjJ??Line 8r B 8 b??Line 9 B 8 d??Line 10 B 8 d??Line 11V VB 8 d??Line 12 B  8 d??Line 13 B !8 d??Line 14 B "8 d??Line 15V VB #8 d??Line 16V V B $8 d??Line 17  B %8 d??Line 18 B &8 d??Line 19 B '8 d??Line 20V V B (8 d??Line 21  B )8 d??Line 22 B *8 d??Line 236B +8 d??Line 246B ,8 d??Line 256B -8 d??Line 26BB6B .8 d??Line 276B /8 d??Line 286B 08 d??Line 296B 18 d??Line 306B 28 d??Line 316B 38 d??Line 32  6B 48 d??Line 33B B fB 58 d??Line 34  6B 68 d??Line 35  6B 78 d??Line 36  6Y 88 f€??Text Box 37  @ D<4___PPT9 ox& " d  Y 98 f€??Text Box 38K&D<4___PPT9 oy& " d   8 `г€??Text Box 39"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!9 drs/downrev.xmlDAK1F!ElְTYb+Hin&Y{z>d60&Q }],Zu &Moo&7xR˲ħJI0D}yj :FG1D(:ss Ff8 b EmV% n#y%J:r,g`Ǐ!ZQ0Jȅ7s_PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!9 drs/downrev.xmlPKp p$ ><4___PPT9 0First, graph x = 3y  1.P " d2   4y `   8C *Group 40#"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!NŠdrs/downrev.xmlDOMk@)V7VJt-=HZ(ކ!&w{]zB@:I@\>_@lL`z|Xbn_c$C\PT0L|K,wԶA]$k5KC-m+*Ǜ3>ఙpl!%cT1Vgy6rH/PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!NŠdrs/downrev.xmlPK`  ,$D >L2 8 d€??Oval 41> D<4___PPT9 d$ " d2  W 8 `€??Text Box 42`  D<4___PPT9 s(-1, 0)$ " d2  2x `V 8C *Group 43#"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!Z!Xdrs/downrev.xmlDOMk@ zM*%u[z(ކ!&߻ԢUēqnuŅ{ ydeRp%h=|!B7./ɠ؆8p'-!24Xqh(uI9?=xmϧ0mcRuX}4a~1K1rqPK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!Z!Xdrs/downrev.xmlPK`V,$D  >L2 8 d€??Oval 44hD<4___PPT9 d$ " d2  V 8 `0€??Text Box 45`VD<4___PPT9 r(5, 2)$ " d2  3x ` h  8C *Group 46#"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!JVdrs/downrev.xmlDOMk@ &J"RŃ6d$ 5 =>b՛JԸҲx ά.9W{ڎ>A8L `|,0նj>!] TJdЍmMm \n*DL,94XӦvGnr~1)>sz/suLftC\>PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!JVdrs/downrev.xmlPKh ` ,$D >L2 8 dP€??Oval 47 h D<4___PPT9 d$ " d2  W 8 `€??Text Box 48` D<4___PPT9 s(7, -2)$ " d2  B 8 jo??Line 49"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!72drs/downrev.xmlDOk0xF5v:o$Ym`Ǐr=VC㬂Yltc+!:K F ^Mkw;X !Gu].e(k2f#yxp4I2iPcG5((ͩ۶?@_?JMD!,}zqyϾ; >PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!72drs/downrev.xmlPK ` ,$D  >  8 `€??Text Box 50"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!|;,drs/downrev.xmlDMK1@!Elְ6-E\AJ?xK7Mp,In=ox3o:\ ^Ө Vq,eG WL0LUo-#O`r+ScЩ4 =zb4Ɩ.$wE1NYOqiޝyjo_u>,Vc#ݰxq˺U՚ZD9zVoU%P!>PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!|;,drs/downrev.xmlPK gp$  ><4___PPT9 6Second, graph 2x  6y =  2.d " d2  4x     8C *Group 51#"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!W^drs/downrev.xmlDOMk@&i"R mȎI0;$ xuo*RJ q8\9pYcez5xYbm?|.Bש.+Ƞۚ8pW6 v!T24Xrh(mAk|umvnIaYӽa~>M&sx PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!W^drs/downrev.xmlPK  ,$D  >L2 8 dP€??Oval 52  D<4___PPT9 d$ " d2  X 8 `€??Text Box 53  D<4___PPT9 t(-4, -1)$ " d2    2x X   8C *Group 54#"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!ddrs/downrev.xmlDOMk@Mb$uR !;&lȮI]AԢUēqnuŅi6<2)b2L۞;Bv*(o2)]^A7 q.5l [CeE4Xqh(uIx3 v=i{9ĤuX}4ӽa~&iC\PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!ddrs/downrev.xmlPKX ,$D  >L2 8 d€??Oval 55X(XD<4___PPT9 d$ " d2  V 8 `€??Text Box 56f D<4___PPT9 r(2, 1)$ " d2    8 `0€??Text Box 57"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!drs/downrev.xmlDOK1G!ElP]bk=xL7M$~{=ox?d6؎(D㝄Q\q)# '0MR6tئeJ)d1|O.S>CUcێ(Eƞ55_o+Ax3Fwo/Tu-0hH}XWTEKQ?`c !́OPK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!drs/downrev.xmlPKp$ ><4___PPT9 !The lines APPEAR to be identical.$" " d2""  2  8  zP€0e0eRectangle 58 `   dFinding a Solution by Graphing   [ 8 fp€-B??Text Box 59=D<4___PPT9 yExample*( " d2  { 8 hp€??Rectangle 60#"@PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!hdrs/downrev.xmlDN0 @ߑHL[JTM\4$.LMRtkz,:ֱb569Y S9$!U:KF Z,Phwt:a Jhb !Bِ0sYf#Y |*?Wy~z뮷~_}{)φkH" 66[<4___PPT9 z Continued.( " d2     -B0fBYt/,',___PPT10,.+AD*' = @B D*' = @BA?%,( < +O%,( < +D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*85%(D' =-s6Bwipe(left)*<3<*85D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*85G%(D' =-s6Bwipe(left)*<3<*85GD' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*8GT%(D' =-s6Bwipe(left)*<3<*8GTD' = @B%(D' = @B%(D7' =4@BBBB%(D' =1:Bvisible*o3>+#.<*8%(D' =-s6Bwipe(left)*<3<*8D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*8%(D' =-s6Bwipe(left)*<3<*8D' = @B%(D' = @B%(D7' =4@BBBB%(D' =1:Bvisible*o3>+#.<*8%(D' =-s6Bwipe(left)*<3<*8D' = @B%(D' = @B%(D7' =4@BBBB%(D' =1:Bvisible*o3>+#.<*8%(D' =-s6Bwipe(left)*<3<*8D' = @B%(D' = @B%(D7' =4@BBBB%(D' =1:Bvisible*o3>+#.<*8%(D' =-s6Bwipe(left)*<3<*8D' = @B%(D' = @B%(D7' =4@BBBB%(D' =1:Bvisible*o3>+#.<*8%(D' =-s6Bwipe(left)*<3<*8D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<* 8%(D' =-s6Bwipe(left)*<3<* 8D' = @B%(D' = @B%(D7' =4@BBBB%(D' =1:Bvisible*o3>+#.<* 8%(D' =-s6Bwipe(left)*<3<* 8D' = @B%(D' = @B%(D7' =4@BBBB%(D' =1:Bvisible*o3>+#.<* 8%(D' =-s6Bwipe(left)*<3<* 8D ' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<* 8%(D' =-s6Bwipe(left)*<3<* 8D' = @B%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*8%(++0+8 ++0+8 ++0+ 8 ++0+ 8 ++0+8 +"G &&<&(  < <  r0€0e0eRectangle 2"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!8@O(drs/downrev.xmlD_O0GM%M:ϤAq|K[槷Ao̗n؅W$ȔV*S `>ΪX昱ٜ.P(1>ƹ/'%:!Ҳiܱąڳ?wCƖtvNg)9PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!8@O(drs/downrev.xmlPKt H$ >  Although the lines appear to be identical, you still need to check that they are identical equations. You can do this by solving for y. First equation, x = 3y  1 3y = x + 1 (add 1 to both sides)I }Z 0 Z< 0 Z     < ^P€??Text Box 6"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!۽Kdrs/downrev.xmlDMK1@!EllѺ6-"V(/ot$K7x7޶H!$ `jk$/#`1SzGa:cɭI +S*cb\f,<,-Eq-/hYSF [ctHٜ/WR^_'`?eU [AyZcL$ܛ9PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!۽Kdrs/downrev.xmlPK  ~$ >JB___PPT9$ Second equation, 2x  6y =  2  6y =  2x  2 (subtract 2x from both sides)  " dZF0 " dZ  W   @ < ^p€??Text Box 7"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!Idrs/downrev.xmlDAK1F!El]b+^M7Mp,Ilz=oxo2mNpP#W{e\#a[=] ;pKnMMjXXRWrkMw2`134\<4___PPT9 The two equations are identical, so the graphs must be identical. There are an infinite number of solutions to the system (all the points on the line).  " }Z  fw p <  <C (Group 8#"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!Xo}drs/downrev.xmlDOMk@)V7Im+A B6d$ 5sz|rt g$E\x[siy b \)rq7RI T1֡aXF]mFgI, 8/9wz<|R2a\4ͽ2?eS ^PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!Xo}drs/downrev.xmlPK :T ,$D >  < HA ??Object 90 <    < `€??Text Box 10p  D<4___PPT9 - y = x + (divide both sides by 3)\. " dZ .  @ < JA ??Object 11 <  ky p <  <C *Group 12#"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!'T{drs/downrev.xmlDOMk@MX%ux*HoCvLِ&߻B^SZWYVO"Ĺ Nϗ95֖IfM"PA}I mwA`[HbM-(z+ %6))~k8ݟw1)< < _0?IgIP@ wPK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!'T{drs/downrev.xmlPK V ,$D >  < JA ??Object 130 <    < `€??Text Box 14p  D<4___PPT9 . y = x + (divide both sides by -6)\/ " dZ /  @  < JA ??Object 15 <  2 <  z€0e0eRectangle 16 `   dFinding a Solution by Graphing   e < fP€-B??Text Box 1700 D<4___PPT9 Example continued*( " d2   -B0fBYt  ___PPT10 .+޾%D' = @B D' = @BA?%,( < +O%,( < +D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*<%(D' =-s6Bwipe(left)*<3<*<D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*<%(D' =-s6Bwipe(left)*<3<*<D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*<%(D' =-s6Bwipe(left)*<3<*<D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*<%(D' =-s6Bwipe(left)*<3<*<D' = @B%(D' = @B%(D7' =4@BBBB%(D' =1:Bvisible*o3>+#.<*<%(D' =-s6Bwipe(left)*<3<*<D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*<%(D' =-s6Bwipe(left)*<3<*<D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*<%(D' =-s6Bwipe(left)*<3<*<D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*<W%(D' =-s6Bwipe(left)*<3<*<WD' = @B%(D' = @B%(D7' =4@BBBB%(D' =1:Bvisible*o3>+#.<*<%(D' =-s6Bwipe(left)*<3<*<D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*<%(D' =-s6Bwipe(left)*<3<*<++0+< ++0+< ++0+< +"= @(  @) @  rн€0e0eRectangle 2"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!$ (drs/downrev.xmlDMO0 @HHXJ(&VHܼk MR%8zֳl1^); ڛ5ry1:wHӓ\AujK\QCҐK,Ɖ1`1iXn{VZ_hqǖjt|*ID1/WRp~r܄ \ȽAPK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!$ (drs/downrev.xmlPK$ > >6___PPT9 ZThere are three possible outcomes when graphing two linear equations in a plane. One point of intersection, so one solution Parallel lines, so no solution Coincident lines, so infinite # of solutions If there is at least one solution, the system is considered to be consistent. If the system defines distinct lines, the equations are independent.zQ " w "  " QwB : [  " @  x€0e0eRectangle 3 `   VTypes of Systems    -B0fBYt ___PPT10.+D' = @B DE' = @BA?%,( < +O%,( < +D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*@Q%(D' =-s6Bwipe(left)*<3<*@QD' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*@Q|%(D' =-s6Bwipe(left)*<3<*@Q|D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*@|%(D' =-s6Bwipe(left)*<3<*@|D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*@%(D' =-s6Bwipe(left)*<3<*@D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*@%(D' =-s6Bwipe(left)*<3<*@D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*@[%(D' =-s6Bwipe(left)*<3<*@[+8+0+@ +" IAD)(  D D  rФ€0e0eRectangle 2"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!(drs/downrev.xmlDN0HHܨSPVT~z"M{moClG&<=8f5fMNvVx#[9m-aS.o&BDu$ `1atZǚ% 9JhbrCՐ0r5WInZ.6-4SCh$wQdf}y=Ky}?>c5WRvx^C$/!āPK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!(drs/downrev.xmlPKpH$ >  Change both linear equations into slope-intercept form. We can then easily determine if the lines intersect, are parallel, or are the same line.I }2  " D  xP€0e0eRectangle 3 `   VTypes of Systems    -B0fBYt___PPT10.+oD2' = @B D' = @BA?%,( < +O%,( < +D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*D8%(D' =-s6Bwipe(left)*<3<*D8D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*D8%(D' =-s6Bwipe(left)*<3<*D8+8+0+D +"= WO H7(  H  H  rP0e0eRectangle 2"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!}Edrs/downrev.xmlD1O0Fw$uHl&)Mk|M ncuNӛ-Fӳp?6NiJ˻G`!U;KN`1aVtXǖ% Jb Cӑ0qv-WInz sڦzY6xwaӸכ73Hc^6/ERK "+O[U! SoPK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!}Edrs/downrev.xmlPK$ > VN___PPT90( How many solutions does the following system have? 3x + y = 1 and 3x + 2y = 6 Write each equation in slope-intercept form. First equation, 3x + y = 1 y =  3x + 1 (subtract 3x from both sides) Second equation, 3x + 2y = 6 2y =  3x + 6 (subtract 3x from both sides)3 0 Z  0 Z-I0 }Z0 tZ3-    C   H ^??Text Box 6"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!$odrs/downrev.xmlDMK1@!El6-E /ަ&I$oǙ7MؑB4I(F+Z XLvޑ3E./X)rk:nR˲ %hG')*)mK!E==ij6VB)[ctLzVR^_ #DC?E=WT"^`c !orog?PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!$odrs/downrev.xmlPK4  p$  ><4___PPT9 ^The lines are intersecting lines (since they have different slopes), so there is one solution. _ " }Z__  w  4 HC *Group 12#"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!V drs/downrev.xmlDOM0 04H("*{ {-6Ķ‚^mzSWZVO"ę% x ydeR"pDێԦ>!] DJdMlMm \n*94Xrh(]A#},޷}̿oOzt?GxPK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!V drs/downrev.xmlPK 4,$D  >} H ^??Text Box 8  D<4___PPT9 ! (divide both sides by 2)2" " dZ!"    H HA ??Object 9 4 $ H  z0e0eRectangle 10 `   VTypes of Systems   [ H f0-B??Text Box 11 @}D<4___PPT9 yExample*( " d2   -B0fBYt##___PPT10#.+D"' = @B D"' = @BA?%,( < +O%,( < +D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*H3%(D' =-s6Bwipe(left)*<3<*H3D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*H3S%(D' =-s6Bwipe(left)*<3<*H3SD' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*HS%(D' =-s6Bwipe(left)*<3<*HSD' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*H%(D' =-s6Bwipe(left)*<3<*HD' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*H%(D' =-s6Bwipe(left)*<3<*HD' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*H%(D' =-s6Bwipe(left)*<3<*HD' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*H%(D' =-s6Bwipe(left)*<3<*HD' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*H%(D' =-s6Bwipe(left)*<3<*HD' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*HC%(D' =-s6Bwipe(left)*<3<*HCD' = @B%(D' = @B%(D7' =4@BBBB%(D' =1:Bvisible*o3>+#.<*H%(D' =-s6Bwipe(left)*<3<*HD' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*H_%(D' =-s6Bwipe(left)*<3<*H_+p+0+H ++0+H +"A. W O 0L7 (  Lt  L  r0e0eRectangle 2"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK! drs/downrev.xmlD=O0@w$uHli"1unb;6b(;[&30e6Ni XhΒX-X*w5ֱcIbCǒd0H6y11,{Mz鵧g7v:๖y5MۛX)Wr{n>UjEzڏkUcMMPK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-! drs/downrev.xmlPK $ > nf___PPT9H@ $How many solutions does the following system have? 3x + y = 0 and 2y =  6x Write each equation in slope-intercept form, First equation, 3x + y = 0 y =  3x (Subtract 3x from both sides) Second equation, 2y =  6x y =  3x (Divide both sides by 2) The two lines are identical, so there are infinitely many solutions.3  t-I }I } tEI }3-                                    Eh  " L  x0e0eRectangle 6 `   VTypes of Systems   Y L d€-B??Text Box 7=D<4___PPT9 yExample*( " d2   -B0fBYtj b ___PPT10B .+D' = @B D' = @BA?%,( < +O%,( < +D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*L3%(D' =-s6Bwipe(left)*<3<*L3D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*L3P%(D' =-s6Bwipe(left)*<3<*L3PD' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*LP}%(D' =-s6Bwipe(left)*<3<*LP}D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*L}%(D' =-s6Bwipe(left)*<3<*L}D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*L%(D' =-s6Bwipe(left)*<3<*LD' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*L%(D' =-s6Bwipe(left)*<3<*LD' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*L%(D' =-s6Bwipe(left)*<3<*LD' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*L%(D' =-s6Bwipe(left)*<3<*LD' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*L#%(D' =-s6Bwipe(left)*<3<*L#D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*L#h%(D' =-s6Bwipe(left)*<3<*L#h+8+0+L +"*   @P (  P  P  r0e0eRectangle 2"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!drs/downrev.xmlDAO1F&fLI"FW 1Tr۰v+v(zzdzv$nG)m[ zq,D {gI™gS,;يز,D ]Cyh:2Fn y1)M !AmBtW#a5~W0q\*Uz~)l[I1۽^ C$/!́~PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!drs/downrev.xmlPK>H$ >  How many solutions does the following system have? 2x + y = 0 and y =  2x + 1 Write each equation in slope-intercept form. First equation, 2x + y = 0 y =  2x (subtract 2x from both sides) Second equation, y =  2x + 1 (already in slope-intercept form) The two lines are parallel lines (same slope, but different y-intercepts), so there are no solutions.3  Z!  Z-I }Z  ZfI }Z3- "<)  " P  xp0e0eRectangle 6 `   VTypes of Systems   Y P d€-B??Text Box 7=D<4___PPT9 yExample*( " d2   -B0fBYtTL___PPT10,.+D' = @B D' = @BA?%,( < +O%,( < +D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*P3%(D' =-s6Bwipe(left)*<3<*P3D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*P3T%(D' =-s6Bwipe(left)*<3<*P3TD' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*PT%(D' =-s6Bwipe(left)*<3<*PTD' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*P%(D' =-s6Bwipe(left)*<3<*PD' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*P%(D' =-s6Bwipe(left)*<3<*PD' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*P%(D' =-s6Bwipe(left)*<3<*PD' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*P%(D' =-s6Bwipe(left)*<3<*PD' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*P'%(D' =-s6Bwipe(left)*<3<*P'D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*P'%(D' =-s6Bwipe(left)*<3<*P'+8+0+P +"2x 5-PT(  T> T ^Ы??Text Box 20@PD<4___PPT9 d$ " d2   T c P€0e0e5%Rectangle 4"@P 3    3Solving Systems of Linear Equations by Substitution4 464    T  j€0e0eTitle 1     BH   -B0fBYty___PPT10Y+D=' = @B + `X(  X# X  r€0e0eRectangle 2 `   ]The Substitution Method   9 X  r€0e0eRectangle 3"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!B_drs/downrev.xmlDN0HQ V(mM lMyz,帚|r;qLm2MU\p"si:4,JQ@Bs4DNcktxr4I\2qŞ[{-` sgEކs$hX= 4bE7gz%Ͳi,rbaPK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!B_drs/downrev.xmlPK H$ >  ]Another method that can be used to solve systems of equations is called the substitution method. You solve one equation for one of the variables, then substitute the new form of the equation into the other equation for the solved variable..I }L   -B0fBYt___PPT10.+AD2' = @B D' = @BA?%,( < +O%,( < +D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*Xa%(D' =-s6Bwipe(left)*<3<*XaD' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*Xa%(D' =-s6Bwipe(left)*<3<*Xa+8+0+X +"B g_p\G(  \  \  rp€0e0eRectangle 2"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!ku*(drs/downrev.xmlDN0HHܨM(B݊)M/ܶ66vd;mcqjVifvlO!$\N0rWƵ6XLvޑ#EXOOfX*pשeYbtJ}yl4YߓlŔr7ܢqyAcO`%lM~nusxl|hL~>,UR\O{9nQDAB.̽PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!ku*(drs/downrev.xmlPKpH$ >  Solve the following system using the substitution method. 3x  y = 6 and  4x + 2y =  8 Solving the first equation for y, 3x  y = 6  y =  3x + 6 (subtract 3x from both sides) y = 3x  6 (divide both sides by  1) Substitute this value for y in the second equation.  4x + 2y =  8  4x + 2(3x  6) =  8 (replace y with result from first equation)  4x + 6x  12 =  8 (use the distributive property) 2x  12 =  8 (simplify the left side) 2x = 4 (add 12 to both sides) x = 2 (divide both sides by 2):  Z!  Z"I }Z  Z4I }ZS  Z:                        "                                          ) \  xp0e0eRectangle 6 `   ]The Substitution Method   Y \ d€-B??Text Box 7=D<4___PPT9 yExample*( " d2  y \ f0€??Rectangle 8#"@PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!azdrs/downrev.xmlDN0 @ߑHL,*&. 6ӸmD$ZxuX { x| Wzm\m}v "&t[HHVܖT X&2 YSߑcV`1jXn[gمh_hۆ]oec^D/sp}"ѐor3>_FsK>2~z?[.^ ?PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!azdrs/downrev.xmlPKJp$><4___PPT9 z Continued.( " d2     -B0fBYt++___PPT10+.+8D"+' = @B D*' = @BA?%,( < +O%,( < +D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*\:%(D' =-s6Bwipe(left)*<3<*\:D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*\:[%(D' =-s6Bwipe(left)*<3<*\:[D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*\[}%(D' =-s6Bwipe(left)*<3<*\[}D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*\}%(D' =-s6Bwipe(left)*<3<*\}D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*\%(D' =-s6Bwipe(left)*<3<*\D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*\%(D' =-s6Bwipe(left)*<3<*\D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*\2%(D' =-s6Bwipe(left)*<3<*\2D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*\2M%(D' =-s6Bwipe(left)*<3<*\2MD' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*\M%(D' =-s6Bwipe(left)*<3<*\MD' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*\%(D' =-s6Bwipe(left)*<3<*\D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*\%(D' =-s6Bwipe(left)*<3<*\D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*\Q%(D' =-s6Bwipe(left)*<3<*\QD ' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*\Q%(D' =-s6Bwipe(left)*<3<*\QD' = @B%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*\%(+p+0+\ ++0+\ +"0   ` (  `  `  rGh0e0eRectangle 2"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!ĺdrs/downrev.xmlDn0xk+V)UEz…/ilGWO3ua'r^Y#੟#SZL%`S,|@#\lڻ`&tZE Ch3}YF߷-i tQ&ɐkT&.GMTk^|&up΅x߀v>.7Zؒ/~y9%s􁜀X{#>PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!ĺdrs/downrev.xmlPKPH$ >  Substitute x = 2 into the first equation solved for y. y = 3x  6 = 3(2)  6 = 6  6 = 0 Our computations have produced the point (2, 0). Check the point in the original equations. First equation, 3x  y = 6 3(2)  0 = 6 true Second equation,  4x + 2y =  8  4(2) + 2(0) =  8 true The solution of the system is (2, 0).7I }# \I } &I } (\    &d  ) `  xp7h0e0eRectangle 6 `   ]The Substitution Method   c ` dSh-B??Text Box 70 aD<4___PPT9 Example continued*( " d2   -B0fBYt#x#___PPT10X#.+D"' = @B D"' = @BA?%,( < +O%,( < +D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*`7%(D' =-s6Bwipe(left)*<3<*`7D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*`7Z%(D' =-s6Bwipe(left)*<3<*`7ZD' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*`Z%(D' =-s6Bwipe(left)*<3<*`ZD' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*`%(D' =-s6Bwipe(left)*<3<*`D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*`%(D' =-s6Bwipe(left)*<3<*`D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*`%(D' =-s6Bwipe(left)*<3<*`D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*`%(D' =-s6Bwipe(left)*<3<*`D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*`%(D' =-s6Bwipe(left)*<3<*`D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*`%(D' =-s6Bwipe(left)*<3<*`D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*`>%(D' =-s6Bwipe(left)*<3<*`>D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*`>d%(D' =-s6Bwipe(left)*<3<*`>d+8+0+` +" |dd(  d d  rDh0e0eRectangle 2"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!drs/downrev.xmlDN0 @HCe$n,ce.ܼkS%Dh?YolXDBF\;mQ7sL F Z^^,%ɒC BPd1L\Oy19m'r)¢Ş[G`o~իi|NjRW#d,UԒws،{ot!W67qPK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!drs/downrev.xmlPK ``$ > NF___PPT9(  MSolving a System of Linear Equations by the Substitution Method Solve one of the equations for a variable. Substitute the expression from step 1 into the other equation. Solve the new equation. Substitute the value found in step 3 into either equation containing both variables. Check the proposed solution in the original equations.<@ Z tZ@N  ) d  x3h0e0eRectangle 3 `   ]The Substitution Method    -B0fBYt ___PPT10.+D' = @B DE' = @BA?%,( < +O%,( < +D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*d@%(D' =-s6Bwipe(left)*<3<*d@D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*d@k%(D' =-s6Bwipe(left)*<3<*d@kD' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*dk%(D' =-s6Bwipe(left)*<3<*dkD' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*d%(D' =-s6Bwipe(left)*<3<*dD' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*d%(D' =-s6Bwipe(left)*<3<*dD' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*dN%(D' =-s6Bwipe(left)*<3<*dN+8+0+d +", h(  ha  h  r1h0e0eRectangle 2"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!Cldrs/downrev.xmlD1O0Fw$uHl!-B eAMk|M nxznفW%TV*S ؔ0Hl!G򰘟07֡fQb|s_5яlG&uC<]ͥ>u$ vPk[n(_}P~?c5~ZؒL' q,rbaPK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!Cldrs/downrev.xmlPK@H$ >  6Solve the following system of equations using the substitution method. y = 2x  5 and 8x  4y = 20 Since the first equation is already solved for y, substitute this value into the second equation. 8x  4y = 20 8x  4(2x  5) = 20 (replace y with result from first equation) 8x  8x + 20 = 20 (use distributive property) 20 = 20 (simplify left side)G # bI }  G/2  !  ) h  xNh0e0eRectangle 6 `   ]The Substitution Method   Y h dpUh-B??Text Box 7=D<4___PPT9 yExample*( " d2  y h f=h??Rectangle 8#"@PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!$drs/downrev.xmlDN0 @ߑHL[J`*&.l mqۈ&)I|=hXgMI )m+ o *l% X.NO+w;:cĆ%1`,yG_ rӈ,MAmC-T|;#+G۵yy~#|Og $|aUk-r=a%{XPK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!$drs/downrev.xmlPKJp$ ><4___PPT9 z Continued.( " d2     -B0fBYt^V___PPT106.+50D' = @B DY' = @BA?%,( < +O%,( < +D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*hG%(D' =-s6Bwipe(left)*<3<*hGD' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*hGj%(D' =-s6Bwipe(left)*<3<*hGjD' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*hj%(D' =-s6Bwipe(left)*<3<*hjD' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*h%(D' =-s6Bwipe(left)*<3<*hD' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*h.%(D' =-s6Bwipe(left)*<3<*h.D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*h.j%(D' =-s6Bwipe(left)*<3<*h.jD ' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*hj%(D' =-s6Bwipe(left)*<3<*hjD' = @B%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*h%(+p+0+h ++0+h +"2   l (  l l  r5h0e0eRectangle 2"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!-drs/downrev.xmlDMO@@&͘x!~* !*R.܆V_ă'oftޛVȇYwV r8"ZB)fڝmNMKlPAcIʚ 2;8o0+=YnZ&ɓ4XPcG/5_Qo-~{b&ݫR7D>//or=q07:+BerPK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!-drs/downrev.xmlPKH$ >  When you get a result, like the one on the previous slide, that is obviously true for any value of the replacements for the variables, this indicates that the two equations actually represent the same line. There are an infinite number of solutions for this system. Any solution of one equation would automatically be a solution of the other equation. This represents a consistent system and the linear equations are dependent equations.I }2  ) l  xP9h0e0eRectangle 6 `   ]The Substitution Method   c l dpAh-B??Text Box 700 D<4___PPT9 Example continued*( " d2   -B0fBYt  ___PPT10 .+DH ' = @B D ' = @BA?%,( < +O%,( < +D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*l%(D' =-s6Bwipe(left)*<3<*lD' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*la%(D' =-s6Bwipe(left)*<3<*laD' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*la%(D' =-s6Bwipe(left)*<3<*la+8+0+l +"F; p(  p  p  r06h0e0eRectangle 2"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!2.drs/downrev.xmlDOO1G&~fLIk  1s\pnunƃ'o~y`;vw Gkj}7:wB9ڟ]IMjXX6<-Y#ߓŔpRGnѸbO*zv~HCy=+u{3<̀%L7[x:F0f|PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!2.drs/downrev.xmlPKP$ > bZ___PPT9<4 `0Solve the following system of equations using the substitution method. 3x  y = 4 and 6x  2y = 4 Solve the first equation for y. 3x  y = 4  y =  3x + 4 (subtract 3x from both sides) y = 3x  4 (multiply both sides by  1) Substitute this value for y into the second equation. 6x  2y = 4 6x  2(3x  4) = 4 (replace y with the result from the first equation) 6x  6x + 8 = 4 (use distributive property) 8 = 4 (simplify the left side)G " t I } t6I } tE$ &                  3                    ) p  x@h0e0eRectangle 6 `   ]The Substitution Method   Y p dVh-B??Text Box 7D<4___PPT9 yExample*( " d2  y p fBh??Rectangle 8#"@PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!drs/downrev.xmlDN0 @ߑHL[J"VM\4$.LMRtkz,:ֱb569Y S+HBDu$`<=Y`谏u B!a:* F}-#M+4 vtPlӫXw;R 7אDmUmd<4___PPT9 z Continued.( " d2     -B0fBYt%%___PPT10%.+BD$' = @B D$' = @BA?%,( < +O%,( < +D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*pG%(D' =-s6Bwipe(left)*<3<*pGD' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*pGi%(D' =-s6Bwipe(left)*<3<*pGiD' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*pi%(D' =-s6Bwipe(left)*<3<*piD' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*p%(D' =-s6Bwipe(left)*<3<*pD' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*p%(D' =-s6Bwipe(left)*<3<*pD' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*p%(D' =-s6Bwipe(left)*<3<*pD' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*pG%(D' =-s6Bwipe(left)*<3<*pGD' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*pG_%(D' =-s6Bwipe(left)*<3<*pG_D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*p_%(D' =-s6Bwipe(left)*<3<*p_D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*p%(D' =-s6Bwipe(left)*<3<*pD ' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*p%(D' =-s6Bwipe(left)*<3<*pD' = @B%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*p%(+p+0+p ++0+p +" ~ v t^ (  t t  r0;h0e0eRectangle 2"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!>~Hdrs/downrev.xmlDAO1F&fLIY)h9.KB am׶¯ɛ/o2LvVH#[;m#a]oVa,I8QbrG[a$6(/8uKd9o07\y<&x&D jZ饥k7:[㹬ބr>R^_ O" {zU*dxρתKH7qPK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!>~Hdrs/downrev.xmlPK0H$ >  RWhen you get a result, like the one on the previous slide, that is never true for any value of the replacements for the variables, this indicates that the two equations actually are parallel and never intersect. There is no solution to this system. This represents an inconsistent system, even though the linear equations are independent.SI }2SS  ) t  x0h0e0eRectangle 6 `   ]The Substitution Method   c t dBh-B??Text Box 700 D<4___PPT9 Example continued*( " d2   -B0fBYt  ___PPT10 .+DH ' = @B D ' = @BA?%,( < +O%,( < +D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*t%(D' =-s6Bwipe(left)*<3<*tD' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*t%(D' =-s6Bwipe(left)*<3<*tD' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*tS%(D' =-s6Bwipe(left)*<3<*tS+8+0+t +")x ,$x (  x> x ^PWh??Text Box 20@PD<4___PPT9 d$ " d2   x c PMh0e0e5%Rectangle 4"@ 00    2Solving Systems of Linear Equations by Elimination3 363    -B0fBYty___PPT10Y+D=' = @B +' |(  |" |  rEh0e0eRectangle 2 `   \The Elimination Method   k |  r;h0e0eRectangle 3"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!f:drs/downrev.xmlDN0EH5H쨓  u+*eiܷ=i;IlgƐNqzE\2iǑzjV{-`5~KdzXƖdj)6{=m5@N@*LPK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!f:drs/downrev.xmlPK H$ >  Another method that can be used to solve systems of equations is called the addition, arithmetic or elimination method. You multiply both equations by numbers that will allow you to combine the two equations and eliminate one of the variables.\I }2L|   -B0fBYt___PPT10.+^ D2' = @B D' = @BA?%,( < +O%,( < +D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*|x%(D' =-s6Bwipe(left)*<3<*|xD' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*|x%(D' =-s6Bwipe(left)*<3<*|x+8+0+| +"L2 B:"(      rP4h0e0eRectangle 2"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!Mdrs/downrev.xmlD1O0FwXuPB@)eKӅ_؎lMX t<^H+k܎`d*+l#0Hl!yFS̥=P(1>GM]ιǶ#:!)u$ɸFeB4T}ZV)DOwa՗׫W@}8?_gT2Y0[[d> cogPK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!Mdrs/downrev.xmlPKpH$ >  Solve the following system of equations using the elimination method. 6x  3y =  3 and 4x + 5y =  9 Multiply both sides of the first equation by 5 and the second equation by 3. First equation, 5(6x  3y) = 5( 3) 30x  15y =  15 (use the distributive property) Second equation, 3(4x + 5y) = 3( 9) 12x + 15y =  27 (use the distributive property)F % MI } FM   {  (   x4h0e0eRectangle 6 `   \The Elimination Method   Y  d0Oh-B??Text Box 73D<4___PPT9 yExample*( " d2  x  f6h??Rectangle 8#"@PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!`idrs/downrev.xmlDN0 @ߑHL,*&. 6ӸmD$ZxuX{8Vp>@WNn.AsL & Z-[b#C RPd1]OXŘFH䶓yҢŞn[>wU0Գ}~z~/̝R'Hc?xUZ"/S?L-H^AڤA.PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!`idrs/downrev.xmlPKJp$ ><4___PPT9 z Continued.( " d2     -B0fBYt___PPT10b.+4D' = @B D' = @BA?%,( < +O%,( < +D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*F%(D' =-s6Bwipe(left)*<3<*FD' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*Fk%(D' =-s6Bwipe(left)*<3<*FkD' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*k%(D' =-s6Bwipe(left)*<3<*kD' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*%(D' =-s6Bwipe(left)*<3<*D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*%(D' =-s6Bwipe(left)*<3<*D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*%(D' =-s6Bwipe(left)*<3<*D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*,%(D' =-s6Bwipe(left)*<3<*,D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*,B%(D' =-s6Bwipe(left)*<3<*,BD ' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*B{%(D' =-s6Bwipe(left)*<3<*B{D' = @B%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*%(+p+0+ ++0+ +"# (     rCh0e0eRectangle 2"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!e>drs/downrev.xmlDMO0 @HHXV,0>]w5^$Um-8Zz[F''A;+a>K!!8m+a[! Y0Qjrg[iۄ%6(q(MG d7yP,7Ҵ QsζSoyXs_/RތODtpyz-Ys~>VH^r/s_PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!e>drs/downrev.xmlPKp H$ >  dCombine the two resulting equations (eliminating the variable y). 30x  15y =  15 12x + 15y =  27 42x =  42 x =  1 (divide both sides by 42)BI }Zq  Z>    (   xMh0e0eRectangle 6 `   \The Elimination Method   c  dOh-B??Text Box 700 D<4___PPT9 Example continued*( " d2  y  fpFh??Rectangle 8#"@PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!tI޳drs/downrev.xmlDN0EH5Hl* BnC"P 0ē"m,Ggtl1NX4A\9mQ<"1))b~x0R=iH%*hcK)CՒ0u=qbc:}#}NYVHB=ݶT}nVӫ\[)u|4\_4܌Wҩ%/ΊD0~x"y0&rPK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!tI޳drs/downrev.xmlPKJp$><4___PPT9 z Continued.( " d2     -B0fBYt2*___PPT10 .+CeDr' = @B D-' = @BA?%,( < +O%,( < +D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*B%(D' =-s6Bwipe(left)*<3<*BD' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*BW%(D' =-s6Bwipe(left)*<3<*BWD' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*Wl%(D' =-s6Bwipe(left)*<3<*WlD' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*l%(D' =-s6Bwipe(left)*<3<*lD ' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*%(D' =-s6Bwipe(left)*<3<*D' = @B%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*%(+p+0+ ++0+ +"+  (      rP>h0e0eRectangle 2"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!*{Qdrs/downrev.xmlD?o0GJUb+NhBXvdku(Oot.伲F8FRZ<΀Fbk n) 5sЄ!~l;2j.Q[&I5*hP;k(ϡ<^= /s`p{^o{?S}ؒfl;_N} ' ȁ/PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!*{Qdrs/downrev.xmlPKp0H$ >  Substitute the value for x into one of the original equations. 6x  3y =  3 6( 1)  3y =  3 (replace the x value in the first equation)  6  3y =  3 (simplify the left side)  3y =  3 + 6 = 3 (add 6 to both sides and simplify) y =  1 (divide both sides by  3) Our computations have produced the point ( 1,  1).?I } 3I }%     " 3o  (   xpPh0e0eRectangle 6 `   \The Elimination Method   c  d8h-B??Text Box 700 D<4___PPT9 Example continued*( " d2  y  f0Jh??Rectangle 8#"@PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!j[drs/downrev.xmlDN0 @ߑHL[J5*TM\40mqۈ)I|=hgLymY,Js-m=de0dGa$J( +eCvđU qtPQnZi. j计s }5yݘXw=OD}<4___PPT9 z Continued.( " d2     -B0fBYt^V___PPT106.+50D' = @B DY' = @BA?%,( < +O%,( < +D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*?%(D' =-s6Bwipe(left)*<3<*?D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*?S%(D' =-s6Bwipe(left)*<3<*?SD' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*S%(D' =-s6Bwipe(left)*<3<*SD' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*%(D' =-s6Bwipe(left)*<3<*D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*%(D' =-s6Bwipe(left)*<3<*D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*<%(D' =-s6Bwipe(left)*<3<*<D ' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*<o%(D' =-s6Bwipe(left)*<3<*<oD' = @B%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*%(+p+0+ ++0+ +"%   0 (     rQh0e0eRectangle 2"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!bdrs/downrev.xmlDN0H}k+qN ԭ8mmۑ)OYͧYFӳ#0e6NiJ-*% g ZN.X(w%7eIbCtd0@6c:}˕SY&AmB=v|mFV5~sfm')/=Hc߃U\!^[U! So?PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!bdrs/downrev.xmlPKpH$ >  4Check the point in the original equations. First equation, 6x  3y =  3 6( 1)  3( 1) =  3 true Second equation, 4x + 5y =  9 4( 1) + 5( 1) =  9 true The solution of the system is ( 1,  1)..+I } (I }+    (  (   x8h0e0eRectangle 6 `   \The Elimination Method   c  d?h-B??Text Box 700 D<4___PPT9 Example continued*( " d2   -B0fBYt>6___PPT10.+D' = @B Dq' = @BA?%,( < +O%,( < +D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*+%(D' =-s6Bwipe(left)*<3<*+D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*+=%(D' =-s6Bwipe(left)*<3<*+=D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*=V%(D' =-s6Bwipe(left)*<3<*=VD' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*Vw%(D' =-s6Bwipe(left)*<3<*VwD' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*w%(D' =-s6Bwipe(left)*<3<*wD' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*%(D' =-s6Bwipe(left)*<3<*D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*%(D' =-s6Bwipe(left)*<3<*D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*%(D' =-s6Bwipe(left)*<3<*+8+0+ +"  B : @" (     rWh0e0eRectangle 2"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!t<=drs/downrev.xmlDMO0 @HHXJ*-h|Hnn㵁&lXh?YoMI )m[ j}sIhΒ3X-//X(w%7MXbCBtd0L@yG rӋ,MsaP[@5_Ple.a$0$?_ի,_εתK 2PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!t<=drs/downrev.xmlPK``$ > NF___PPT9(  Solving a System of Linear Equations by the Addition or Elimination Method Rewrite each equation in standard form, eliminating fraction coefficients. If necessary, multiply one or both equations by a number so that the coefficients of a chosen variable are opposites. Add the equations. Find the value of one variable by solving equation from step 3. Find the value of the second variable by substituting the value found in step 4 into either original equation. Check the proposed solution in the original equations.@K Z tZK   (   x9h0e0eRectangle 3 `   \The Elimination Method    -B0fBYt( ___PPT10.+D' = @B D[' = @BA?%,( < +O%,( < +D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*K%(D' =-s6Bwipe(left)*<3<*KD' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*K%(D' =-s6Bwipe(left)*<3<*KD' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<* %(D' =-s6Bwipe(left)*<3<* D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<* %(D' =-s6Bwipe(left)*<3<* D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*_%(D' =-s6Bwipe(left)*<3<*_D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*_%(D' =-s6Bwipe(left)*<3<*_D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*%(D' =-s6Bwipe(left)*<3<*+8+0+ +"% JBP*(     r3h0e0eRectangle 2"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!Įdrs/downrev.xmlD1o0FJUb+im J c ؎l_xzMfn؉W`dJ+l0Hl!0>?M0lr:mBŢ !羬IۖLd4xK( Od5*jli^S9j{n?\uWw !z/X.<rwJƖt8 ށ~/{d> co7PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!Įdrs/downrev.xmlPK 7H$ >  ESolve the following system of equations using the elimination method. F% FF    HA ??Object 3"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!OWdrs/downrev.xmlDOMk@ zӍ"Т =xfIhv6dW}/׽ŕZWYV0E s+.|~s#k-_r^= j񉮙/Da&%t#۶}m!u]7(+ %6-).F+'Y%s<.Y+o^AxC|w0?NfrPK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!OWdrs/downrev.xmlPK@ 0 8 $D >  ^`h??Text Box 4"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!Akdrs/downrev.xmlDOK1G!Elٖib*ot$K7x㍧i؉| 3`d+'Jl% 0\_l7tƚ% % P1%Rd0]K6c:}ͥsYVpڦ-=)F@;UٜBt`>j)SK^ rZn0DRaMPK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!Akdrs/downrev.xmlPK P`p$ ><4___PPT9 jFirst multiply both sides of the equations by a number that will clear the fractions out of the equations.$k " }kk  (   x0hh0e0eRectangle 8 `   \The Elimination Method   [  feh-B??Text Box 10=D<4___PPT9 yExample*( " d2  |  hPkh??Rectangle 11#"@PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!*,drs/downrev.xmlDN0EH5Hl*DB݊ E>'E<&|}-ѹ:Eo[#c qZ D[Ǥ` &"I8+ eCuĉU[In[gDZ4计k ue_H?^ӓD>?fxzN-d:/@Tç7z!W So PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!*,drs/downrev.xmlPKJp$><4___PPT9 z Continued.( " d2     -B0fBYt1 ) ___PPT10 .+.~D9 ' = @B D ' = @BA?%,( < +O%,( < +D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*F%(D' =-s6Bwipe(left)*<3<*FD' = @B%(D' = @B%(D7' =4@BBBB%(D' =1:Bvisible*o3>+#.<*%(D' =-s6Bwipe(left)*<3<*D ' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*k%(D' =-s6Bwipe(left)*<3<*kD' = @B%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*%(++0+ ++0+ ++0+ +"j4   `  (     r\h0e0eRectangle 2"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!TlRdrs/downrev.xmlDMO0 @HHXJ'-h|\۸mFTIuz,p\ONvV$l픶]C"Z$B֫%ʝmAmlؐ.!" 7e8o0[  hMultiply both sides of each equation by 12. (Note: you don t have to multiply each equation by the same number, but in this case it will be convenient to do so.) First equation,6I }I }    HA ??Object 6"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!C0drs/downrev.xmlDOk0?76UR2&x[y64ŽzR kr91)mv7kjCRʐE?vqή"k{p4I2i`CեMprJ͉7EJ>@C{|6My("PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!C0drs/downrev.xmlPK P 8 $D >w R F  C (Group 7#"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!;5hdrs/downrev.xmlDOj@-tV%uZ\Jw5 f̘;y/VDK+-+shydeR0i=\v)*(S)]VA75q.1lrB$`ɡ6e(['g^6Iח~SN|ߡ@.PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!;5hdrs/downrev.xmlPK  ,$D >   HA ??Object 8R F  m   ^gh??Text Box 9P  D<4___PPT9 (multiply both sides by 12)( " d2  y z @  C *Group 10#"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!+Bdrs/downrev.xmlDOj@ p I:.B*.k i)\{L-:j]eYA<@VW\(ߞV GX[&#9n5I"KQA}J mwA`[HbM-QHڗN?F~vݏ痏,&+O:̟'"CPK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!+Bdrs/downrev.xmlPK{ z@u,$D >   JA ??Object 11z r  i   `0mh??Text Box 12 @ D<4___PPT9 (simplify both sides)( " d2  *   zPah0e0eRectangle 13 `   \The Elimination Method   e  f0^h-B??Text Box 1400 D<4___PPT9 Example continued*( " d2  |  hlh??Rectangle 15#"@PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK! drs/downrev.xmlDJ1F!M.ijht7I ^pq6@> & qJb"DdcR0PdvGa+$ TPR&aZ[IngLZ4jl鶦sY~Orv<)u~__ Wѩ%/+H7qPK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-! drs/downrev.xmlPKJp$><4___PPT9 z Continued.( " d2     -B0fBYt ___PPT10.+0DK' = @B D' = @BA?%,( < +O%,( < +D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*%(D' =-s6Bwipe(left)*<3<*D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*%(D' =-s6Bwipe(left)*<3<*D' = @B%(D' = @B%(D7' =4@BBBB%(D' =1:Bvisible*o3>+#.<*%(D' =-s6Bwipe(left)*<3<*D' = @B%(D' = @B%(D7' =4@BBBB%(D' =1:Bvisible*o3>+#.<*%(D' =-s6Bwipe(left)*<3<*D' = @B%(D' = @B%(D7' =4@BBBB%(D' =1:Bvisible*o3>+#.<*%(D' =-s6Bwipe(left)*<3<*D' = @B%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*%(+p+0+ ++0+ +"F &&p &(  !   r^h0e0eRectangle 2"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!'drs/downrev.xmlDN0HHܨC u+*?'nxb;8q54M=׋ δh$֐  FCombine the two equations. 8x + 3y =  18 6x  3y =  24 14x =  42 x =  3 (divide both sides by 14)I }$I }N    l  ^ah??Text Box 6#"@PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK![8{ drs/downrev.xmlDAN0EH$vԡ@ӺTB)bA84Ɓ؎lIoψ,Goj3^); ε >?^AĄNc)Pj\VBg~ gh8re9^08P]Ǧ>}v٫R7D)?/PW2yX^DAr/sPK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-![8{ drs/downrev.xmlPK @p$ ><4___PPT9 xSecond equation,  " }    HA ??Object 7"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!cdrs/downrev.xmlDOKk@ zӍR zhoCv4;jw ǏXQ'Bx"ν4p\g [l Y!Hilj#ΡDvѓ$Úc-V˳մؾ7<=/Ϡzo'|NPDWPK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!cdrs/downrev.xmlPKDW@ 8 $D >w /S03  C (Group 8#"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!]/drs/downrev.xmlDOMk@ zXUDjA B6d$ 5s(x|jt g4E\x[si}|"3RnRI T1֡aXF]mFϒ$k [TTgcais=~w)p?@EM2_2Y,^PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!]/drs/downrev.xmlPKN/0. ,$D >   HA ??Object 9/Sh 3  o   `oh??Text Box 10 0D<4___PPT9 (multiply both sides by 12)( " d2  x   0  C *Group 11#"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!bodrs/downrev.xmlDOMk@)xMFWŃ6d$ 5{L)]aYA<@V);~?A8L `|[`m˿|&BW.ɠ؊8p[֙5!ܔrEiАcEf|خG_/89cRѭ i   `С??Text Box 12 0 D<4___PPT9 (simplify both sides)( " d2     JA ??Object 13   *   zmh0e0eRectangle 14 `   \The Elimination Method   e  fP-B??Text Box 150 {D<4___PPT9 Example continued*( " d2  |  hP??Rectangle 16#"@PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!dadrs/downrev.xmlD_K0G~p_K-i]626 k[_=^8E'5& 02T>W7|@# a1?0`6߆E9 Chs}QF?-J4xK( O$ 5PSvZ@WΌy{8w|2t+o} ' ȁϏPK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!dadrs/downrev.xmlPKJp$ ><4___PPT9 z Continued.( " d2     -B0fBYt___PPT10s.+I7D' = @B D^' = @BA?%,( < +O%,( < +D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*%(D' =-s6Bwipe(left)*<3<*D' = @B%(D' = @B%(D7' =4@BBBB%(D' =1:Bvisible*o3>+#.<*%(D' =-s6Bwipe(left)*<3<*D' = @B%(D' = @B%(D7' =4@BBBB%(D' =1:Bvisible*o3>+#.<*%(D' =-s6Bwipe(left)*<3<*D' = @B%(D' = @B%(D7' =4@BBBB%(D' =1:Bvisible*o3>+#.<*%(D' =-s6Bwipe(left)*<3<*D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*%(D' =-s6Bwipe(left)*<3<*D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*-%(D' =-s6Bwipe(left)*<3<*-D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*-?%(D' =-s6Bwipe(left)*<3<*-?D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*?S%(D' =-s6Bwipe(left)*<3<*?SD ' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*S%(D' =-s6Bwipe(left)*<3<*SD' = @B%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<* %(++0+ ++0+ ++0+  +"    (     rQh0e0eRectangle 2@Pp`  Substitute the value for x into one of the original equations. 8x + 3y =  18 8( 3) + 3y =  18  24 + 3y =  18 3y =  18 + 24 = 6 y = 2 Our computations have produced the point ( 3, 2).PI }G 2I }) 2  (   x]h0e0eRectangle 6 `   \The Elimination Method   c  dXh-B??Text Box 700 D<4___PPT9 Example continued*( " d2  y  fhh??Rectangle 8#"@PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!6drs/downrev.xmlDN0EH5Hl* nC"P 0ē"i,Ggt|9VX4A\:mV:"1))rqx0B=ohH*hb )Cِ0uqbc:}-}VYv.-N vtPӫ\u=Od}e:>@DMn?ZԒϲez? TzPK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!6drs/downrev.xmlPKJp$><4___PPT9 z Continued.( " d2     -B0fBYt___PPT10+6GxD' = @B DD' = @BA?%,( < +O%,( < +D{' = @B%(%(D' = @B%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*%(+8+0+ +"UZ 9 98(     ry0e0eRectangle 2"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!H&drs/downrev.xmlDMO0 @HHX"M||S]v 4IIcqgr4}r I )m[ 뺼B"Z$D9mEUlP.ơ"4 7es`ѷBy<"OaP[@595wu|Iˋ~I1W-m6͸grz* p/sPK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!H&drs/downrev.xmlPKPH$ >  Check the point in the original equations. (Note: Here you should use the original equations before any modifications, to detect any computational errors that you might have made.)I }  t  ^py??Text Box 6"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!rdrs/downrev.xmlDOk1G~0B/fwVVJz6nM&Y]}Cqx4J>hg Imk1Jl%7 0M{b͒ĆےP)2%y1cE=sڦ-)F@{,Fˣ֪ٞ֫x^,f'à_LEZuUdj)^q>v{ Tz>PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!rdrs/downrev.xmlPKVvp$ ><4___PPT9 First equation,, " }Z     HA ??Object 7"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!bCKdrs/downrev.xmlDO]k0}66Ln륹k˒dZ>'}X:±\+<@8&P,F ̵;'P>.'#܏-Z1%ɋذ׆*ʰ߷Nq<  .wouMY C\^PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!bCKdrs/downrev.xmlPK(n 8 $D >  HA ??Object 8"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!ddrs/downrev.xmlDO]k0}MZ3ۘ/\۲6{g\kYA:K@WV\+8|O GY&`z,v1] RJW5dlO Zn:%I! 鵡wgl$ٍySqڼ>r)^Ax .?uEZPD WPK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!ddrs/downrev.xmlPK  8 $D >v k \ C (Group 9#"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!_drs/downrev.xmlDO]k0}|4SQDLcodmF|  JA ??Object 10k(\8 $D >Z  `0y??Text Box 11 0 PD<4___PPT9 vtrue* " d2  v  `x??Text Box 12"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!9Sdrs/downrev.xmlDMO1@&͘x!ҥ1JV 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Dl%u""mq!Pdn2$}   JA ??Object 16I8 $D >Z  `P y??Text Box 17  D<4___PPT9 vtrue* " d2    `x??Text Box 18"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!jdrs/downrev.xmlDOK1G!ElԲ6-"Vxnf$~{=ox?dֻV)DYpP (x>ݎAĄL a6`tܤFd JXr#ÔH宕eQCy`GͷSPnʻZӾ},~+D}:q9qx>k\{39PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!jdrs/downrev.xmlPKPzp$  ><4___PPT9 H The solution is the point ( 3, 2). % " }2%%  *   z y0e0eRectangle 19 `   \The Elimination Method   e  fx-B??Text Box 2000 D<4___PPT9 Example continued*( " d2   -B0fBYt  ___PPT10 .+  D' = @B DO' = @BA?%,( < +O%,( < +D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*%(D' =-s6Bwipe(left)*<3<*D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*%(D' =-s6Bwipe(left)*<3<*D' = @B%(D' = @B%(D7' =4@BBBB%(D' =1:Bvisible*o3>+#.<*%(D' =-s6Bwipe(left)*<3<*D' = @B%(D' = @B%(D7' =4@BBBB%(D' =1:Bvisible*o3>+#.<*%(D' =-s6Bwipe(left)*<3<*D' = @B%(D' = @B%(D7' =4@BBBB%(D' =1:Bvisible*o3>+#.<*%(D' =-s6Bwipe(left)*<3<*D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<*%(D' =-s6Bwipe(left)*<3<*D' = @B%(D' = @B%(D7' =4@BBBB%(D' =1:Bvisible*o3>+#.<*%(D' =-s6Bwipe(left)*<3<*D' = @B%(D' = @B%(D7' =4@BBBB%(D' =1:Bvisible*o3>+#.<* %(D' =-s6Bwipe(left)*<3<* D' = @B%(D' = @B%(D7' =4@BBBB%(D' =1:Bvisible*o3>+#.<* %(D' =-s6Bwipe(left)*<3<* D' = @B%(D' = @B%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+#.<* %%(D' =-s6Bwipe(left)*<3<* %++0+ ++0+ ++0+ ++0+  +" (  F   rpx0e0eRectangle 2"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!+mBdrs/downrev.xmlD1O0Fw$uHl&Em uP4]خ51vdm¯bNӛ/{Ӳ3p?VNi[Kؗ*l% X.+wwfIbCP5d0\G6c:}͕KgBLAmBv'#ᠳ=~0q|dz7 X>?X_VlkU`%ԛ8PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!+mBdrs/downrev.xmlPK``$ > JB___PPT9$ In a 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՜.+,0X    On-screen Show (4:3)< I' 0Times New RomanArial Wingdings Arial NarrowSymbol Martin Gay 1_Martin GayMicrosoft Equation 3.0MathType 5.0 EquationSystems of EquationsPowerPoint PresentationSystems of Linear EquationsSolution of a SystemSolution of a SystemFinding a Solution by GraphingFinding a Solution by GraphingFinding a Solution by GraphingFinding a Solution by GraphingFinding a Solution by GraphingFinding a Solution by GraphingFinding a Solution by GraphingTypes of SystemsTypes of SystemsTypes of SystemsTypes of SystemsTypes of SystemsPowerPoint PresentationThe Substitution MethodThe Substitution MethodThe Substitution MethodThe Substitution MethodThe Substitution MethodThe Substitution MethodThe Substitution MethodThe Substitution MethodPowerPoint PresentationThe Elimination MethodThe Elimination MethodThe Elimination MethodThe Elimination MethodThe Elimination MethodThe Elimination MethodThe Elimination MethodThe Elimination MethodThe Elimination MethodThe Elimination 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