' +JJJJ ?\>m0M='+l> /+l   d]@ŵLҦ]]LF L}BBL] X  ` 鷎귭෍ᷩ췩緈JJJJx Lȿ L8ᷭ緍췩 緍i 8 `巬 췌`x (`(8`I`B` ``>J>J>VU)?`8'x0|&HhHh VY)'&Y)xꪽ)' `Hh`V0^*^*>&` aI꽌ɪVɭ&Y&&Y& 꽌ɪ\8`&&꽌ɪɖ'*&%&,E'зЮ꽌ɪФ`+*xS&x'8*3Ixix&& 8  '  & x)*++`FG8`0($ p,&"ųųೳŪŪųųij  !"#$%&'()*+,-./0123456789:;<=>?   1 '" *"( (9"1 ( ,.(0# 2  /#0/#0 *?'#07#00/0/'#07#0:"4<*55/**5/*%5/)1/)1/)1/)'#0/#0*5/*75/**5/*:5//#0/#0'#07#0:::*::'#07#0"):$(%"%:$(%"%$$2%4%$$2%4%$(2()!)E(!8b $!H(+ "@H !D)"E` @ $ C ` DQ &J80^݌Hh ü ü݌ ռ ռ ռA ļD ļ? ļAEDE?HJ>h Լ ռ ռ ռ`HJ>݌h Hh݌`葠葠ȔЖȔЖȠHIHHHHhHH݌hHhHh݌H6 VDP (ED Z $0x˵*̵+뷮浭õL H8`BĵC µLµBõĵܭɵBʵCȱBȱB BȱBȌµõĵĵµ ˵̵ LLҦ  `   LDcpq` [` ~  LӜu`".Q`pNФbptťܥm2<(-Py0\|e<6e<g< JJJJj귍hI  aUL@ kU8 L8f֭lX Q^R(jQ0l^l\  wUuW ԧ H h@ [_ /QSIRb_L`LLLL`ª`LQLYLeLXLeLee ўQH\(h0L& Ꝥ$`( R \ZLl8 ўR HH\`\Z[YS6`LxQɿu3'RͲʎRʎ]]]ɍuL͟ɍ}RLRɍg^H8 ^hZLɍR LͲɊRR% QLܤͲ Z@ -^ ş\[Z QY\[Z8`l6Lş_Ȍb_Ͳ] )Y h( ֭ͲLɍ [LLĦ__ ^ 9 LҦ3 9 a   0LjLY u< (_9 ˭ɠuɠK_9 ?LˆʎõĵL徭 õ ĵµ aµ`` L̦µ_bJLuLz`  ȟ QLȝJ̥KlV  ȟ QlV eօ3L e3L &RL &QL d L4 Ne)n `@-eff L f`L . tQLѤ LҦL` OPu d L Ne)noon 8ɍ` ^f\õL ^NR  RΩLҦ)\Z ʽ LHv 3h`0h8` [L NС õ`A@` ŵL^Lõ`  \ 濭0 \  ȟ Q ^\lZl^?cqH şch`fhjõĵ@OAP`u@`@&`QR`E Ls  @DAE@u`8` %@ @A@`@`@A`Mµ ) LЦ`8@AWc@8@-@HAȑ@hHȑ@ȑ@hHȑ@Ȋ@ch8&ȑ@Hȑ@Ah@LHȑ@ȑ@ htphso`hMhL`9V8U897T6S67`INILOASAVRUCHAIDELETLOCUNLOCCLOSREAEXEWRITPOSITIOOPEAPPENRENAMCATALOMONOMOPRINMAXFILEFINBSAVBLOABRUVERIF!pppp p p p p`" t""#x"p0p@p@@@p@!y q q p@  LANGUAGE NOT AVAILABLRANGE ERROWRITE PROTECTEEND OF DATFILE NOT FOUNVOLUME MISMATCI/O ERRODISK FULFILE LOCKESYNTAX ERRONO BUFFERS AVAILABLFILE TYPE MISMATCPROGRAM TOO LARGNOT DIRECT COMMANč$3>L[dmx- (  yխŠ@跻~!Wo*9~~~~ɬƬ~_ j ʪHɪH`Lc (L ܫ㵮赎 ɱ^_ J QL_Ls贩紎 DǴҵԵƴѵӵµȴ 7 ַ :ŵƴѵǴҵȴµ納贍﵎ٵ്ᵭⳍڵL^ѵ-I `  4 ò-յ!  8صٵ紭ﵝ 7L (0+BC  7L HH`LgL{0 HH` õL H hBL BH [ h`Lo õ ڬL B ڬ LʬH hB@ յյ [L (ȴ) ȴ 7L L ( L (ȴL{ƴѵ洩ƴǴҵ 7 ^* B0 HȱBh ӵԵ 8 L8 ݲ` ܫ  / / ED B / / ]ƴS0Jȴ ȴ)  紅D贅E B ƴ  / 0L Ν `HD٤DEEhiHLGh ` ŵBѵ-` ѵB-` ܫ XI볩쳢8 DH E𳈈췍Ȍ X0 · JLǵBȵC`,յp` 䯩 R-յյ`յ0` K R-յյ`ɵʵӵԵ` 4 K ( ѵҵLBȱBL8` DBHBH : ַ޵BȭߵBhhӵԵ RBܵmڵ޵ȱBݵm۵ߵ` 䯩LR˵̵ֵ׵`êĪLR E( 8` R` ELRŪƪ`췌 յյI뷭鷭귭ⵍ㵍跬ª 뷰` Lf ݵܵߵ޵ ^`8ܵ i B8` 4L ֵȱB׵ ܯ䵍൭嵍 ` DȑB׵Bֵ  ַ յյ`굎뵎쵬 뵎쵌``õĵBCõĵ`µµ`L õBĵCصص Qƴ0"Bƴ 󮜳` 0۰ϬBƴ8`i#`ЗLw!0>ﵭ` m ﳐ 7i볍 8 ЉLw`H h ݲL~ `浍국䵍뵩嵠Jm赍嵊mjnnn浈ۭm浍浭m䵍䵩m嵍`"L ŵ8ŵH ~(` d ֠Ġz# u`to check your work.":113 X,YX,TO,TO,YX,Y:E P167,169,55200:"@6V2H@The absolute value of any number is@D2H@either the number itself or its@D2H@additive inverse, whichever is@D2H@positive. The absolute value of 0@D2H@is 0.":Y107:T155:h each "I$"y in the@D6H@system on the same coordinate@D6H@plane using the same axes.@2D2H@<2> Select several equations" K"@6H@from the area where all of the@D6H@"I$"ies overlap.@2D2H@<3> Substitute the values from each@D6H@equation into each@D6H@"I$"y air of "I$"ies can be solved@D2H@by graphing the "I$"ies in the@D2H@same coordinate plane system and@D2H@observing the overlap of the@D2H@solution sets.":11w JX10:Y35:T155:O269:152:"@5V2H@To solve a system of "I$"ies@D2H@graphically:@2D2H@<1> Grap798,C:2:(4)"RUNAM4.5.1"3 GC3160:P72,74,55 HX10:Y35:T75:O269:152:Y83:T131:152:"@5V2H@The solution set of a system of@D2H@"I$"ies is the intersection of@D2H@the solution sets of the individual@D2H@"I$"ies in the system." I"@2D2H@A p47:P55 "@1V5H@"P:71- (S)155ıI 16368,0:"@40X40YN@";p K(S):K12819:K155K20521: 16368,0:23 :22 158 47:K155İ2:"RUNALGEBRA 4" 55 .31051:48 /30976 0"@LI21V1H@"19)"@RI@": 724n7C(24798):I$"inequalit":35339:P1:S16384:71O24798,C:0:1002: "@I3H21V@Press "(1)" Key to View the Next Page@D3H@Press "(2)" Key to View the Last Page@I@":16368,0 K(S)128:17:K21K812:K21PP1:46 K8PP1:     >>6>>"">"0666>aD1H@the range X "(94)" 0, use X in the@2D1H@sentence; for the range X < 0, use the@2D1H@additive inverse of X instead.@2D1H@Combine these two solution sets to@2D1H@solve the original sentence.":11X10:Y35:O269:T99:152152{"@4D2H@The absolute value of a number@D2H@represents the number line distance@D2H@between that number and 0.":11"@4V1H@To solve an open sentence which@2D1H@includes the absolute value of a@2D1H@variable X, solve two sentences: for""@ RY:X95:T155:Y51:120:YR:"@8V2H@Y"(Q3)A2"X+"F2"@14H@Y"(Q2)H"X+"F1:41:="@10V2H@<2>"A$(2)(123)"X,Y"(125)".":145:41:137:"@15V2H@<3>"A$(3):19>p"from zero on a@2D1H@number line (negative distance does@2D1H@not make sense, so the value $:37:"of the "I$".":7C<6:X5517:Y2A2XF2:Y25Y25R<Y25Y25m<Y1HXF1:Y15Y15|<Y15Y15<Q235Y2HXF1Q294Y2HXF1Q335Y1A2XF2Q394Y1A2XF2Ă:412<227X7,107Y18227X7,107Y28:<3:O=:X$"system":J$"overlap":K$"absolute value":M$"additive inverse":C(24798):L$"@I@":Y$(4):Z16384:Q16368:;(36251)6İ2:Y$"RUNALGEBRA 4";<40:"@2D2H@Similarly, a ";:T$X$" of "I$:37:"@2D2H@is solved by finding the@2D2H@";:T$Z"@I16V2H@true@14H@true"L$:7:JR(5)1R(2):YR(5)1R(2):W0(V35YHJF)(V94YHJF)(Q335YA2JF2)(Q394YA2JF2):W2162:I16:"@138C"32J"H"13Y"V@$@133CB@";:150:"$":150::s;Q$"inequality":W$"positive":P$"sentence"268)1:159F9"@I27H7V@WRONG, IS: "L$:400:A2Hĺ"@I37H7V@"(16)L$919:T5:B18:L27:R11:39::34:(36251)İ2:Y$"RUNALGEBRA 4":39:3000:"@11V2H@"Y(Q3)A2"$"J"+"F2"@14H@"Y(Q2)H"$"J"+"F1:41:"@14V2H@"Y(Q3)A2JF2"@14H@"Y(Q2)HJF1:41:LT5:117:G2S1:J2SQ:I2W1:T2B:FF1:AH:VQ2:117:X0:"@5V29H@Y"(Q3)A2"X+"F2"@D29H@Y"(V)H"X+"F1:21:TA0:G78156:Q335TA2OF2THOF1TA18Q394TA2OF2THOF1TA18G78AA2TA1 9TA1ĺ"@I27H7V@CORRECT "L$:36268,(36cludes the @D2H@"E$" for the @D2H@"X$" and press P. @D2H@Press N to indicate @D2H@that the "E$" @D2H@is the empty set, "(16)". "L$:P195((E)0)7P1P1:33:114:121:117:NNR(2):V35:Q294:NN2V94:Q2358F2F:A2A:Q3V:FT5:155:120:X185:O269:120:"@I5V2H@The "O$"s of a@D2H@"X$" of "I$" @D2H@are graphed at the @D2H@right. Use the keys: @D2H@ D:Down R:Right @D2H@ U:Up"7)"L:Left @D2H@to move the cursor";7" to @D2H@the part of the graph @D2H@which in1,Y::3:U5YR(3)1R(2):JR(3)1R(2):V35JFAYV94JAYF146:145p5"@"32Y"H"13J"V@$":5T6:B10:L2:R17:38:T13:B15:385X10:Y35:O269:T99:1205X185:Y35:O269:T155:1205K150:11::6X10:Y35:O178:TI55.2:D192:A0V35A0V94D266g4AIF5AIF5ē227I7,107(AIF)8D,107(AIF)8r4:3:4I55.8:D64:V35D1524Y107(AIF)8:Y64Y644Y152Y1524DYĂ:3: 5227I7,D227I7,Y:227I71,D227I719:39:"@7V2H@<9> Pick a sample point@D6H@in the "J$" and@D6H@substitute the"3"@6H@values of the@D6H@"R$" into@D6H@each "Q$" to@D6H@check that each is@D6H@a true statement.@D6H@This verifies that@D6H@your work is@D6H@correct.":162:"@15C@":161*4.":40:140:"@15V2H@<6>"A$(3):19:R24:B18:39:"@7V2H@<7> If the graphs do not@D6H@"J$", the "X$"@D6H@"E$" is the@D6H@empty set "(16)".":A2HQ1Q37O3"@12V2H@<8> If the graphs do@D6H@"J$", the "J$"@D6H@portion is the@D6H@"E$" for the@D6H@"X$".":5:LT5:117:"@5V19HI@Y"(V)A"X+"F;L$"@7V2H@<1>"A$(1)(V)A"X+"F".":41:50001T7:B17:L2:R12:38:"@5V19H@Y"(V)A"X+"F";@I@Y"(Q2)H"X+"F1;L$:AH:FF1:41:5:117:"@7V2H@<4>"A$(1)(Q2)H"X+"F1:41:VQ2:145:"@10V2H@<5>"A$(2)(123);2"X,Y"(125)"Y":A$(2)" Select a value for@D6H@X and plot a true@D6H@value for the@D6H@"R$" ":A$(3)" Fill the graph on@D6H@the side of the@D6H@"O$" where@D6H@the sample point is."M1F2F:A2A:Q3V:"@5V2H@Solve the system Y"(V)A"X+"F";Y"(Q2)H"X+"F1:121:6:FT)IJ125A/|2:L,IR,I:1:L1,IX,I:IJR228RR1A:XX1AQ/}:3:121:W/~c/P443/X10:Y35:O269:T155:120:Y51:120:X185:120:114:V35:Q294:P2V94:Q235/P3V35:Q235/P4V94:Q2940A$(1)" Plot "O$"@D6H@for .y"@E27H13V@<@9F@>@5U6B@"(95)"@10DB@"(126)"@E@":191,107263,107:227,64227,151:I1992587:I,103I,111::I751418:224,I231,I::.zL227:L1L:RLW7:B107F83:XLW7:V35V60B152:L266:L1189 /{I67B:J107(AWF)8:(V35V60)1R(2):AHFF1AH(FF13FF13)114:AHıI-s(A)(H)114:w-uW0:IFTLT.1:XAIF:X5X5W0WI-vX5X5SI-w:W1227W7:B107(AWF)8:S1227S7:SQ107(ASF)8:W1,BS1,SQ:W11,BS11,SQ:-xX,YO,YO,TX,TX,Y:269,(36269)1:113:,p"@I5V29H@WRONG"L$" @D27H@"11),q"@29H6V@Solution @D29H@set is:":122:I01:"@E"32O(I)"H"13T(I)"V@$@E@"::19:T4:B18:L1:R19:38::34:164:(E)0151:434-rAR(3)1R(2):FR(3)1R(2):HR(3)1R(2):F1R(3"@D1H@ "(91)"ESC]:Cancel last point "L$:I2W1(W1227)2:T2B:G2S1(S1227)2:J2SQ:X01:21::AA:AN0:V62109:I01:ANAN(T(I)FA(O(I)))::111+mI01:T(I)FA(O(I))AN1+n,oAN1ĺ"@I5V29H@CORRECT"L$" @D27H@"11):AQ0113:362D1H@The "O$"s for "*k"@1H@both partial "N$" @D1H@sets are shown. Use the @D1H@keys shown below to show @D1H@2 points which are part @D1H@of the "E$". @D1H@ D:Down R:Right @D1H@ U:Up"6)"L:Left"6)"@D1H@ P: PLOT ";+l9)29H@Y"(V)"-"A"X+"FO)iI16:"@"12I"V2H@"16)::"@13V2H@"A$(1)"@D2H@"A$(5)*j"@5V30H@Y"(V)A"X+"F"@D29H@Y"(V)"-"A"X+"F:19:T7:L1:B19:R25:39:B18:L27:R11:39:121:FT0:LT5:117:FT5:LT0:AA:117:"@7V2H@<2> Plot the "N$"s@D5H@for both.@IH@"B$(I)::Q,0:K279000:98(gAQ0:"@5V28H@"10)"@"13G1"V2H@"16):8000:B$(G)A$(1)B$(L)A$(5)B$(L)A$(1)B$(G)A$(5)AQ1:"@I13V18H@CORRECT"L$:105)h"@I13V18H@WRONG";L$",@D18H@these@D18H@are the@D18H@ones to@D18H@use.@5V30H@Y"(V)A"X+"F"@DV2H@"16):98:'a"@13V@";:I16:"@2H@"B$(I)::Q,0:93U'b"@I2H"12G"V@"B$(G)L$'cK(Z)128:K27K8K21K1399:GG(K8)(K21)(K21G1L)(K8G1L):G1(L1)G6'dG6G1(L1)'eK13103:"@13V@";:I16:ILII1:(f"@2I120:DR(6):BR(6):T$B$(D):B$(D)B$(B):B$(B)T$::"@13V@";:I16:"@2H@"B$(I)::L1:Q,0{&]"@I2H"12L"V@"B$(L)L$&^K(Z)128:K8K21K1394:LL(K21)(K8):L1L6&_L6L1 '`K13ĺ"@5V28H@"(B$(L),7):G1(L1):Q,0:"@"13L1":A$(1)"X "V$" 0;Y"(V)A$"X+"F$:A$(2)"X "V$" 0;Y"(V)(A)"X+"F$%[A$(3)"X "V$" 0;Y"(V)A$"X-("F$")":A$(4)"X < 0;Y"(V)A$"X+"F$:A$(5)"X < 0;Y"(V)(A)"X+"F$:A$(6)"X < 0;Y"(V)(A)"X-("F$")"`&\I16:B$(I)A$(I)::7H@Use the "(2)","(1)"@D27H@keys to @D27H@light your @D27H@choice and @D27H@"(91)"RETURN] to@D27H@select it. @D27H@"(91)"ESC] will @D27H@cancel yourM%Z"@27H@choice. @D27H@"11)"@D27H@"11)L$:X10:Y83:O179:T155:120:Y99:120:F$(F):A$(A):FR(3)1R(2):V60:GR(2):VV2(G1):F2V60#XX185:Y35:O269:T155:120:"@4V1H@Plot the "E$"@D1H@for Y"(V)A;U$"X"U$"+"F".@2D1H@<1> Select two "P$"s@D5H@to plot for partial@D5H@"E$"s:@2D2H@RANGE;SENTENCE":7,52182,52$Y"@I7V27H@"11)"@D2T5:LT0"V117:AA:"@6V30H@Y"(V)"-"A"X+"F"@8D1H@<3> The "E$" for@D5H@the original "P$"@D5H@is the conjunction@D5H@(combination) of the@D5H@range-restricted@D5H@"P$" "N$"s.@I7V29H@Y"(V)A;U$"X"U$"+"F;L$:122:75#WP195((E)0):P1P1:33:AR(3)H5V@Y"(V)A"X+"F"@ID30H@Y"(V)"-"A"X+"F:41:"@I9V9H@X < 0" "U"@30H6V@Y"(V)"-"A"X+"F:41:"@I11V1H@<2> Plot the "N$"s@D5H@for both.":FT0:LT5:"@8V20H@X "V$" 0@D9H@X < 0@6V30H@Y"(V)"-"A"X+"F:117:"@5V30H@Y"(V)A"X+"F"@ID30H@Y"(V)"-"A"X+"F;L$:AA:FH@ Y"S$U$"2X"U$" @D27H@"11)L$:41:7\ QP443:149:121:FR(3)1R(2):AP:A4A1w SGR(2):V35:G1V94;!T"@4V1H@Plot the "E$"@D1H@for Y"(V)A;U$"X"U$"+"F:41:"@7V1H@<1> Identify "P$"s@D5H@for the ranges @I@X "V$" 0@IRD5H@and X < 0.@I3007:120J"@14V6H@X"V$"0@D5H@Y"S$"2X":1:I133153:12,I92,I::G227:I152641:G,I266,I:GG(I113)2::41:"@14V18H@X<0@D16H@Y"S$"-2X":2( KI133153:97,I177,I::G226:I152641:G,I187,I:GG(.5(I113))::3:121:"@I5V27H@"11)"@D27@D1H@the domain X < 0) substitutes its@2D1H@additive inverse.":7I121:X185:Y35:O269:T155:120:"@5V1H@In order to solve the@2D1H@"Q$" Y "S$" "U$"2X"U$" we@2D1H@really are solving these@2D1H@two "P$"s:":41:X10:Y107:O179:120:Y131:120:X94:Y1:2:189,64227,107:3:7G"@4V1H@Solving an "R$" or "Q$"@2D1H@that contains the "K$" of a@2D1H@variable X requires solving two@2D1H@"P$"s: one "P$" (valid for@2D1H@the domain X "V$" 0) uses the variable@2D1H@itself, while the second (valid for"AH"@2D1H@X's additive@2D1H@inverse, which@2D1H@is "W$".@9V20H@-X@11V17H@ 0@23H@ 0"F"@17H@ 1@23H@ 1@D17H@ 3@23H@ 3@D17H@ 5@23H@ 5@D17H@-1@I20H@ 1@23H@ 1@I12V31H@"H$:41:"@16V17H@-3@I20H@ 3@23H@ 3@I29H10V@"H$:41:"@17V17H@-5@I20H@ 5@23H@ 5@I8V27H@"H$:410:Y67:X136:O157:120:"@9V18H@X@24H@Y@I17H11V@ 0@23H@ 0":41:"@17H@ 1@23H@ 1@I33H@"H$:41:"@13V17HI@ 3@23H@ 3@I35H10V@"H$:41OE"@I14V17H@ 5@23H@ 5@8V37HI@"H$:41:1:266,64227,107:19:"@9V1H@But when X is @2D1H@negative, Y has@2D1H@the value of20220,136:7C149:121:"@4V1H@Consider what a graph@D1H@looks like when an@D1H@"K$" is in the@D1H@"R$": Y="U$"X"U$"@30H6V@Y="U$"X"U$:41:"@9V1H@Here, when X is@2D1H@"W$" (or@2D1H@zero), Y is@2D1H@equal to X."DX115:Y67:O179:T155:120:Y83:12"@D2H@For example, "U$"4"U$" = "U$"-4"U$" = 4@15V8H@<@18H@>@20H@<@30H@>@16V7H@"(126)"@19H@"(126)"@31H@"(126)"@15V13H@4@25H@4@I18V6H@-4@31H@4"L$ B52,12052,135:57,12391,123:98,123130,123:136,120136,136:141,123175,123:182,123214,123:220,1-3)@D11H@< 3 >@D10H@"(126)"@19H@"(126)@78,12398,123:105,123130,123:73,12073,135:136,120136,135:"@18V9HI@-3"L$:19:147:"@18V9H@ @2B@-3@5V2H@This means that the "K$"@2D2H@of any number Z is exactly the same@2D2H@as for its "M$", -Z."A20H@<@23H@3@27H@>@D19H@"(126)"@28H@"(126):136,120136,135:140,123161,123:193,123168,123:199,120199,136+?19:147:"@18V28H@ @B@3@6V2H@The "K$" of a negative@2D2H@number Y is -Y. So the absolute@2D2H@value of -3 ("U$"-3"U$") is also 3.@14V12H@-(6000=148:"@2H17V@<@34F@>@6V2H@The "K$" of any "W$"@2D2H@number X is X. For example, the@2D2H@"K$" of 3 ("U$"3"U$") is 3.":5:14,139259,139:I3124121:I,136I,144:I1,136I1,144::3o>"@18V3H@";:N55:N1(N2))::41:"@I18V28H@3@I15Vnot "J$", so the@2D1H@"E$" is the empty@2D1H@set, "(144)".":3:7[82:Y$"RUNAM4.5.1.1"n:M59,56,81,87;P60,61,67,71,73,43<"@6V1H@The ";:T$K$:37:"of a number is a@2D1H@"W$" value which indicates the@2D1H@number's ";:T$"distance":37:1:245,64203,152:245,66203,154:G245:I641522:189,IG,I:GG1:G203G2036:41:"@30H6V@Y "S$" 2X-1":5:252,64210,152:252,66210,154:G152:I2112644:I,GI,152:GG8:G64G64B7:41:"@9V16H@The graphs@2D1H@of the two "I$"@2D1H@do ,154245,66:543:121:189,64266,152:191,66268,154:191,66268,154:189,64266,152:6:G107:I147:227I.5,G227I.8,G:GG1::3:19R5L1:R19:T4:B18:38:"@5V1H@Here is a graph of@2D1H@another "X$" of@2D1H@"I$".@5V30H@Y "(94)" 2X":121:H@Y "S$" -X@8V1H@When we plot both of them@2D1H@together, the "J$"@2D1H@area is where both are@2D1H@true.@6V1H@"26):121:3:5340:"@14V8H@This is the@2D1H@"E$" for the@2D1H@"X$".":T7:B18:L26:R6:0:269,30269,155:38:121:1:204,152247,64:202e@2D1H@"X$" is the area where the@2D1H@"I$" in the "X$" "J$".":71121:3:"@5V5H@Here is the graph of:@30H@Y "S$" 2X":19:T4:B6:L2:R17:38:B18:L25:R7:38:121:5:"@6V5H@Here is the graph of:@30H@Y "S$" -X219:T4:38:"@5V30H@Y "S$" 2X@D30of "R$"s":37:"@2D2H@is solved by finding the@2D2H@";:T$Z$:37:"of the "R$"s.";:200A0"@6V1H@The "E$" for a "X$" of@2D1H@"I$" consists of all ordered@2D1H@pairs for which all of the@2D1H@"I$" in the "X$" are true.@2D1H@The graph of the "E$" for thISCUSSION@9V10H@<2> RULE@2D10H@<3> EXAMPLE@10H2D@<4> SAMPLE PROBLEM@10H3D@<0> Return To CONTENTS@6DI10H@WHICH (0-4) ??"L$:6:Kİ35:2:Y$"RUNAM4.5",MK:P1:33:34-C358:M46,56,127,151.P47,48,49,43Y/"@6V2H@We know that a ";:T$X$" 82:X(36)72:OX4(T$)7:TY12:T$" ";:120^&"@L@":XTB:X1:L1:R)::"@R@":|'XTB:X1:L1:R)::(I1225:11::)I1150:11::*35339:R(X)((1)X)1:(E)87+M0:P0:33:"@14H5V@LEARNING MODE@10H7V@<1> DO(I)"H"13T(I)"V@$":e M4G27ĺ"@"32O"H"13T"V@ ":XX1:X021:"@"32O(X)"H"13T(X)"V@ ":21m 22 O(X)O:T(X)T:O(1)O(0)T(1)T(0)X122: !"@2H1V@"C"@5H@"P"@2HD@"M: "31051:36 #30976 $"@21V1HLI@"19)"@RI@":2%Y(37),103T8227O7,111T8@ TT(G85)(G68):(T)6T5(T)c G80ĺ"@"32O"H"13T"V@$":32 OO(G76)(G82):(O)6O5(O) W1,BS1,SQ:I2,T2G2,J2:W11,BS11,SQ:I21,T2G21,J2:G78M332:(M4(E)5(E)6)X0āI0X1:"@"32Ȱ111 K(Z):11:K16020:"@I22V1H@"36)L$:G T5:O5:X0X0e "@"32O"H"13T"V@!":Q,0 G(Z)128:G85G68G80G76G82G78G2724:23 C2M4G7832 "@"32O"H"13T"V@ ":T0O0ē224O7,107T8231O7,107T8:227O7PP1:34# K8PP1:35:P43/ 33:45@ (Z)155ıW Q,0:"@40X40YN@";~ K(Z):K12813:K155K20514: Q,0:16 :15 126 35:K155İ2:Y$"RUNALGEBRA 4" 43 L$"@22V6H@Press SPACE BAR to Continue"L$:Q,0:4 :3:O 5:189,64266,152:G189:I641522:189,IG,I:GG1.7::3:o 11:K(Z)176:K0K46: L$"@3H21V@Press "(1)" Key to View the Next Page@D3H@Press "(2)" Key to View the Last Page"L$:Q,0 K(Z)128:11:K21K88:K21630000:24576:163:U$(9):E36251:V$(94):S$(35):H$(36):O$"boundary line":I$"inequalities":Z$"intersection":N$"solution":E$N$" set":R$"equation":42Q,0:0:1002: 1:204,152247,64:G152:I2052664:I,GI,152:GG8:G64G6%24798,C:C4Ĺ36251,5:I14:36266I,0::74J%2:(4)"RUNAM4.5.1"%C$(3)"ABSOLUTE VALUES":21,14042,14049,14842,15621,15614,14821,140:V TEST,IN TWO VARIABLES,OF LINEAR INEQUALITIES,INVOLVING ABSOLUTE VALUES,, `$K1R:12::$(36251)İ2:(4)"RUNAM4.5.1"$46$D0SDU:$U62SD60$U94SD35$U60SD62$U35SD94$$X34:2:B$"RUNALGEBRA 4"1105#i"@"(X7)"H14V@<@7F@>@"((X31)7)"H10V@"(95)"@8DB@"(126):X1,115X60,115:X31,80X31,150:IX10X567:I,111I,119::I911448:X27,IX36,I::#jM$k GRAPH OF AN INEQUALITY,GRAPHS OF SYSTEMS,GRAPHING OPEN SENTENCES,INEQUALITIES.125:D64(U35U60)88:Q1AIG:Q15Q15>"`Q15Q15x"a(Q15D64Q15D152)ē227I7,107Q18227I7,D"c:3:"e31,12031,138:60,3660,154:59,3659,154:X31:Y50:I13:X,YX,Y10:YY24::"fX,YO,YO,TX,TX,Y:#gX210:8227W7,107(AWG)8:$!ZI12![C(I)B(3)1B(2):D(I)B(3)1B(2):((U35U60)D(I)AC(I)G)((U94U62)D(I)AC(I)G)94:91!^:O13:I12:"@"13D(I)"V"32C(I)"H@$"::41:I12:"@"13D(I)"V"32C(I)"H@ "::41::/"_6:I552U62 TU3U35" UU4U94( V W"@27H13V@<@9F@>@5U6B@"(95)"@10DB@"(126):191,107263,107:227,64227,151:I1992587:I,103I,111::I751418:224,I231,I:: XW0:I55.3:AIG5AIG5RI:WWI!Y:227R7,107(ARG)94E$"no":D$"N"EO25:I13:B1B1A(I)::B13Ĺ36267,(36267)1P21:T13:B19:L1:R26:37:T7:B18:L27:R38:37::33:200Q1000RAB(3)1B(2):GB(3)1B(2):DB(3)1B(2):XB(3)1B(2):WB(3):YWX:UB(4):U1U60 SU$"N":U94U35E$"yes":D$"Y"MI1:B10:25:VV2:HH2:"@15V1H@Should it include the@D1H@area above the line?":E$"no":D$"N":U62U94E$"yes":D$"Y"N25:VV2:"@17V1H@Should it include the@D1H@area below the line?":E$"yes":D$"Y":U62Uyou. Complete the @D2H@graph by answering the "K"@2H@following questions @D2H@with Y (yes) or N (no):"A$:A195((36251)0) LP1A1:32:82:87:"@29H7V@Y"(U)A"X+"G:88:"@13V1H@Should the graph include@D1H@the "R$"?":H21:V15:I1:E$"no":Dhe graph@D6H@of the "N$" is@D6H@the partial plane on@D6H@the side of the@D6H@"R$" where@D6H@the points lie.":95:60JX10:Y35:T99:O178:102:X185:O269:Y51:T155:102:"@5VI2H@The "R$" for @D2H@the "N$" on the @D2H@right has been plotted @D2H@for tted@D6H@line).":0:88:3:U35U94İ88:728G2:88H38:21:T7:B18:L2:R26:37:"@7V2H@<4> Find two solutions@D6H@for the "N$"@D6H@by picking a value@D6H@for X and plotting@D6H@two points for@D6H@which the "N$"@D6H@is true.":90sI"@13V16H@Taph the boundary@D6H@line.":USDE88:38:21:T7:B18:L2:R26:37:"@7V2H@<3> If the "N$" is@D6H@of # or ^ form, then@D6H@include the boundary@D6H@line in the graph.@D6H@If it is of < or >"-F"@6H@form, then omit the@D6H@"R$" (show@D6H@it with a do2:T52:102:"@5V2H@GRAPH THE INEQUALITY "A$D"Y"(U)DA"X+"DG:500:"@7V29H@Y"(SD)A"X+"G D"@I7V2H@<1> Transform the@D6H@"N$" into a@D6H@form which has Y@D6H@alone as one@D6H@member.":40:"@5V23H@"D"Y"(U)DA"X+"DG"@2D29H@Y"(SD)A"X+"G"@15V2H@<2> Gr47:102:"@13V2H@However, if the "N$" is of the@D2H@form 'less than' or 'more than,'@D2H@then the partial plane is bordered@D2H@by, but does not include, the line"A"@2H@which the limiting "S$" defines.":6ZBP446:82:87:X10:Y35:T155:O269:10mpty set.":21?T6:B10:L2:R38:37:T12:B19:L1:R39:37:"@6V2H@The edge(s) of the "N$"'s@D2H@partial plane may be defined by an@D2H@"S$" if the "N$" is of the@D2H@form 'less than or equal to' or"@"@2H@'greater than or equal to.'":39:Y99:T1ntence which defines a@D2H@partial plane made up of the ordered@D2H@pairs for which the "N$" is@D2H@true.">39:Y99:T147:102:"@13V2H@If there are no ordered pairs for@D2H@which the "N$" is true, we say@D2H@that the solution set is "(16)", the@D2H@e7H@Y=2X is @D17H@included"A$:21:L2:R38:T5:B7:37:"@5V2H@Y#2X means Y is less than OR equal@D2H@to 2X.";:38:" Y^2X means Y is "G$"@D2H@OR equal to 2X.":6<P61,46j=X10:Y43:O269:T92:102:"@6V2H@An "N$" in two variables is an@D2H@open se33H@Y^2X":38:5:98,8063,152:98,8263,154:38:2:G152:I67102:I,GI,152:GG2::I102110:I,80I,152::38:6:231,80196,152:231,82196,154:38:1:G80:I2281921:I,80I,G:GG2::I1921851:I,80I,152::3:38;"@I13V17H@The line@D1:"@I6V27H@ The line @D27H@ Y=2X is @D27H@NOT included"A$:69X10:O269:Y35:T67:102:"@5V2H@If we want to include the ordered@D2H@pairs for which Y is equal to 2X@D2H@(Y=2X) we write the inequalities as:":X49:105:X182:105:"@12V5H@Y#2X@ine @D27H@ Y=2X is @D27H@NOT included"A$:21:T4:B19:R39:L1:37:"@7V1H@If Y>2X (Y is "G$"@2D1H@2X) then opposite half plane@2D1H@is defined.":39:103A85:259,80224,152:1:G80:I2562201:I,GI,80:GG2::I2202131:I,80I,152::3:38@a partial plane consisting@2D1H@of all the ordered pairs in@2D1H@which the value of Y is less@2D1H@than twice the value of X.":39:10365:259,80224,152:1:G152:I2272642:I,GI,152:GG4::I264267:I,80I,152::3:38:7"@I27H6V@ The ld pairs in which the"4"@D1H@value of Y is twice the@2D1H@value of X.":39:103:"@10V36H@$@2D2B@$@4D3B@$@2D2B@$":5:255,83228,145:254,84227,146:3:21:T4:B19:R39:L1:375"@7V1H@The ";:F$N$:36:" Y<2X@2D1H@(Y is less than 2X) defines@2D1HTS@I10H6D@WHICH (0-4) ??"A$:3:Kİ34:43>/MK:P1:32:33Q1M50,60,66,74a2P51,57,463"@6V1H@An "N$" is similar to an@D1H@"S$": both are open sentences.":38:"@D1H@The ";:F$S$:36:"Y=2X defines@2D1H@a line consisting of all@2D1H@orderenu@2V7HI@"31)"@11H22V@WHICH (0-5) ??"A$:101:3:CK:32:24(C$(K))2:"@2VI@"C$(K)A$:C600o-33:C181*.M0:P0:32:"@14H5V@LEARNING MODE@10H7V@<1> DISCUSSION@9V10H@<2> RULE@2D10H@<3> EXAMPLE@10H2D@<4> SAMPLE PROBLEM@10H3D@<0> Return To CONTEN:32:"@20H5V@CONTENTS@6V@":I14:"@10H@<"I"> "C$(I)"@D14H@"C$(I4):::"@7V4H@";:I13:(22)"@3DB@";::"@5V3H@";:I14:"<"I">@3D3B@";::X17:O45:Y38:T48:I14:102:YY24:TT24:_,5000:"@17V4H@"(31)"@3HD@<0>@10H@<0> Return To ALGEBRA Me30976# #"@21V1HLI@"19)"@RI@":a $Y(37)82:X(36)72:OX4(F$)7:TY12:F$" ";:102 %JTB:L1:J1:RL):: &R200:150 'R300:150 (R600:150 )R50:150 *35339:I18:C$(I)::B(X)((1)X)1:106 +P0:M0:C0(E)12889ĖH:V:"yes":30+ H:V:"no"r (E)128(D$)A(I)1:II1:"@19V1H@Correct":39:"@19V1H@"8): "@19V1H@The correct answer is "A$E$A$:A(I)0:39:"@19V1H@"25):II1:H:V:E$" ": "@2H1V@"C"@5H@"P"@2HD@"M: !31051:35 "106( 34:K155İ2:B$"RUN ALGEBRA 4"0 46u 35399,1:"@22V6H@Press SPACE BAR to Continue":35399,0:N,0:12 K(E):12:K12822:N,0:K16022:"@I22V1H@"36)A$: N,0 H:V:A$" ":(E)12889(E)1287828:41:26 35399,0:Press "(2)" Key to View the Last Page":35399,0m K(E)128:K07:12:N,0:K21K87:K21PP1:33 K8PP1:34:P46 32:49 (E)155ı N,0:"@40X40YN@"; K(E):K12814:K155K205ı N,0:18 :17 }24800:27903:A$"@I@":B$(4):E16384:N16368:S$"equation":N$"inequality":R$"boundary line":G$"greater than":420:1002:12:K(E):K1283:N,0:KK176:K0K43:0 35399,1:"@3H21V@Press "(1)" Key to View the Next Page@D3H@          1A(2)):C1(1)XY:D1(YT)(ZV):E1C1TA1V8<.]<(36251)6İ2:(4)"RUNAM4.5.1"u<(36251)0129:50<:2:"++ERROR++"(222)" AT LINE # "(218)(219)256::N4:H12:P18:27:G10:ATG11';x;TA(9)B(1):VA(9):A(A(4)1):B1(A(4)1):EA(4)1:AEV2T0200:2<,TA(9)1A(2):VA(9)1A(2):XA(9)1A(2):Y(A(8)1)(A(1)1A(2)):Z(A(8)1)(A(1)1A(2)):A1(A(8)1)(A(1)V18H@The answer is@I@Y = @3B@"V:23:S12:B18:L2:R17:40:R38:L18:40::37:(36251)Ĺ36266,(I12):2:(4)"RUNALGEBRA 4":50:N1Z:14::!;"@14V2H@If "Y"X+ "Z"Y = "D1"@D2H@And "C1"X+ "A1"Y = "E1"@2D2H@Then X =@15H@, Y ="L11ĺ"@17V18H@Try again":L1L11:135S9"@I16V18H@The answer is@I@X = @3B@"Tk9L10:"@31H18V@Y ="9H36:P19:N3:27:"@17V18H@"9):AVĺ"@17V18H@Correct":45:"@17V18H@"13):I1I11:1429L11ĺ"@17V18H@Try again":L1L11:139":"@I18V:"@13V3H@X + "A"Y = "G1"@D2H@"B1"X + "E"Y = "H1"@12V18H@Enter the value of@D18H@the "L$" shown,@D18H@then (RETURN) or ("(2)")@D18H@to erase.@16V31H@X = "8H36:N3:P17:27:"@17V18H@"9):ATĺ"@17V18H@Correct":45:"@17V18H@"13):I1I11:138&9 eliminate one of the @D2H@"L$"s by substitution, then use @D2H@the value you find for the remaining"7I10:"@2H@"L$" to find the value for the @D2H@"L$" you first eliminated. "V$:K195((36251)0):O1K1:36:L10:2008G1TAV:H1B1TE "E"("V")";p6~"@30H@= "H1" @16V23H@"B1"X + "EV" @30H@= "H1"@D27H@"B1"X @30H@= "TB1"@D28HI@X @30H@= "T;V$:86T,VU,VU,WT,WT,V:f7T10:U269:V35:W92:127:W155:127:V92:T122:127:"@I5V2H@Solve the system of "Q$"s shown @D2H@below. First"@8V2H@<4> Substitute@D2H@the Y value@D2H@"; 6}"into both@D2H@"Q$"s and@D2H@solve for X":43:"@9V23H@X - "A"Y @30H@= "G1"@D20H@X - "A"("V") @30H@= "G1"@D24H@X - "AV" @30H@= "G1"@D28HI@X @30H@= "T;V$:43:"@14V22H@"B1"X + "E"Y @30H@= "H1"@D19H@"B1"X +3V@"B1"X + "E"Y = "H1"@D18H@"B1"("G1" + "A"Y)+"E"Y@32H@= "H1:23:"@17V2H@<3> Solve for Y":43-5|"@22H15V@"B1G1"+"B1A"Y+"E"Y @32H@= "H1"@D22H@"B1G1" + "B1AE"Y @32H@= "H1"@D28H@"B1AE"Y @32H@= "H1B1G1"@D30HI@Y = "V"@I@":23:S8:B18:L2:R38:40:1B1TEV:"@5V23H@X - "A"Y = "G1"@D22H@"B1"X + "E"Y = "H1:43:"@8V2H@<1> Solve one@D2H@"Q$" for X@D2H@in terms of Y":43:"@21H9V@X - "A"Y = "G1"@D26H@X = "G1" + "A"Y":23^4{"@12V2H@<2> Substitute@D2H@the solution in@D2H@the other@D2H@"Q$:43:"@24H1e second "Q$" for one@D5H@"L$".@2D1H@<4> Substitute value into first"2u"@5H@"Q$"; solve for other "L$"@2D1H@<5> Check your work.":82vO450:T10:U269:V35:W155:127:W59:127:T129:127:"@5V2H@Solve the system@D2H@of "Q$"s:":2003zG1TAV:Hs in the system.@D5H@If your work was accurate the@D5H@resulting sentences will be@D5H@arithmetically true.":8E2t"@L4V1H10C@SUMMARY:@R143C7V1H@<1> Solve one "Q$" in terms of one@D5H@"L$".@2D1H@<2> Substitute solution into other@D5H@"Q$".@2D1H@<3> SolvSubstitute the value of the"0r"@5H@"L$" obtained in step <3> into@D5H@any "Q$" containing both@D5H@"L$"s, and solve for the@D5H@remaining "L$".";:43:"@2D1H@<5> Check your work by substituting@D5H@both "L$" values into the"p1s"@5H@original equationtitute this solution into the@D5H@other "Q$" by replacing the@D5H@"L$" for which you solved with"0q"@5H@the solution you found.":43:"@17V1H@<3> Solve the resulting "Q$"@D5H@for the value of its single@D5H@"L$".":23:L1:R39:S8:B19:40:"@8V1H@<4> = "G1:8.nO111,116,50.o"@4V1H@The SUBSTITUTION METHOD allows you to@D1H@solve a system of "Q$"s in two@D1H@"L$"s using these steps:":43:"@8V1H@<1> Solve one of the "Q$"s for one@D5H@"L$" in terms of the other@D5H@"L$"."a/p43:"@12V1H@<2> Subs-l"@10V23H@Use Found Value@D23H@of X to find Y":43:"@13V2H@"A"("T") + "B1"("V") = "G1"@13V23H@Substitute X and@D23H@Y in the Other@D23H@Equation to see@D23H@if your answer@D23H@is correct.":43.m"@15V3H@"AT" + "B1V" = "G1:43:"@17V10H@"G1" bine Factors@2D23H@Isolate Variable@2D23H@Simplify":8,k1:157,56157,144:3:"@138CL4V1H@NOW CHECK YOUR WORK@R143C7V2H@"A"X + "B1"Y = "G1"@D8H@Y = "E"X - "H1"@7V23H@Original System@D23H@of Equations":43:"@6H10V@Y = "E"("B1") - "H1"@D6H@Y = "EB1H1X + "B1"("E"X - "H1")="G1"@2D1H@"A"X + "B1E"X - "B1H1"="G1"@2D8H@"AEB1"X - "B1H1"="G1"@2D13H@"AEB1"X="G1B1H1"@2D15HI@X = "T;V$6,j1:157,56157,152:3:"@7V23H@Original System@D23H@of Equations@2D23H@Substitution@2D23H@Simplify Factors@2D23H@Com6V11H@"A"X + "B1"Y = "G1;V$*h5:143,112143,152:"@14V21H@"A"("T") + "B1"(Y) = "G1:44:"@16V22H@"AT" + "B1"(Y) = "G1:44:"@18V22H@"AT" + "B1"(@I@"V"@I@) = "G1:8+i"@4V1HL138C@IN SUMMARY:@R143C7V5H@"A"X + "B1"Y="G1"@D11H@Y="E"X - "H1"@2D1H@"A"@10V1H@Since we know now that X="T", we can@2D1H@substitute that value and solve for Y:":44*g"@I18H8V@Y = "E" + "H1"@I14V7H@Y = "E"("T") - "H1:44:"@16V7H@Y = "ET" - "H1:44:"@18V7HI@Y@I@ = "ET"-"H1" = @I@"ETH1;V$:44:"@18H8V@Y = "E"X - "H1"@IH@Finally, we solve for this variable."(e"@13V18H@"AEB1"X@22H@= "G1B1H1:43:"@I15V20H@X@I@ = @I@"T;V$:5:115,99193,99193,131115,131115,99:43:8\)f"@4V1H@Here is our system of two "Q$"s:@2D11H@"A"X + "B1"Y = "G1"@2D18H@Y = "E" + "H1:44:" sides of the equation.":43:0:91,6191,9011,9011,6191,61'(d"@I15H11V@+ "B1H1"@23H@+ "B1H1"@ID8H@"AEB1"X@15H@+ 0@22H@= @I@"G1B1H1"@I@":5:103,75103,107199,107199,75103,75:43:23:0:103,75103,107199,107199,75103,75:SS1:40:"@18V1urther.":43:0:147,46147,7349,7349,46147,46:5:91,6191,9011,9011,6191,61>'c"@10V8HI@"AEB1"X@I15H@- "B1H1"@22H@= "G1:43:23:SS3:40:"@15V1H@Now we isolate the variable term by@2D1H@adding or subtracting";:" each other term@2D1H@on boths so@2D1H@";%a"we can reduce the "Q$" to simplest@2D1H@form.":43:"@8V2H@"A"X + @I@"B1E"X@I15H@- @I@"B1H1"@I22H@= "G1:5:147,46147,7349,7349,46147,46:3:43:SS3:23:40S&b"@13V1H@Next, we add the common factors to@2D1H@simplify the "Q$" f15C@ @I8V23H@("C"X - "H1")":I23111:"@"I"H8V0C@("E"X - "H1")@"I1"H15C@("E"X - "H1")":46: %`"@8V3HI@"20)"@I10H7V@("E"X - "H1")":46:"@I7V3H@"20)"@I6V10H@("E"X - "H1")":44:23:S7:B19:L1:R39:40:"@11V1H@Now we need to multiply the factor4H8V@"E"X - "H1"@I10V1H@Since Y is the same in both "Q$"s,@2D1H@we can replace Y in the first "Q$"@2D1H@with the value of Y we see in the@2D1H@second "Q$".":45:I1231b$_"@0C"I"H6V@"A"X + "B1"@15C"I1"H@"A"X + "B1;:46::"@0C6V12H@"A"X + "B" @2D18Hs what the@2D1H@value of Y is, expressed in terms of@2D1H@X.";"]" Because these "Q$"s are a@2D1H@system, we already know that the value@2D1H@of Y is the @I@same@I@ in both "Q$"s.@6V12H@"A"X + "B1"@I@Y@I2D20HI@Y":44:23:S9:B19:L1:R39:40#^"@I2118,129 !YO90,102,105,107,50n!ZTA(9):VA(9):A(A(4)1)B(1):B1(A(4)1):EA(4)1:AEV2T090>"\G1ATB1V:H1VET:"@4V1H@A system of two "Q$"s:@2D12H@"A"X + "B1"Y @22H@= "G1"@2D20H@Y = "E"X - "H1:43:"@10V1H@The second "Q$" tells uind the value of the first variable@D2H@Then enter your answers below. "V$:O1F1:36:300 T145:H22:27:G11AVĺ"@18V33H@RIGHT":36265,(36265)1:86 U"@18V10H@WRONG. X="T" AND Y="V V23:S14:B18:L2:R38:40::37:400!XM89,110,10:U269:V35:W99:127:W155:V107:127:"@5V2HI@Below is a system of simultaneous @D2H@linear "Q$"s. Use the addition @D2H@property of equality to eliminate @D2H@one "L$" so that you may solve "` S"@2H@for the other, and substitution to @D2H@fstituting@D2H@them into the@D2H@second "Q$".@18H9V@"C1"("T")+"A1"("V")="E1"@10V20H@"C1T" + "A1V" ="E1P"@11V25H@"E1" = "E1"@D25H@True@15V2H@The solution is X="T", Y="V"@D2H@The solution set is "(123)"("T","V")"(125):8QF195((36251)0)RTor X.@23H@X@30H@= "T:41:"@14V2H@<6> Check the@D2H@values by@D2H@substituting@D2H@them into the@D2H@first "Q$".@19H14V@"Y"("T")+"Z"("V")= "D1"@21HD@"YT"@24H@+"ZV"@29H@= "D1gO"@16V25H@"D1"@29H@= "D1:23:40:"@9V2H@<7> Check the@D2H@values by@D2H@sub3:41:"@17V2H@<3> Solve for the@D2H@"L$" Y.@26H17V@Y = "VM23:S8:B18:L2:R38:40:"@8V2H@<4> In the first@D2H@"Q$", replace@D2H@Y by its value.@22H8V@"Y"X+"Z"(Y)= "D1"@D22H@"Y"X+"ZV"@30H@= "D1"@D22H@"Y"X@29H@ = "D1ZV:41N"@12V2H@<5> Solve f"A1"Y="E1"@8V2H@<1> Multiply all@D2H@members of the@D2H@first "Q$"@D2H@by "X".@21H8V@"Y"X+"Z"Y="D1"@D21H@"C1"X+"A1"Y= "E1:41:"@21H8V@"YX"X+"ZX"Y= "D1X:L41:"@13V2H@<2> Add and@D2H@eliminate the@D2H@"L$" X.@13V25H@"ZXA1"Y = "D1XE1:161,83252,8@D6H@substituting the values of the@D6H@"L$"s in each of the given@D6H@"Q$"s to verify that the@D6H@resulting sentences are true.":8IO450:T10:U269:V35:W59:127:W155:127:300K"@5V2H@Solve the system of "Q$"s:@D2H@"Y"X+"Z"Y="D1"@16H@"C1"X+ that "L$".":8GT10:U269:V35:W155:127:"@5V2H@<5> Substitute the value of the@D6H@"L$" obtained in step 4 into@D6H@any "Q$" containing both@D6H@"L$"s. Solve the resulting@D6H@"Q$" for the remaining@D6H@"L$"."H"@12V2H@<6> Check the solution by If one of the coefficients is@D6H@positive and the other negative,@D6H@then add the two "Q$"s;@D6H@otherwise subtract them. One of@D6H@the "L$"s will then be@D6H@eliminated."F"@12V2H@<4> Solve the resulting "Q$"@D6H@with one "L$" for the value@D6H@ofs appear on one side and@D6H@the constant is on the other.@2D2H@<2> Multiply both members of each@D6H@"Q$" by numbers that will@D6H@make the coefficients of one of@D6H@the "L$"s the same in@D6H@absolute value.":8ET10:U269:V35:W155:127:"@5V2H@<3>).":8BO67,69,71,50CT10:U269:V35:W155:127:"@5V2H@To solve a system of simultaneous@D2H@linear "Q$"s in two "L$"s by@D2H@the addition or subtraction method:@2D2H@<1> Transform each "Q$" into an@D6H@equivalent one in which the"D"@6H@"L$"4:B19:L1:R38:40:"@5V2H@Now replace X for its value in either@2D2H@"Q$" and solve for Y."A"@14V13H@6X + Y = 5":41:"@15V12H@6(1)+ Y = 5":41:"@16V13H@6 + Y = 5":41:"@17V19H@Y = -1@10V2H@Since X=1 and Y=-1, the solution set@2D2H@is (1,-1ow. By@2D2H@multiplying the first "Q$" by 3,@2D2H@you can add the two "Q$"s and@2D2H@remove the "L$" Y.@14V14H@6X + Y = 5@D14H@5X - 3Y = 8":45:"@14V31H@($3)"]@"@U13H@18X + 3Y = 15":41:"@17V13H@23X@22H@= 23@D15H@X@22H@= 1":91,131182,131:23:Siminate "L$"s when the@2D2H@coefficients of a pair of@2D2H@corresponding terms do not have the@2D2H@same absolute value. We can use the@2D2H@multiplication";>" property of equality@2D2H@to remedy this problem.":8?"@5V2H@Look at the two "Q$"s bel<"@16V1H@Y in one of the two@D1H@"Q$"s.@D1H@The solution set is@D1H@(2,3).@4H11V@X+2 Y=8":42:"@10V9H@3";:42:"@B@ @2B@3";:42:"@B@ @D2B@$3@D4H@X+6 =8@D4H@X =2":6:212,48212,152:3:8="@5V2H@But adding or subtracting "Q$"s@2D2H@will not el62,112:252,41175,126:41:"@15V1H@We can subtract the@D1H@second from the first";"@17V1H@to get an "Q$"@D1H@that does not involve@D1H@X.@8V7H@3Y=9@D8H@Y=3":35,5977,59:5:161,83266,83:3:23:S11:B19:R22:L1:40:"@15V1H@Now substitute 3 for"8H17V@";:F15:3:" ";::"@37H@X@5V25H@Y@17V24H@";:F1:3:T183:V51:I15:T,VT5,V:VV16::T171:V129:I17:T,VT,V5:TT14:R:"@5V5H@X+2Y=8@D5H@X-Y=-1@11V1H@The coefficients of@D1H@the "L$" X are@D1H@the same in both@D1H@"Q$"s.":164,562tems made up of the@2D2H@"Q$"s of horizontal and vertical@2D2H@lines may be found by using it.":88T157:U269:V35:W155:127:"@5V26H@"(95)"@26H18V@"(126)"@16V23H@<@37H@>":185,41185,151:162,131263,131:"@25H14V@";:F15:3:"@B2U@";:9"@2 ??"V$:K0:N4:5:Gİ38:2:(4)"RUNAM4.4"@3MG:O1:36:37L4C388_5M54,66,73,81u6O55,56,61,63,50b7"@5V2H@The Addition Property of Equality may@2D2H@be used for solving a system of@2D2H@linear "Q$"s in two "L$"s.@2D2H@Equivalent sys144 .Z10:144Y /35339:A(T)((1)T)1:B(T)1A(2):(36251)081:C(24798),2M0:O0:36:"@14H5V@LEARNING MODE@10H7V@<1> DISCUSSION@9V10H@<2> RULE@2D10H@<3> EXAMPLE@10H2D@<4> SAMPLE PROBLEM@10H3D@<0> Return To CONTENTS@I10H22V@WHICH (0-4):I0Q1:A$A$((256I))::A(A$):AQ27:H:" ";:H:A:b $"@2H1V@"C"@5H@"O"@2HD@"M:q %31051:39| &30976 '"@21V1HLI@"19)"@RI@": (JSB:L1:J1:RL):: )H170:144 *Z50:144 +Z150:144 ,Z100:144 -Z250:"@I22V1H@"36)V$:? H:P:N):Q0:I0N:256I,32::J1,0Z HQ:V$((256Q))V$; G(ZZ):14:GXX29:J1,0:HQ:((256Q));:G14135:QQ(G149QN1)(G136Q0):GGXX:G47G58G45ĖHQ:(G);:256Q,G:QQ1:QN35 "28B #A$""J1,0:"@40X40YN@";; G(ZZ):GXX16:G155G205ıJ J1,0:20T :19` 63900 38:G155İ2:(4)"RUN ALGEBRA 4" 50 35399,1:"@22V6H@Press SPACE BAR to Continue":35399,0:J1,0:14 G(ZZ):14:GXX24:J1,0:G16024::GKGN5: 35399,1:"@3H21V@Press "(1)" Key to View the Next Page@D3H@Press "(2)" Key to View the Last Page":35399,0 G(ZZ)XX:G09:14:J1,0:G21G89:G21OO1:37 G8OO1:38:O50 36:52 AA(1):(ZZ)155ı 4 63900c24800:27903:XX128:V$"@I@":ZZ16384:J116368:Q$"equation":L$"variable":47r0:1002:"@G15C0KE@";:E(F):A(E10):BEA10:EE9:F0ĺ(12E)E)"@B@";(20A)"@B@"(B)"@ER@";: 14:G(ZZ):GXX5:J1,0:GG176                                  N1IN2B-:AA213AA14:BB107BB8:CC213CC14:DD107DD8:VROR++ "(222)" AT LINE # "(218)(219)256:~,6:V43:H2521611:H,V:H,V1:VV1:57::"@33H8V@Y=X+1":58:5:V135:,2:"RUNAM4.4.1":8,,43:2:(4)"RUNALGEBRA 4"-AA0:I33.5:IN1N25IN1N25CCI:DDN1IN2:AA0AAI:BBX,YX5,Y:YY8::"@8V29H@Y@14V37H@X@29H17V@";:N44:N1NN2:"@2U@";+~3:"@UB@";::"@14V24H@";:N33:N0ĺ"@2F@";:NN1+3:"@F@";::+GRAPHIC METHOD,ADDITION AND SUBTRACTION,SUBSTITUTION METHOD,SOLVING SYSTEMS TEST-,:2:"++ERF@";>*{3:"@F@";::"@16V29H@";:N23:N0NN1:"@2U@";o*|3:"@2UB@";::X157:O269:Y35:T155:121:L+}"@8V30H@"(95)"@30H18V@"(126)"@13V23H@<@37H@>":162,107262,107:213,65213,149:X171:Y105:I17:X,YX,Y5:XX14::X211:Y75:I19:50:I13:X,YX,Y10:YY24::;)yX,YO,YO,TX,TX,Y:*z"@5V30H@"(95)"@30H18V@"(126)"@12V23H@<@37H@>":162,99262,99:213,41213,149:X211:Y51:I16:X,YX5,Y:YY16::X171:Y97:I17:X,YX,Y6:XX14::"@13V24H@";:N33:N0NN1:"@24)1K(rT5:B18:L2:R19:23:46:L21:R38:46::42:(UU)Ĺ24798,C:200S(s64(t24798,C:C4ĹUU,4:I14:36263I,0::98(u200(vN5R(5)3:N6R(7)4:N1R(7)4:N3R(7)4:N1N3118: )x31,12031,138:60,3660,154:59,3659,154:X31:Yo5:N1N3:N2N4:400:AA,BBCC,DD:3:"@5V21H@Enter coordinates@D21H@of intersection @D21H@( , )":H23:V8:MX3:47:A(1)A:H27:47:A(2)A:A(1)N5A(2)N6ĺ"@18V33H@RIGHT":ANAN1:113'p"@18V36H@ @25H@WRONG@F@("N5","N6")(qAN2Ĺ36264,(3626@5V21H@Graph first point@D21H@of Y="N3"X+"N4:27:R302S:Y13U:J1:"@27H5V@second @D21H@point of Y="N3"X+"N4:27:J0:H0:I1NE:X1(I)SY1(I)UJ1&lX1(I)(R30)2Y1(I)13YH1&m:H1J1ĺ"@18V33H@RIGHT":AN1:111&n"@18V33H@WRONG'5)"@D2H@ Y = "N3"X+"N45)"@17V17H@ @D17H@ "X$}%hW0:"@13V10H@";:I33:N1IN25N1IN25WW1:X(W)I:Y(W)N1IN2%i:125:EW:E3E3%j6:400:AA,BBCC,DD:3:F0:I33:N3IN45N3IN45FF1:X1(F)I:Y1(F)N3IN4:NENE1&k:"wo @D2H@linear "Q$"s,@D2H@the first of @D2H@which is already @D2H@graphed. Graph @D2H@two points of the0%g"@2H@second "Q$". @D2H@Use the keys: @D2H@D:Down L:Left @D2H@U:Up R:Right@D2H@ P:Plot @D2H@"17)"@D2H@ Y = "N1"X+"N2(125)".":8,#bP195((UU)0):P1P1:41#d118:J0:AN0:NE0:N2N6N5N1:N4N6N3N5:F0:I33:FF(N1IN2(N1IN2))::F2100:F0:I33:FF(N3IN4(N3IN4))::F2100p$fX10:O137:Y35:T155:121:X143:O269:121:"@5V2HI@Below are t@<4> Check that X="N5" and Y="N6"@2D6H@Y="N1"X+"N2"@21H@Y="N3"X+"N4"@2D6H@"N6"="N1"("N5")+"N2 #a"@U21H@"N6"="N3"("N5")+"N4"@2D6H@"N6"="N6" (true)@21H@"N6"="N6" (true)@2D2H@The common solution is ("N5","N6").@D2H@The solution set is "(123)"("N5","N6")"56:5:AA,BBCC,DD!_3:23:46:"@8V2H@<3> Read the@D6H@coordinates of@D6H@the point of@D6H@intersection.":56:"@10C"13N6"V"30N52"H@$@15C@":56:"@L13V6H@("N5","N6")@R@":23:R38:46]"`I15:0:154,51I161,51I:154,155I161,155I:3::"@8V2H:46:X67:O108:Y83:T131:121:O87:121:O108:T99:121:T7:"@11V10H@X@13H@Y@8V2H@<2> Graph Y="N3"X+"N4:EW:E3E3!^JJN1:HHN2:N1N3:N2N4:400:N1JJ:N2HH:"@10H13V@";:I1E:X(I)"@13H@"Y(I)"@D10H@";::56:I1E:"@"30X(I)2"H"13Y(I)"V@$"::1:X(W)I:Y(W)N1IN2.[:125:EW:E3E3\"@10H13V@";:I1E:X(I)"@13H@"Y(I)"@D10H@";::56:I1E:"@"30X(I)2"H"13Y(I)"V@$"::56:6:400:AA,BBCC,DD:W0:I33:N3IN45N3IN45WW1:X(W)I:Y(W)N3IN4w ]:3:23:T7:B18:L2:R21YW0:F0:I33:FF(N3IN4(N3IN4))::F288:"@5V2H@Solve graphically:Y="N1"X+"N2"; Y="N3"X+"N4"@8V2H@<1> Graph Y="N1"X+"N2:X67:O108:Y83:T131:121ZO87:121:T99:121:O108:121:"@11V10H@X@2F@Y@10H13V@";:I33:N1IN25N1IN25WWhs@D6H@to find the common solution.@D2H@<4> Check that the solution@D6H@satisfies both "Q$"s.":8mWP464XX10:O269:Y35:T51:121:T155:121:X157:O269:Y51:T155:121:118:N2N6N5N1:N4N6N3N5:F0:I33:FF(N1IN2(N1IN2))::F288:@2D2H@<1> Graph one "Q$" in a@D6H@coordinate plane.@D2H@<2> Graph the second "Q$" in a@D6H@coordinate plane using the sameaV"@6H@set of coordinate axes.@D2H@<3> Read the ordered number pair@D6H@associated with the point of@D6H@intersection of the grap linear "Q$"s can be@D2H@solved by graphing the "Q$"s in@D2H@the same coordinate plane system and@D2H@finding the coordinates of all@D2H@points common to the graphs.":8{UX10:O269:Y35:T155:121:"@5V2H@To solve a pair of linear "Q$"s@D2H@graphicallyt":45:"@2D2H@";:T$"equations":45:".":8<RP83,85,64SX10:O269:Y35:T75:121:Y83:T131:121:"@5V2H@The "S$" set of a system of two@D2H@linear "Q$"s is the intersection@D2H@of the "S$" sets of the@D2H@individual "Q$"s. T"@2D2H@A pair of":6:219,42161,112P219,43161,113:5:266,56182,152:266,55182,151:3:"@15V23H@Y=X-1":23:I113:2:5I:21)::"@5V2H@A set of linear@2D2H@"Q$"s that has,Q"@D2H@no "S$" set is@2D2H@called a ";:T$"system of":45:"@2D2H@";:T$"inconsisten=X); they are@2D2H@";:T$"independent":45:"@2D2H@"Q$"s as well.":8O122:"@5V2H@The two lines have@2D2H@no points in common.@2D2H@They are parallel.@2D2H@The "S$" set of@2D2H@this system has no@2D2H@members. It is the@2D2H@empty set.":"@6V23H@Y=X+3have a common@2D2H@"S$" of (1,1),@2D2H@they are called@2D2H@";:T$"consistent":45:"@2D2H@equations.":23:I114:2:5I:21)::"@5V2H@Because the "S$"@2D2H@sets are notEN"@D2H@identical (for@2D2H@example, (0,2) is on@2D2H@Y=-X+2, but not on@2D2H@Y, and their@2D2H@common point is@2D2H@called the ";:T$"point":45:"@2D2H@";:T$"of intersection":45:".L6:175,144259,48:175,145259,49:5:184,40260,120:184,41260,121:23:I114:2:5I:21)::"@5V2H@Because the two@2D2H@";:3M"lines H@coincide. This is@2D2H@called a ";:T$"system of":45:"@2D2H@";:T$"dependent "Q$"s":45:"@2D2H@The two "Q$"s'@2D2H@"S$" sets are@2D2H@identical.":8jK122:"@5V2H@The two lines have@2D2H@one point in common.@2D2H@The lines intersect@2D2H@at (1,1) of the":45:"@2D2H@";:T$"system":45:".":8I122:150:H161252:H,V:H,V1:VV1:57::"@10V31H@2Y=2X+2@6V25H@Two@D24H@Lines@7V32H@>@11V27H@"(126):3:203,59228,59:192,66192,95:58J"@5V2H@All of their points@2D2H@are in common; they@2D2G"@5V2H@To solve such a system, find the@2D2H@ordered pairs of numbers that satisfy@2D2H@both "Q$"s. Each such ordered@2D2H@pair is called a ";:T$S$" of the":45:"@2D2H@";:T$"system":45/H"; the set of all "S$"s@2D2H@is called the ";:T$S$" setThe graph of a linear "Q$" in two@2D2H@variables is a straight line. When we@2D2H@graph two lines, we impose two@2D2H@conditions on the variables at the@2D2H@same time. So, these "Q$"s areF"@D2H@called a system of simultaneous@2D2H@linear "Q$"s.":8:P0:41:"@14H5V@LEARNING MODE@10H7V@<1> DISCUSSION@9V10H@<2> RULE@2D10H@<3> EXAMPLE@10H2D@<4> SAMPLE PROBLEM@10H3D@<0> Return To CONTENTS@I10H6D@WHICH (0-4) ??"X$:5:Kİ43:60AMK:P1:41:42CM68,82,87,98DP69,71,73,75,79,64E"@5V2H@I14:121:YY24:TT24:>21,14042,14049,14842,15621,15614,14821,140:"@17V4H@"(31)"@D3H@<0>@4U10H@<0> Return To ALGEBRA Menu@2V7HI@"31)"@11H22V@WHICH (0-5) ??"X$:120:5:CK:41:24(C$(K))2:"@2VI@"C$(K)X$?C300:42:C1116@M0:B;35339:I14:C$(I)::R(X)((1)X)1:(UU)698:129<P0:M0:C0:41:"@20H5V@CONTENTS@6V@":I14:"@10H@<"I"> "C$(I):I2ĺ"@14H9V@METHOD=:"@7V4H@";:I13:(22)"@3DB@";::"@5V3H@";:I14:"<"I">@3D3B@";::X17:O45:Y38:T48:,0:HS:((256S));:K1415523K136SSS1J4K149SMX1SS15KK128:(K47K58)K45ĖHS:(K);:256S,K:SS1:SMX556487I$"":I0S:I$I$((256I)):I:A(I$):8Q1150:14::9I12:14:::I1100:14: *31051:44 +309765 ,"@21V1HLI@"19)"@RI@":s -Y(37)82:X(36)72:OX4(T$)7:TY12:T$" ";:121 .JTB:L1:J1:RL):: /V:H:MX1):S0:I0MX:256I,32::QQ,0 0HS:X$((256S))X$; 1K(ZZ):14:K128492QQ1S0ĺ"@"302S"H14V@";:NS:3O "S1U1U0ĺ"@"13U"V29H@";:NU:3w #UU(K85)(K68):(U)5U4(U) %K80:"@"302S"H"13U"V@$": &SS(K76)(K82):(S)4S3(S) '6:AA,BBCC,DD:3:28 )"@2H1V@"C"@5H@"P"@2HD@"M: 6)X$: U0:S0:QQ,00 J1ĺ"@"R"H"Y"V@$"Q "@"302S"H"13U"V@!":QQ,0 K(ZZ)128:14:K030:K85K68K80K76K8230 "@"302S"H"13U"V@ ":U0S0ē210S14,107U8216S14,107U8:213S14,104U8213S14,112U8" !U"@40X40YN@";3 K(ZZ):K12816:K155K205ıB QQ,0:20L :19V 129y 43:K155İ2:"RUNALGEBRA 4" 64 35399,1:"@22V6H@Press SPACE BAR to Continue":35399,0:QQ,0:14 K(ZZ):14:K12824:QQ,0:K16024:"@I22V1H@"35:v 35399,1:"@3H21V@Press "(1)" Key to View the Next Page@D3H@Press "(2)" Key to View the Last Page":35399,0 K(ZZ)128:K09:14:QQ,0:K21K89:K21PP1:42 K8PP1:43:P64 41:67 D(1):(ZZ)155ı QQ,0:%Y24800:27903:ZZ16384:UU36251:QQ16368:Q$"equation":X$"@I@":S$"solution":59h0:1002:"@G15C0KE@";:F(N):A(F10):BFA10:FF9:N0ĺ(12F)F)"@B@";(20A)"@B@"(B)"@ER@";: 14:K(ZZ):K1285:QQ,0:KK176:K0K4                       1B1C1:E1A1C1:F1A1B1C1:D1B1D1A13000:=XH1A(3):D1A(3):E1A(3):C1A(3):G1H1E1:A1H1D1:F1C1E1:D1F1C1D17000:=:2:"++ERROR++"(222)" AT LINE #"(218)(219)256>>C1E1:C1D1E1G1C1E1C1G1D1E1D1G1500:7":162,99263,99:213,41213,151:"@29H6V@";:F321:4:"@2DB@";:F1ĺ"@2D@";:FF1:F:"@13V24H@";:F33:4:"@F@";:F1ĺ"@2F@";:FF1 ;:T171:R98:I17:T,RT,R5:TT14::T211:Rals "E1".@2D2H@If B equals "C1" and C equals "F1", what@D2H@is the value of A?@I14V2H@COMBINED VARIATION@L16V14H@A =":H23:O17:28:"@R@":QH1ĺ"@18V2H@RIGHT":2009"@18V2H@WRONG. ANSWER IS "H1:24:9T,RU,RU,VT,VT,R::T157:U269:R35:V@RIGHT":200;8"@R18V2H@WRONG. ANSWER IS "G1C1F1:24:8"@I@":S6:E12:X36:H3:42:"@I@":S15:E19:42:7000:"@I5V2H@A varies directly as B and inversely@D2H@as C. 9"@7V2H@A equals "A1" when B equals "D1" and@D2H@C equ19:42:"@I5V2H@A varies jointly as B and C." 8"@7V2H@A equals "A1" when B equals "D1" and@D2H@C equals "E1".@10V2H@If B equals "C1" and C equals "F1", what@D2H@is the value of A?@I14V2H@JOINT VARIATION@L14H16V@A = ":H23:O17:28:QG1C1F1ĺ"@18V2HR10,91224,91:245,91266,91:182,139196,139:217,139238,139:96}K4:Z9:1005:T10:U269:R35:V99:133:R107:V155:133:L1Z2:NL:37:127:NL1:N10İ38:1010:496~37:130:L:38:1010:49-7400:"@I@":S6:E12:X36:H3:42:"@I@":S15:Es "C1" and C equals "F1".@11V2H@<1> Find the constant@D6H@of proportionality@16V2H@<2> Find the value@D6H@of A@10V30H@AC@35H@"A1"$"E1"@D26H@K =@33H@=@D30H@B@36H@"D1"@28H14V@= "G1"@16V26H@KB@31H@"G1"$"C1"@D22H@";@6|"A =@29H@=@35H@= "H1"@D27H@C@32H@"F1:20,91224,91:245,91266,91:94zH1A(3):D1A(3):A1H1D1:E1A(3):G1H1E1:C1A(3):F1C1E1:"@5V2H@A varies directly as B and inversely@D2H@as C. If A equals "A1" when B equals "D1"@D2H@and C equals "E1", find the value of A@D2H@";5{"when B equalen@D2H@B equals "C1" and C equals "F1".@11V2H@<1> Find the constant@D6H@of proportionality.@16V2H@<2> Find the value of@D6H@A.@30H10V@A@35H@"A1"@26HD@K =@33H@=@D30H@BC@35H@"D1"$"E1"@28H2D@= "G14y"@26HD@A = KBC@D28H@= "G1"$"C1"$"F1"@D28H@= "G1C1F1:212vN4119:N6122:492wT10:U269:R35:V155:133:G1A(5):D1A(5):E1A(5):A1G1D1E1:C1A(5):F1A(5):B1D1C1F1:"@5V2H@A varies jointly as B and C. If@D2H@A equals "A1" when B equals "D1" and C@D2H@equals "E1;3x", find the value of A wh@.@D14H@x@21H@x@28H@z@33H@x y@G14H15V@"Q$"@16H@"Q$"@21H@"W$"@23H@"W$"@29H@"Q$"@34H@"Q$"@36H@"W$2u"@17V15H@"Q$"@22H@"W$"@29H@"W$"@34H@"W$"@36H@"Q$"@R@":49,10763,107:168,107182,107:91,131119,131:140,131168,131:196,131210,131:231,131259,131:9 variable z varies@D2H@directly as a variable x and@D2H@inversely as another y. For a@D2H@non-zero constant k,@16V2H@Therefore@7H12V@kx@24H@zy@D3H@z =@10H@or zy = kx or"_1t"@13V27H@= k@D8H@y@25H@x@15V13H@z y@20H@z y@28H@z@33H@x y@D18H@=@25H@or@31H@=@37H@"Q$"@36H@"Q$"@17V13H@"Q$"@15H@"Q$"@20H@"W$"@22H@"W$"@29H@";/rW$"@34H@"W$"@36H@"W$"@R@":84,9998,99:84,131112,131:133,131161,131:196,131210,131:231,131259,131:"@37H16V@.@13V12H@xy@R@":90s133:"@5V11H@COMBINED VARIATION@2D2H@occurs when aiable z varies@D2H@directly as the product of variables@D2H@x and y. For a non-zero constant k,@11V12H@z@15HD@= k or z = kxy@15V2H@Therefore"; Find the constant@D6H@of proportionality.@14V2H@<2> Solve for Y when@D6H@XY = K.@10V27H@XY = K@D26H@("D1")("A1")= K@D28H@K = "F1"@2D27H@XY = K@D25H@("B1")Y = "F1"@D28H@Y = "E1:9$[Z9:(36251)Z2H@";>"W"@D@<5> The graph of XY = K is a@D6H@hyperbola.":9J"XN449#YB1A(9):C1A(9):T10:U269:R35:V67:133:V155:133:A1A(9):D1B1C1:"@5V2H@If Y varies inversely as X,@D2H@and if Y="A1" when X="D1",@D2H@find the value of Y when X = "B1"R29H17V@=":182,139196,139:217,139231,139:9"V133:"@6V2H@<3> If X is multiplied by a non-zero@D6H@number, Y must be divided by@D6H@that same number.@2D2H@<4> If X is divided by a non-zero@D6H@number, Y must be multiplied by@D6H@that same number.@2D of values for the@D6H@variables, is equal to the@D6H@product of X@G@"W$"@R@ and Y@G@"W$"@R@, any other@D6H@pair of values. So:@17V6H@X@G@"Q$"@R@Y@G@"Q$"@R@ = X@G@"W$"@R@Y@G@"W$"@R@ or"/!U"@16V26H@X@G@"Q$"@R@ Y@G@"W$"@R2D26H@X@G@"W$"@R@ Y@G@"Q$"@3,86,49ST10:U269:R35:V155:133:"@5V4H@PRINCIPLES OF INVERSE VARIATION@2D2H@<1> The product of X and Y is@D6H@constant: XY = K, where K is the@D6H@constant of proportionality.@2D2H@<2> The product of X@G@"Q$"@R@ and Y@G@"Q$"@R@, any" T"@6H@pairLP"Y@G@"Q$"@R@ X@G@"W$"@R@Y@G@"Q$"@R@":56,11584,115:105,115133,115Q44:"@14V20H@or@23HU@X@G@"Q$"@R@ Y@G@"W$"@R2D23H@X@G@"W$"@R@ Y@G@"Q$"@R26HU@=":161,115175,115:196,115210,115:"@LE10H13V@$@15V10H@$@13V15H@$@15V15H@$@RE@":9RN8W$"@R4B@";:43:" @6B@";::"@13V15H@X@G@"W$"@R@Y@G@"W$"@R@":24OI18:2:5I:37)::"@13V1H@ @5V2H@Since neither Y@G@"Q$"@R@ nor X@G@"W$"@R@ is zero, you@2D2H@can divide both members by X@G@"W$"@R@Y@G@"Q$"@R@ and@2D2H@get:@15V8H@X@G@"W$"@R@";W$"@R@Y@G@"W$"@R@";M" = K":44:"@13V1H@SO:@11V8H@";:I12:"X@G@"Q$"@R@Y@G@"Q$"@R4B@";:43:" @D4B@";::"@13V8H@X@G@"Q$"@R@Y@G@"Q$"@R@ =@11V21H@";:I12CN"X@G@"W$"@R@Y@G@"W$"@R4B@";:43:" @D4B@";::"@21H13V@";:I13:"X@G@"W$"@R@Y@G@"@D28H@ @D28H@ @D28H@ @27H16V@ ":0:168,112175,112:168,111175,111:168,110170,110:3:L"@5V2H@If (X@G@"Q$"@R@,Y@G@"Q$"@R@) and (X@G@"W$"@R@,Y@G@"W$"@R@) are ordered@2D2H@pairs of an inverse variation, then@3D8H@X@G@"Q$"@R@Y@G@"Q$"@R@ = K and X@G@"4,151:213,41213,151:0:I14:32:19I:I)::"@14V33H@ @31H16V@ @DB@ @DB@ @13V35H@ @32H17V@ ":253,104253,112:254,104254,112:255,104255,112:252,104252,112:3:I16:6I:24(6I):I)::9ZKI16:6I:32:I)::"@14V24H@"4)"@D27H@ 4,100R16:(I)(I10)10ĺ"@11V3H@ 2@D2B@ -@D2B@ "((I))"@L7H11V@"(I)"@R13V7H@ "I45::24:1:162,99263,99:"@5V2H@If K=0, then the @D2H@graph is along the@D2H@two axes."9)"@D2H@"9)"@L11V2H@"9)"@D2H@"9)"@R@"J212,41212,151:214,4121T14,100R16:(I)(I10)10ĺ"@11V3H@ 2@D2B@--@D2B@ "(I)"@7H11VL@"(I)"@R13V7H@ "G45::254,110:I31.1:TI:R2I:212T14,100R16:(I)(I10)10ĺ"@L11V3H@"T;Z$"@R@ 2@D2B@--@D2B@ "I"@13V3H@ "VH45::I2.13.1:T2I:RI:212T1E45::24:75:"@6V2H@second and fourth@8V2H@negative.@L11V2H@"8)"@11V3H@X"(19)"Y = -2@R13V3H@ ":170,88:I31.1:TI:R2I:212T14,100R16:(I)(I10)10ĺ"@L11V3H@"(I)(19)"@R11V7H@ 2@D2B@--@D2B@ "(I)VF45::I2.13.1:T2I:RI:212:I31.1:TI:R2I:212T14,100R16:(I)(I10)10ĺ"@11V1HL@"(T)Z$"@R11V7H@ 2@D2B@--@DB@"((I))"@13V3H@ "D45::I2.13.1:T2I:RI:212T14,100R16:(I)(I10)10ĺ"@11V1H@ 2 @D4B@ -- @D2B@"((I))"@11VL7H@"(I)"@R7H13V@ "@R@"qA254,90:I31.1:TI:R2I:212T14,100R16:(I)(I10)10ĺ"@11V3HL@"I;Z$" = 2@R8H@2@DB@-@DB@"IB45::I2.13.1:RI:T2I:212T14,100R16:(I)(I10)10ĺ"@11V3H@2 @D2B@- @D2B@"(I)"@11V7HL@"(R)"@R13V7H@ "vC45::170,110B@"((I))"@L4H11V@"Z$(I)" = 1@R@"p?45::"@16V2H@This shape is called@2D2H@a ";:B$"hyperbola":41:".":9@134:"@6V24H@II@34H@I@34H17V@IV@24H@III@5V2H@The curve is in the@D2H@first and third@D2H@quadrants if K is@D2H@positive.@L3H11V@X"Z$"Y = 21.1:TI:R1I:212T14,100R16:(I)(I10)10ĺ"@L2H11V@"((T))Z$"@R8H11V@ 1@2D3B@ "((T))"@U3B@ --@11HL11V@ = @15H@1@R13V3H@ "#>45::I1.13.1::T1I:RI:212T14,100R16:(I)(I10)10ĺ"@11VL1H@"8)"@D1H@"8)"@R11V2H@1@D2B@--@D258,94:I31.1:TI:R1I:213T14,100R16:(I)(I10)10ĺ"@L11V3H@"T"@R7H11V@ 1@2D2B@ "T"@U2B@ -"<45::I1.13.1:T1I:RI:213T14,100R16:(I)(I10)10ĺ"@11V3H@ 1@D2B@ -@D2B@ "R"@L11V5H@"Z$;R" = @R8H13V@ "=45::165,104:I3"@5V2H@The graph of the@D2H@equation XY = K is@D2H@not a straight line.@D2H@The equation is not@D2H@linear.":134:"@L11V3H@X Y = 1@R18H11V@LET@D18H@K=1@11V5HL@"Z$"@R@":45j;T171:R98:I17:T,RT,R5:TT14::T211:R51:I16:T,RT5,R:RR16::o that@2D2H@the product is equal to the constant@2D2H@of proportionality. We say that Y@2D2H@";:B$"varies inversely as":41:" X or Y is@2D2H@";:B$"inversely proportional":419" to X, because@3D15H@Y = K $ .@23H15V@1@2DB@X":161,131168,131:9:oordinates of its ordered pairs is a@2D2H@non-zero constant. For any ordered@2D2H@pair (X,Y) of a function@3D2H@";7"XY = K, or Y = , where K is a@D20H@non-zero constant.@15V17H@K@17H17V@X":119,131126,131:98"@5V2H@As X increases, Y decreases sz#f" x@  c6`<`@x   %p"p>>"><"">"","">> >~"<,<"*" ">p@pp@pp@p@99>*1,<Acx>``~x?> ><,<"*" "> >>"&8x2* "" "" >""""""2" "-"0"&2" *"", "*"P@@@@P@P@)!> 62*>~Ac>|>" ~>`~0|xw>0"&2" *"", "*0=">"*<x`0>, " > "" >>""2" "2"* ""-0"<&2>"*""&2""*AP@ppppp@pp)9*(,~Ac>|" ~>x`|xxxx0<&2>"*""&2""*?I">"28>x*`9 > >" <>"">**""""!0A" &2"& *&"&2&""*P@@@P@PP0) ?6 ><Ac>|<< >x`x pxxx0 &2"& *&"&2&""*0>I">p""""" *&""""""!"0`",,<>"""">p@ppPpppp99*  6Ac>|>"" >`p`x>0",,<>""""> R>>"@ """6&""""""!"" 0"  6<& * Acx"" >``@x0   <xA06 >>8>>>><""""<>""!"">>> 8  >6 `>`@x>> 8 D "LU :F`F`$L"%e%`$e($`%80%`$80$`'$L:{|0L_`F)׭F F)L i)`) qp`<) Ji L? Ji L? ((d d22(LI GIG`RH`GH`LH`E yIy`OF) ELSSzI zU`TR`P EI E`NLUX F{LY F|LTIF eiLȱ|ȱ{ U`FFB DLF .L`% ImE8Ie$e΅ϩeeυ GIy) e$) =}м`0:)F FmFeF`@D`C r)׍} ~  `K r)׍ `H F($LV F%ʹ-ʹ7ʹ' ʹ6ʹʹ8 ʹ խŠӳ6 ʹӠʹ'ʹ2ʹ-10H3D@<0> RETURN TO CONTENTS@I10H22V@WHICH (0-4) ??@I@":J0:K4:6:Gİ39:48b2MG:N1:37:38n3C4974M53,82,88,915N54,56,58,64,76,49r6"@5V2H@An ";:B$"inverse variation":41:" is a function@2D2H@in which the product of the@2D2H@c:H:F:"@"W"K@"X)::, +Y1300::15:B ,Y12000::15:V -Y150::15:g .35339:2000~ 02:(4)"RUNAM4.3"N1M0:N0:C(24798):37:"@14H5V@LEARNING MODE@10H7V@<1> DISCUSSION@9V10H@<2> RULE@2D10H@<3> EXAMPLE@10H2D@<4> SAMPLE PROBLEM@PK36 #29` $A$"":I0P1(P6):A$A$((256I))::Q(A$):QP28:H:" ";:H:Q: %"@2H1V@"C"@5H@"N"@2HD@"M: &31051:40 '30976 ("@21V1HLI@"19)"@RI@": )R(37)82:T(36)72:UT4(B$)7:VR12:B$" ";:133 *FSE16025 "@I22V1H@"36)I$:E O:H:P0:I0K:256I,32::YY,0c HP:I$((256P));:I$;} G(ZZ):15:G12830 YY,0:HP:((256P));:G14136 G136PPP2 !G149PKPP2 "GG128:G47G58ĖHP:(G);:256P,G:PP2:(ZZ)155ı' YY,0:"@40X40YN@";= G(ZZ):G12817R G155G205ıa YY,0:21k :20u 137 39:G155İ2:(4)"RUN ALGEBRA 4" 49 "@22V6HI@Press SPACE BAR to Continue"I$:YY,0 G(ZZ):15:G12825 YY,0:G6 YY,0:GG176:GJGK6%  "@3H21VI@Press "(1)" Key to View the Next Page@D3H@Press "(2)" Key to View the Last Page"I$ 15:G(ZZ):G12810 YY,0:GG128:G21G810 G21NN1:38 G8NN1:39:N49 37:51 5 63900n24800:27903:A(D)((1)D)1:Z$(19):Q$(21):W$(22):ZZ16384:YY16368:I$"@I@":460:1002:::"@G15C0KE@";:E(F):A(E10):BEA10:EE9:F0ĺ(12E)E)"@B@";(20A)"@B@"(B)"@ER@";: 15:G(ZZ):G128                            ,B4,E5,B5,E6,B6,E7,B7,E8,B8,E9,B9,E0,B0,F1,C1,F2,C2,F3,C3,F4,C4,F5,C5,F6,C6,F7,C7,F8,C8,F9,C9,F0,C0,G1,D1,G2:35339: "@0V2H133C0KL@EDU-WARE@15C@"(14)"@IR@":I35:I:28:11)::"@3V28H@ALGEBRA 4@I@":ZOPENAM4.PROGRESS":(4)"READAM4.PROGRESS":I16:U(I)::G3,I0,G4,J1,G5,J2,G6,J3,G7,J4,G8,J5,G9,J6,G0,J7,H1,J8,H2,J9,H3,J0,H4,K1,H5,K2,H6,K3,H7,K4,H8,K5,H9,K6u A1,D2,A2,D3,A3,D4,A4,D5,A5,D6,A6,D7,A7,D8,A8,D9,A9,D0,A0,E1,B1:(4)"CLOSE":E2,B2,E3,B3,E405,40,77,88,56 :N1,N2,N3,N4,N5,N6,N7,N8,N9,N0,M1:2,3,0,6,1,0,2:I06:C(I): 37,166,42,175,45,181,52,185,56,187,64,186,72,184,82,182,88,191,104,193,107,190,112,183,114,184,120,186,124,184,128,185,132,183,141,179,145,156,150,159,168 2:(4)" 35339:7:9:3:K12:I05:C(I):5:I,K:C(6):5:36251,0:I120:36251I,0::I3629936351:I,0::2:(4)"RUNALGEBRA 4" :"@0V0HG10C0K@Q";:8:"E":I223:I:1:1;:40:1::"Z";:8:"C":15 I239:"R";::O 91,24,64,104,126,64,1,F0C0,G1D1,G2A1,D2:G3,I0G4,J1G5,J2G6,J3G7,J4G8,J5G9,J6G0,J7H1,J8H2,J9H3,J0H4,K1H5,K2H6,K3H7,K4H8,K5H9,K6G3,I0: I302:I3I3:N1,N2N1I3,N3N1,N4N1I3,N3N1,N2:N5,N3N1,N3I3N6,N3N1,N3I3N5,N3::N7,N8N9,N0:N9,N8N7,N0:;::255:ZZ(0):24800:27903:(4)"BLOADEWS3":3:6J0:1002:A1,D2A2,D3A3,D4A4,D5A5,D6A6,D7A7,D8A8,D9A9,D0A0,E1B1,E2B2,E3B3,E4B4,E5B5,E6B6,E7B7,E8B8,E9B9,E0B0,F1 C1,F2C2,F3C3,F4C4,F5C5,F6C6,F7C7,F8C8,F9C9     = 3:6@R@":42:"@9V2H@EXTREMES":73,75196,75:83,7583,81:196,75196,81:42:"@2D18H@MEANS!C111,97111,99118,99:162,99167,99167,97:42:"@D2H@The first and fourth "N$"s are@2D2H@called the ";:E$"extremes":40:", and the@2D2H@second and third "N"s and when@2D2H@there is no whole "N$" other than@2D2H@one which is a common divisor of theA"@D2H@terms:@L2H2D@8:4@9H@=@12H@6:3@19H@=@22H@4:2@29H@=@32HI@2:1@RI@":8\B"@5V2H@A "P$" is an "Q$" which@2D2H@states that two "R$"s are equal:@L11H3D@1:2 L@34H@PENNY@L19H@:";:42:"@132C13H@c@130C25H@c";:42:"@9H@";:K15:"c";::42:"@143C13H17V@5"(17)"@19H@:@25H@1"(17)"@R@":8\@"@5V2H@A "R$" is said to be expressed in@2D2H@";:E$"simplest form":40:"when both terms of@2D2H@the "R$" are whole "N$17,91231,91:X10:L126:W75:B106:150>"@5V2H@But both "N$"s must be expressed@2D2H@in the same units of measure. For@2D2H@example, you can't compare a nickel@2D2H@to a penny "Y$", but you can@2D2H@convert the nickel to 5 pennies: ?"@15V2H@NICKE ways:@3D2H@USING A@D2H@DIVISION@D2H@SIGN@2D2H@USING A@D2H@RATIO@D2H@SIGN@6U20H@AS A@D20H@FRACTION@3D20H@AS AN@D20H@ORDERED@D20H@PAIR":B1A(6)1:B14B1659'=C1(A(8)1)B1:"@11V13H@"C1(5)B1"@4D13H@"C1":"B1"@5U31H@"C1"@2D31H@"B1"@3D30H@"C1","B1:2H@A ";:E$R$:40:"of one "N$" to another@2D2H@(not zero) is the quotient of the@2D2H@first "N$" divided by the second.@2D2H@The two "N$"s are called the@2D2H@";:E$"terms":40:"of the "R$".":8;"@5V2H@You may write a "R$" in a "N$" of@2D2H@differentsay that "A1B1" is "A1" times as@2D2H@large as "B1". This comparison is made@2D2H@by computing the quotient:9"@3D2H@You may also say that the two "N$"s@2D2H@have a "R$" of "A1" to 1.@6U29H@"B1A1"@2D29H@"B1"@32HU@= "A1".":203,99217,99:8:"@5V2> SAMPLE PROBLEM@10H3D@<0> RETURN TO CONTENTS@I10H6D@WHICH (0-4) ??"A$:Q4:5:Kİ38:46m4MK:P1:36:375C293:M55,70,76,867P56,58,59,62,64,66,68,51m8A1A(8)1:B1A(8)2:"@5V2H@When you compare the "N$"s "B1A1" and@2D2H@"B1", you :5:CK:36:24(C$(K))2:"@2VI@"C$(K)A$:Cİ38:2:B$"RUNALGEBRA 4"u137:C5Ĺ36251,3:I14:36259I,0::862C2İ2:24798,C:B$"RUNAM4.3.1"Y3M0:P0:36:"@14H5V@LEARNING MODE@10H7V@<1> DISCUSSION@9V10H@<2> RULE@2D10H@<3> EXAMPLE@10H2D@<4?"A$"@7V4H@";:I14:(22)"@2DB@";::"@6V3H@";:I15:"<"I">@2D3B@";::X19:L42:W46:B56:I15:128:WW16:BB16:G0"@16V4H@"(31):21,13242,13249,14042,14821,14814,14021,132:"@17V3H@<0>":59,3259,158:60,3260,158:31,12031,129:O0:Q5"@"Y"K@"Z):: *43:43:434 +L1300::14:@ ,42:42[ -35339:I15:C$(I): .C0:36:"@20H5V@CONTENTS@6V@":I15:"@10H@<"I"> "C$(I):I4ĺ"@14H11V@VARIATION"y/:"@10H14V@<0> Return to ALGEBRA Menu@2V7HI@"31)"@11H22V@WHICH (0-5) ?((256I)):II(K$" "K$"0"):D$D$((256I))::V(D$):VU28:H:" ";:H:V:x $"@2H1V@"C"@5H@"P"@2HD@"M: %31051:39 &30976 '"@21V1HLI@"19)"@RI@": (W(37)82:X(36)72:LX4(E$)7:BW12:E$" ";:128 )NSF:H:N:I22V1H@"36)A$:> T:H:Q1);:U0:I0Q:256I,32::J,0\ HU:A$((256U));:A$; 400:KK128:K029:J,0:HU:((256U));:K1335:UU2(K21UQ)2(K8U0):K47K58ĖHU:(K);:256U,K:UU2:UQ35 "28X #D$"":I0U2:K$8:P51 36:53% (G)155ı< J,0:"@40X40YN@";b K(G):K12816:K155K205ıp J,0:20z :19 130 38:K155İ2:B$"RUN ALGEBRA 4" 51 "@22V6HI@Press SPACE BAR to Continue"A$:J,0 400:J,0:K16024:"@2F)F)"@B@";, (20A)"@B@"(B)"@ER@";:S 14:400:J,0:KK176:K0KQ5: "@3H21VI@Press "(1)" Key to View the Next Page@D3H@Press "(2)" Key to View the Last Page"A$ 14:400:KK128:J,0:K21K89:K21PP1:37 K8PP1:3+24800:27903:A(E)((1)E)1:G16384:J16368:A$"@I@":UU36251:N$"number":B$"":P$"proportion":R$"ratio":V$"variable":W$"variation":Y$"directly":Q$"equation":450:1002: "@G15C0KE@";:F(N):A(F10):BFA10:FF9:N0ĺ(1D2H@we say that one "V$" ";Z'eE$"varies":40:"@2D2H@";:E$Y$:40:"with the other.":8+(f"@5V2H@A ";:E$"linear direct "W$:40:"is a@2D2H@function in which the ratio between a@2D2H@"N$" y of the range and the@2D2H@corresponding "N$" x of the domai1"V2H@"C1"@"4C12"H"14C1"V@"C1:"@"3C1"H"14C12"V@"C1:C13ıd&c44:S9:F18:H3:Y0:Z9:41:'d"@5V2H@When two "V$"s are related so@2D2H@that the "R$" of one value of one@2D2H@"V$" to the corresponding value@2D2H@of the other "V$" is constant,@2@10V5H@= =@U24H@SIDE PERIMETER@2D25H@1@7F@4@2D25H@2@7F@8@2D25H@3@7F@12@L15H5U@4@R@":14,8328,83:49,8356,83:77,8384,83:8A&bX21:B120:LX(C12)8:WB(C12)8:128:26:10C1(C11)(C12):C1"@7F@"B1"@6V5H@"C1"@18V25H@"B1"@"3C1"H16V@"C1"@"14C:C11:B14:98:C12:B18:98:C13:B112:98:23$`S5:F19:Z37:H3:41:"@5V1H@Notice that the "R$" of the perimeter@D1H@to the side is always the same.@17V1H@We say that the "R$" is a ";:E$"constant":40:".@9V2H@12 8 4@2D3H@3 2 1@U@ ="%a"NG, ANSWER IS "B(F)J#[23:K17:3:12K:36):::37:(UU)0118:51`#]M94,109,115,118#^P95,100,102,104,105,106,51/$_"@5V2H@Suppose we had a square with a length@D2H@of 1 on the side.@24H8V@SIDE PERIMETER@18V2H@Its perimeter would be 4 .":P1F1:36:500:"@12V2H@FIND THE MISSING NUMBER":C(1)3:C(2)12:C(3)23:C(4)31:FA(4):I14:IFĂ:89"X"@L14V"C(I)"H@"B(I):"YT15:HC(F)1:Q4:"@10H14V@:@19H@=@29H@:":27:"@R@":VB(F)ĺ"@17H17V@RIGHT":UU9,(UU9)1:91#Z"@11H17V@WRO123:8!VX10:L269:W35:B83:128:W92:B155:128:"@I5V2H@An "Q$" of two "R$"s is given @D2H@below. One of the four "N$"s is @D2H@missing. Fill in the missing "N$" @D2H@so that the "Q$" will become a ":F195((UU)0)l"W"@9V2H@true "P$"."20)A$8 T"@5V2H@Find the "R$" of "C1" "F$" to@D2H@"B1" "G$".@9V2H@<1> Put the@D6H@measures@D6H@into the@D6H@same unit.@2D2H@<2> Divide the@D6H@measures.@9V20H@"C1" "F$"@D21H@EQUALS@D20H@"A1" "G$!U"@20H2D@"A1" "G$"@20H2D@"B1" "G$"@31HU@= "E1:140,123210,@"P$".@3D2H@<1> Find the product@D6H@of the extremes.@2D2H@<2> Set the result@D6H@equal to the@D6H@product of the@D6H@means.@2D2H@<3> Solve for x. S"@10V25H@"C1"$"A1" = "E1"@25H3D@"B1"$X = "E1"@25H4D@X = = "D1"@29HU@"E1"@2D29H@"B1:203,139217,139:ounds":G$"ounces":D1A(5):E1A(5):C1E1D1:B1D116:A1C116:84Q0:12,59267,59:3:10,67269,67:C1A(5):B1A(9):A1A(5)B1:E1C1A1:D1E1B1:C1B181:"@5V2H@Find the value for X that will makeR"@2H@the "Q$" "C1":X = "B1":"A1" a true@D2H2H@"P$".":8:LP9X10:L269:W35:B59:128:B156:128\MP78,79,80,81,81,81,81,81,51NF$"nickels":G$"pennies":D1A(5):E1A(5):C1E1D1:B1D15:A1C15:84OF$"feet":G$"inches":D1A(5):E1A(4)1:C1E1D1:B1D112:A1C112:84CPF$"pth neither b nor c equal@D2H@to zero, then their "R$" is:@2D17H@a:b:c@3D2H@If a:b = c:d, with neither b nor d K"@2H@equal to zero, then a$d = b$c.@2D2H@a and d are called the extremes.@D2H@b and c are called the means.@D2H@The "Q$" itself is called a@D@a@25H@x@2DB@x@32H@bx@2U2B@ax@L7H9V@=@R12H@$@L14H@1@17H@=@23HR@$@28HL@=@R@":35,7542,75:70,7577,75I147,75154,75:224,75238,75:175,75182,75:8mJX10:L269:W35:B83:128:W91:B155:128:"@5V2H@In general, if a, b, and c are real@D2H@"N$"s wie "N$"s with@D2H@neither b nor x equal to zero, then@6D2H@and a:b = ax:bx@2D2H@To find the "R$" of two quantities@D2H@of the same kind:eH"@D2H@<1> Find measures in the same unit@D2H@<2> Divide the measures@8V5H@a@2DB@b@10H@b@2UB@a@21H@a@2DB@b@21H@b@2UB:H:T:" ":HH2::"@12H16V@"(19)"6":42:"@18H16V@=@14V22H@3":43:"@14V22H@ @22H16V@3":43:H15:T15:R17:H:T:"2":43:H:T:" ":HH2::"@24H16V@"(19)"2@R@":8FP71,74,51GX10:L269:W35:B107:128:B156:128:"@5V2H@If a, b, and x ar$"s the ";:E$"means":40:".":8D"@5V2H@In any true "P$", the product@2D2H@of the extremes is equal to the@2D2H@product of the means.@L3D10H@1:2 = 3:6":42:X11:T15:D12:T15ĖX:T2:" "EX:T:"1":43:TT2::H27:T15:L17:H:T:"6":43:L199:X137:128:L269:128:W75:B106:128:L199:128:8340:"or the@2D2H@";:E$"constant of "W$:40:".":14,8321,83:83K(G):14:K128400:3B(1)A(5):B(3)A(9):B(4)A(5)B(3):B(5)B(1)B(4):B(2)B(5)B(3):B(1)B(3)500:2:B$"RUNAM4.3.1"2|V2D1A(10):C1A(9):A1A(9):B1D1C1:E1D1A1:C1A1127:q2X,WL,WL,BX,BX,W:2RATIO AND PROPORTION,DIRECT VARIATION,INVERSE VARIATION,JOINT AND COMBINED,VARIATION TEST,:3128:L73:128:W109:B140:128:L126:128nter your answer in the space"6)"@D2H@provided."27)"@6V37H@ "A$1y"@L6D4H@Y = ";:Q4:H13:T14:27:"@R@":VE1ĺ"@17V2H@RIGHT":36261,(36261)1:1231z"@17V2H@WRONG. ANSWER IS "E12{23:S6:F10:Z36:H3:A$:41:A$:S13:F18:41::37:(UU)İ kx@D27H@y = ("D1")("A1")@27HD@y = "E1:8G0vF195((UU)0):120:51A1xX10:L269:W35:B83:128:W91:B155:128:P1F1:36:127:"@I5V2H@y varies directly as x."13)"@D2H@If y="B1" when x="C1","19)"@D2H@what is the value of y when x="A1"? @D2H@E128:127:"@5V2H@If y varies "Y$" as x, and if@2HD@y="B1" when x="C1", find the value of y@D2H@when x="A1". )0u"@10V2H@<1> Find the constant@D6H@of "P$"ality.@3D2H@<2> Solve for y when@D6H@x="A1".@5U27H@y = kx@D27H@"B1" = k("C1")@D27H@k = "D1"@2D27H@y =me "N$".@2D2H@<4> When x is divided by a "N$",@D6H@y will be divided by the same@D6H@"N$". .r"@D2H@<5> The graph of y=kx is a straight@D6H@line whose slope is k and which@D6H@passes through the origin.":8l/sP451:X10:L269:W35:B67:128:B139:","-p"@6H@any other pair of values.@16V17H@y";:N1:3:" y";:N2:3:"@2D17H@x";:N1:3:" x";:N2:3:"@20H17V@=":119,139132,139:147,139161,139:8\.qB155:128:"@6V2H@<3> When x is multiplied by a@D6H@"N$", y will be multiplied by@D6H@the sa2D2H@<1> The "R$" y:x is constant.@D6H@That is, y=kx, where k is the@D6H@non-zero constant of@D6H@"P$"ality.-o"@D2H@<2> The "R$" of y";:N1:3:" and x";:3:", any pair@D6H@of values for the "V$"s, is@D6H@equal to the "R$" of y";:N2:3:" and x";:3:X215:W98:I14:X,WX,W4:XX14::"@27H5V@y@9D36H@x":44:"@31H7V@k@8D9H@and having a@2D2H@slope equal to the constant of@2D2H@"P$"ality.":214,84235,84235,54:8+mP110,113,51l,nX10:L269:W35:B155:128:"@5V5H@PRINCIPLES OF PROPORTIONALITY@s a straight line@2D2H@passing through the@2D2H@origin":X185:L269:W35:B123:128*k"@5V29H@"(95)"@37H7D@>@5V28H@";:206,41206,99262,99:206,99245,41:N1422:3:"@DB@";::"@13V30H@";:N14:3:" ";::X204:W51+lI16:X,WX4,W:WW8::0)i"@5V2H@In a direct "W$", you can say@2D2H@that y varies "Y$" as x, or@2D2H@y is "Y$" "P$"al to x.@3D2H@y = kx, where k is a nonzero@2D2H@constant.":8S*j"@5V2H@The graph of a linear@2D2H@direct "V$" with "(6)"@2D2H@as the domain and range@2D2H@in@2D2H@is the same for all ordered pairs ofU(g"@D2H@the function except (0,0).":8)h"@5V2H@This definition can be rewritten in@2D2H@mathematical terms:@9V2H@y@2D2H@x@4H10V@= k , x@G@ = @R@0. The "V$" k is the@3D2H@";:E$"constant of "P$"ality":16(B)155ı! I,0:"@40X40YN@"G K(B):K12817:K155K205ıU I,0:21_ :20i 150 47:K155İ3:A$C$ 60 "@22V6HI@Press SPACE BAR to Continue"B$:I,0 K(B):15:K12825:I,0:K16025:"@I22V1H@"36)B$:_ K2 15:K(B):K1286:I,0:KK176:KJKU6: "@3H21VI@Press "(1)" Key to View the Next Page@D3H@Press "(2)" Key to View the Last Page"B$ 15:K(B)128:K010:I,0:K21K810:K21PP1:46 K8PP1:47:P60 45:62 + 150ZZ36251:E110:A(E)::I16368:B16384:A$(4):B$"@I@":C$"RUNALGEBRA 4":D$"RUNAM4.2":35339:B(J)((1)J)1:590:1002:"@G15C0KE@";:F(N):A(F10):EFA10:FF9:N0ĺ(12F)F)"@B@"; (20A)"@B@"(E)"@ER@";:                @@``@|@@@bx6 @|8`0p@ @0 0|| @L << @ 0 0|~ff@@@@@@ |@@@`@xp@@@?6px@0 ~@x@@ |p||||0|p0@| |||| | p| | | | |@|@ @|||@~x@L@@@@|@ |@@`@x@@@&pF@0`0L@ @@ < @@ < @ < @ 0 0||||||||@@@@@@@@@p@@~pp@X@?`Fx|0 p`L@ @ 0 @@@| @ @ 0< @ p```````@L@@@@ 0<00@@@@p@@?d`6x@00p`L@ p @ 0 @@ 0 @ @ |xxxxxxx@@@@@@@@@x`x@@f|p||@86~000L@0 @ 0 @@@@| @ @ 0<0@ |xxxxxxx@L@@@@ 0<00@@@@`@? `p@00x@x@|p|||p~||0||p|||||| p |||||| || |0p```````@@@@@@@0```|@`|||xp @ `|@|00p@0 @<| 0 @0|@| @L LL@ L0@0 ~||||||||@L@@@@ L@p@@@@@@8`~@00`@p@|||@~pp0p|@|p|||p p p|p|p| || |@@@@@@@@@0`x@@`~|x`pp6 p`@|00p@0 @0| 0 0|@|| @L L L@ L0@0 |~f>@L@@@@ L@|@@@``~@@@@@@@@~~@@|`~pppp|F p@0@|||@ @@@ |@pp@ ||| || |@< LL | |p@ @@@ @x~ff>>fff@|@@ 0 0`0??"?6?30>>0<0~<13a<?33>>?3?~6? ?#6?33080<0 1? 21 6g~6?<'~8<<0x?p <411???016?30>>33?`?y$%GΩϩ  %%%GΩϩ  %%$$'к`&$%GΩϩ  %%%GΩϩ  %%$$м`  p6 6619 1991119 19910 99 9999999 9999997 97799100 100010011110 110011111120 14 14           71,119:"@12V12H@(X";:N1:4:",Y";:4:") $":50:"@8V27H@(X";:N2:4:",Y";:4:")@9V27H@$":50:"@10V23H@M":143,99192,75:50!M1:193,75193,99:"@9V29H@Y";:N2:4:"@29H12V@Y";:N1:4:3:"@10V2H@=@4H9V@Y";:N2:4:"-Y";:N1:4:28,8363,83:50:2X@2H2D@values.":9K"@5V2H@SLOPE M = @U12H@DIFFERENCE OF Y VALUES@2D12H@DIFFERENCE OF X VALUES":84,43238,43:50:J80:O270:U59:T123:148:"@12H11V@<@24F@>@3U24H@"(95)"@24H14V@"(126):172,64171,119:"@8V25H@Y@12V37H@X"L85,91262,91:171,641 of a line@2H2D@which passes through two points@2D2H@P";:N1:4:"(X";:4:",Y";:4:") and P";:N2:4:"(X";:4:",Y";:4J") is the@2D2H@quotient of the differences of Y@2H2D@values of these points divided by the@2H2D@difference of the corresponding rence of the ordinates, and the@2D2H@horizontal change is the accompanying@2H2D@change of abscissas, the slope of a@2H2D@line is:"H"@16V13H@DIFFERENCE OF ORDINATES@13H2D@DIFFERENCE OF ABSCISSAS":91,139252,139:9zI"@5V2H@In general, the slope, m,51:1J:17Q:"@128CL@"(20)"@143C@":QQ1:DD2:HH4::"@11V17H@"(20)"@12V26H@20 1@15V26H@80 4@31H14V@=@36H@.@R@":185,115207,115:240,115250,115:9G"@5V2H@Because the vertical change in moving@2H2D@from one point to another is the@2H2D@diffeio:":50:"@16V19H@";:N5:4:"@UB@";:N1:4:N0:4:"@U2B@";:N1:4:N5:4:"@U2B@";:N2:4:N0:4:"@18V9H@RUN@4U21H@RISE"DH7:V18:D0:"@L@":Q1:J1204:1J:17Q:(20):H:V:D0HH5:70END:4:N0:4F21,136133,136133,10421,136:D@of a hill, you must determine the@2D2H@amount of vertical ";:F$"rise":49:" for the@2D2H@amount of horizontal ";:F$"run":49:".":9|C"@5V2H@For example, if a hill rises 20 feet@2D2H@over 80 feet of horizontal distance,@2H2D@its slope is the rat:"F"B$E)B$::"@143CR2H17V@S@UR@L@UR@O@UR@P@UR@E@4H18V@RUN@8H16V@RISE@14V13H@SLOPE = @U@VERTICAL CHANGE@2D21H@HORIZONTAL CHANGE@2D21H@RISE@D19H@=@D21H@RUN"B147,115266,115:147,147175,147:"@5V2H@To find the steepness, or ";:F$"slope":49:",@2H2CUSSION@9V10H@<2> RULE@2D10H@<3> EXAMPLE@10H2D@<4> SAMPLE PROBLEM@10H3D@<0> Return To CONTENTS@I11H6D@WHICH (0-4) ??"B$:J0:U4:6:Kİ47:3:A$D$=MK:P1:45:46>C4116?M64,79,82,103@P65,67,71,73,75,60A"@130CG13V@";:E04:8E))DA(A1)(DA(A1))NNA(A1):DDA(A1):54R7:W0ND0ĢV:H:"@L@-":HH2u8W(N)ĢV:H:"@L@"(W):HH29NĢV1:H:"@R@"(N):H:"---":H:(D):"@R@":;C(31152):(ZZ)2102<M0:P0:45:"@14H5V@LEARNING MODE@10H7V@<1> DIS"M: .31051:48 /30976; 0"@21V1HLI@"19)"@RI@":y 1U(37)82:J(36)72:OJ4(F$)7:TU12:F$" ";:148 215:E175:52:: 315:A1115:52:: 415:K(B)128: 5W((N)(D))(ND):NNWD:A1110-6NA(A1)(NA(A1 &K(B)128:15:K038:I,0:HX:((256X));:K1344:XXD(K8X)D(K21X(U1)D):K47K58K45ĖHX:(K);:256X,K:XXD:XUD44 +37 ,E$"":E0X1:E$E$((256E))::Y(E$):YX36:H:" ";:H:Y: -"@2H1V@"C"@5H@"P"@2HD@(B)128:K028:15:I,0:K85K68K82K76K8028:DU:JJ3(K82)3(K76):J36J24n J24J36 UU(K85)(K68):U7U17 !U17U7 "K80ı #146:28 $V:H:U);:X0:E0UD:256E,32::I,0 %HX:B$((256X))B$;@ JTL J UEPWRz6F GFy 7 ?` !!""## ( !(!"("#(# P !P!"P"#P# ((ˍ) J ?`DLk@D` LH5Ω " .Ωz % $L.jjΩ .@0~~@@|@>f@<@@ 0 0@pp@@@L@@@@p@0`x@@@0@@@p#| `@|x0p@ @0| @0|| @L << @ 00 0~~f>>fff@ @@@@@ |@@@@@@@ppp@0 x@@ pp|||p0pp@0@ |p|| p pp | p p |@p@ @|||@|x@L@@@@p@ ||z234,106234,58:"@10V34H@3@14V30H@3@12V2H@Using the slope@D2H@formula, we can find@D2H@the slope of the line.@2D2H@2-(-1) 3@2D2H@1-(-2) 3@9H17V@=@3F@=@L16V15H@1@R@":14,13956,139:77,13984,139:24({E116:3:4E:22)::"@5V2H@Point (1,2) mu6::J192:U88:E15:J,UJ,U8:JJ14::"@33H7V@"(3)"@13V27H@"(3)&y"@32H5V@Y@BD@(1,2)@37H12V@X@15V25H@(-2,-1)":220,88220,120:221,88221,140:50:248,46190,108:"@8V2H@A line can be drawn@D2H@through any two@D2H@points.":50:192,107234,107'1220,151:221,41220,151:176,91262,91%w"@12V27H@";:N2:4:" ";:N1:4:"@3F@";:N1:4:" ";:N2:4:"@5V29H@";:N3:4:"@2D29H@";:N2:4:"@2D29H@";:N1:4:"@3D29H@";F&x"@3D29H@";:N2:4:"@2D29H@";:N3:4:J217:U43:E17:J,UJ7,U:UU113E:21)::E114:5E:26:13)::144::46:(ZZ)2İ3:A$D$K$s60b$tM117,124,127,132p$uP118,60(%v"@2H5V@Look at the two points@D2H@graphed on the right.":J172:O269:U35:T156:148:"@11V25H@<@37H@>@31H5V@"(95)"@18V31H@"(126):220,4"D1","E1") and@D2H@("C1","B1")?@D11H@M= ":98,139119,139:V17:H15:U3:36:Y0109:AY#oH15:V19:36:Y0111:(B1E1)(C1D1)AYĺ"@18V18H@RIGHT":ZZ6,(ZZ6)1:114#q"@18V2H@WRONG@2U20H@"B1E1"@2D20H@"C1D1:140,140160,140C$r24:E16:3:"i"@5V2HI@The line containing@D2H@the pair of points@D2H@given below is@D2H@graphed on the right.@D2H@Find slope m of the@D2H@line.@I5V32H@Y@7D4F@X":P1F1:45:139:2000:3000X#mD1:"@2H13V@What is the slope of@D2H@the line that passes@D2H@through (133,91154,91:3000N!eN(B1E1):D(C1D1):H9:V16:"@L15V5H@=@R@":53:9j!fC5:45:(36272)1132!gD1:F195((ZZ)2):36272,1:J10:O164:U37:T89:148:U99:T155:148:U37:J171:O269:148:144:B$:E16:3:5E:21)::B$:14,39160,39:10:JE1:147 ^"@12H10V@ @3B@"J:51:H4:V10:A4:JC1:146:14:N7:L1:147:"@12V6H@ @3B@"J:N17:H12((C1))((B1)):L1:JD1:10:147!_H17:V11:A4:146:14:L1:H16:N13:147:"@12V12H@ @3B@"J:51:"@11V17H@ = @U@"B1E1"@2D20H@"C1D1:points@D2H@("C1","B1") and ("D1","E1") is@2D5H@(Y";:N2:4:" )-(Y";:N1:4:" ) @D2H@m=@D5H@(X";:N2:4:" )-(X";:N1:4 ]2000:" )":35,91105,91:51:H5((C1)):N7:L1:10:JB1:147:"@10V6H@ @3B@"B1:51:H13((C1))((B1))((D1)):N13:L12:J10:O164:U83:T115:148:248,40248,152:3:"@11V2H@If a line is parallel@D2H@to the Y-axis, it has@D2H@no defined slope.":9[J171:O269:U35:T155:148:144:139:U15B11591y\"@2H5V@The slope of the line@D2H@that passes through@D2H@the ,U:S,GE2,UZ:SE:GUVX:51:"@11V2H@The slope of the line @D2H@is negative.":9Y5:J11:O163:U35:T65:148:3:144:J171:O269:U36:T156:148:"@5V2H@If a line is parallel@D2H@to the X-axis, the@D2H@slope is zero.":5:175,59266,59Z50: line@2HD@is positive.":9V"@2H5V@As a point moves from@D2H@left to right along a@D2H@line that is falling,@D2H@Y decreases as X@D2H@increases.":J170:O269:U35:T156:148:144:Z.2((1)8):J182:U48:J,U:E183266:S182:G48WUUZ:U140ēEm@D2H@left to right along a@D2H@line that is rising,@D2H@Y increases as X@D2H@increases.":J171:O269:U35:T155:148:144:Z.2((1)8):J183:U140:J,U:S182:G144TE183266:UUZ:U48ēE,U:S,GE2,UZ:SE:GUU:51:"@11V2H@The slope of theght.@D2H@<3> Find the vertical change, the@D6H@change in Y-values, in going@D6H@from the left point to the right@D6H@point.@D2H@<4> Divide the vertical change by@D6H@the horizontal change.":9RP83,86,89,91,91,91,91,60S"@5V2H@As a point moves froPJ10:O269:U35:T156:148:"@5V2H@To find the slope of a line.@2D2H@<1> Select any two points on the@D6H@line.@D2H@<2> Find the horizontal change, the@D6H@change in X values, in going@D6H@from the point on the left to";Q"@D6H@the point on the ri:50:146,99193,99:"@20H13V@X";NN1:4:"@27H@X";:N2:4:"@11V4H@X";:4:"-X";:N1:4:50:"@16V2H@A property of a line is that its@D2H@slope is constant.@17V21H@So, you may use@2HD@any two of its points to compute the@2HD@slope.":3:9OP80,60must be@2D1H@"W$").":7D>@I1100:(Z)12827I150:::Q,0:98L>A:R>Bq>(#G1:L1:"@5V28H@"10):97>>6>>"">6"6"6"               G,D:E150:RR(C1D1)3:ZZ(B1E1)3:Z36Z156E50::3003_3 R172R269E50::3003k3 R,Z:3 G,D:E150:QQ(C1D1)3:AA(B1E1)3:A36A156E50::3 Q172Q269E50::3 Q,A:ZQ,A:LO,UO,TJ,TJ,U:H2 2,3,5,7,11,13,17,19,-2,-3,-5,-7,-11,-13,-17,-192:3:"++ERROR++ "(222)" AT LINE # "(218)(219)256:2Z91B13:R220C13:D91E13:G220D13:R,ZR1,ZR1,Z1R,Z1R,Z:G,DG1,DG1,D1G,D1G,D:A3 QR:AZ:R,Z5:Nĺ"@2D@";:r14:"@2D"30(N0)"H@";::J192:E15:J,89J,95:JJ14::U43:E17:218,U223,U:UU16::1E1A:H:V:J;:51:H:"@128C@"J;:"@143C@";:VV1:"@11V3H@="::1EHNL:H:J;:51:H:"@128C@"J;:HHL:"@143C@";::2J,U(9)1B(2):C1B(9)1B(2):D1B(9)1B(2):E1G1D1I1:B1G1C1I1:H1G1D1:1"@5V31H@"(95)"@31H18V@"(126)"@7U25H@<@37H@>":176,91261,91:220,41220,150:"@26H12V@";:N10:4:" ";:N5:4:"@3F@";:N5:4:" ";:N10:4:"@5V30H@";:N1515 = ("G1")X + ("I1")"O/24:E18:3:11E:36):::46:(36251)2İ3:A$C$W/60/SB(5)1B(2):JB(5)1B(2):USJ:D1B(5)1B(2):E1B(5)1B(2):B1UD1:C1JD1:E115B115E115B115D1C1E1B1139:S0G1B(4)1B(2):I1B0)0:C(1)0:C(2)0:B1E1C1D1134."@11V2H@THE EQUATION THAT PASSES THROUGH@D2H@("D1","E1") AND ("C1","B1") IS:@2D8HL@Y =@22H@X+":H15:V15:U3:36:QY:U3:"@L@":H27:V15:36:QG1YI1ĺ"@3H18VR@RIGHT":36259,(36259)1:137/"@R8H17V@WRONG, Y9:U37:T74:148:U83:T156:148:"@I5V2H@Find the equation of the line that @D2H@passes through the two points given @D2H@below. Then fill in the missing @D2H@portions of the equation."11)"@I@":14,39266,39:14,72266,72:P1F1#.141:45:S0:C(Write the@D6H@equation in@D6H@slope-intercept@D6H@form.@23H10V@Y="G1"X+"I1"@15V2H@<4> Check the@D6H@equation by@D6H@using the other@D6H@point.@15V23H@"B1"="G1"("C1")+("I1")@23HD@"B1"="J1"+("I1")@23HD@"B1"="B1:9,D2:F19:(ZZ)2F14:M5-J10:O26 = @U@"B1"-("E1")@26H11V@"C1"-("D1")@10V35H@="G1:182,83238,83+"@13V2H@<2> Find the@D6H@Y-intercept by@D6H@using one of@D6H@the points.@13V22H@b ="E1"-("G1")("D1")@24H14V@="E1"-("H1")@15V24H@="I1:24:E110:3:9E:36)::J1G1C1,"@10V2H@<3> :4:")@2D2H@<3> The equation is Y=mX+b":97*P460v*J10:O269:U36:T156:148:T67:148:141:D1C1E1B1128?+"@5V2H@Find the equation of the line that@D2H@passes through the points@D2H@("D1","E1") and ("C1","B1").@10V2H@<1> Find the slope: m2H@that passes through points (X";:N1:4:",Y";:4:")@D2H@and (X";:N2:4:",Y";:4:"):"+*~"@2D2H@<1> Find the slope: m = @U@Y";:4:"-Y";:N1:4:"@2D26H@X";:N2:4:"-X";:N1:4:182,83217,83:"@2D2H@<2> Find the Y-intercept:b=Y";:N1:4:" -m(X";st@D2H@satisfy the equation@2D7H@Y=mX+b":50:"@10V2H@So, 2=m(1)+b,@D7H@2=1+b@D7H@b=1":50:"@14V2H@So, the line's":"@15V2H@Y-intercept must be 1,@2HD@which it is.":9(|P125,60[)}J10:O269:U36:T132:148:"@2H5V@To find the equation of the line@D LINEAR6INTERCE" ">"">0>>>>>>>od@2D2H@practice to plot a@2D2H@third point as a@2H2D@check on the first@2H2D@two.":E151:V270:U36:X156:147@116:117:"@36H6V@"D$:41:"@10V32H@"D$:41:265,40168,152:41:"@34H8V@"D$:265,40168,152:9AP66,68,48zBE10:V270:U36:X132:2D@and C are real "Y$"s, with A and B@2H2D@not both zero. The graph of the@2H2D@"Q$" will be a "S$", so">"@D2H@it is called a ";:A$"linear "Q$:40:".":9o?"@5V2H@You need plot only@2H2D@two points to graph@2H2D@a linear "Q$",@2H2D@but it is goe set of all@2H2D@those points whose "U$"s@2H2D@satisfy the "Q$". The line is@2H2D@called the ";:A$"graph of the "Q$:40:".":9h="@5V2H@A first degree "Q$" in two@2D2H@variables, like X+Y=3, can be written@2H2D@in the form AX+BY+C=0, where A, B,@2H@ @R27H@-1@DB@4":41:"@31H5V@0@DB@3@I34H5V@1@DB@2@I13V31H@"D$:41:"@34H5V@1@DB@2@I37H5V@2@DB@1@33H14VI@"D$:41:"@37H5V@2@DB@1":168,79259,131:9;"@5V2H@Each root of the "Q$" does give@2D2H@"U$"s of a point along this@2H2D@line. This line is th7:E249:147:114:115:"@5V22H@X@1F@-2@F@-1";:I02:"@2F@"I;::"@6V22H@Y@14F@";:I15:I"@4B@";:9"@I5V24HL@ @R24H@-2@D25H@5@25H10VI@"D$:41:"@5V24HL@ @R24H@-2@D25H@5@27H5VLI@ @R27H@-1@D28H@4@I11V27H@"D$:41:"@5V31HI@0@DB@3@I29H12V@"D$" @5V27HLhat are solutions of"+6"@D2H@X+Y=3.":97"@5V2H@If we plot the@2D2H@points associated@2H2D@with the ordered@2H2D@pairs from the@2H2D@table, the points@2H2D@seem to lie on a@2D2H@"S$".":E151:U36:V270:X60:147:E165:V186:147f8E207:V228:141,86,883M52,65,70,7734P53,55,59,61,63,485"@5V2H@Every root of the "Q$" X+Y=3 is@2D2H@an "O$" of "Y$"s (X,Y). If@2H2D@the set "(6)" (real "Y$"s) is the@2D2H@replacement set for X and Y, then we@2D2H@can find an infinite "Y$" of@2H2D@"O$"s t9,3659,154:150:37:C2C4C5860M0:P0:36:"@14H5V@LEARNING MODE@10H7V@<1> DISCUSSION@9V10H@<2> RULE@2D10H@<3> EXAMPLE@10H2D@<4> SAMPLE PROBLEM@10H3D@<0> Return To CONTENTS@I11H6D@WHICH (0-4) ??"F$:I4:6:Hİ38:451MH:P1:36:372C55:"<"I">@2D3B@";::E19:V42:U46:X56:I15:147:UU16:XX16::"@10H16V@<0> Return To ALGEBRA Menu@2V7HI@"31)"@11H22V@WHICH (0-5) ?? "F$$/"@16V4H@"(31)"@D3H@<0>":21,13242,13249,14042,14821,14814,14021,132:31,12031,132:60,3660,154:55339:I18:B$(I)::I15:C$(I)::(HH)0107 -P0:M0:C0:36:"@20H5V@CONTENTS@6V@":I12:"@10H@<"I">@2D3B@";::"@U@<3>@2D3B@";:I45:"<"I">@2D3B@";::"@7V14H@";:I18:B$(I)"@D14H@";.:"@7V4H@";:I14:(22)"@2DB@";::"@6V3H@";:I1CEJ28: #122:287 $"@2H1V@"C"@5H@"P"@2HD@"M:F %31051:39Q &30976n '"@21V1HLI@"19)"@RI@": (U(37)82:E(36)72:VE4(A$)7:XU12:A$" ";:147 )15:I175:43:: *I1150:15:: +15:H(Y)S1:1 ,Y$"number":3< H(Y):15:HS125:R1,0:H16025:"@I22V1H@"36)F$:~ H(Y)S1:H028:15:R1,0:H85H68H82H76H8028 TE:DU:EE3(H82)3(H76):E36E24 E24E36 UU(H68)(H85):U17U7 !U7U17 "H80ĴCC0J0UCP1:37" H8PP1:38:P48. 36:50? (Y)155ıW R1,0:"@40X40YN@";| H(Y):HS117:H155H205ı R1,0:21 :20 159 38:H155İ3:E$"RUN ALGEBRA 4" 48 "@22V6HI@Press SPACE BAR to Continue"F$:R1,0FF9:G0ĺ(12F)F)"@B@";< (20I)"@B@"(TT)"@ER@";:n 15:H(Y):HS16:R1,0:HH176:H0HI6: "@3H21VI@Press "(1)" Key to View the Next Page@D3H@Press "(2)" Key to View the Last Page"F$ 15:H(Y)S1:H21H810:R1,0:H21P0 15924800:27903:A(E)((1)E)1:D$"$":E$(4):Y16384:R116368:F$"@I@":S1128:Q$"equation":O$"ordered pair":U$"coordinate":K$"Y-intercept":HH36251:S$"straight line":440:1002: "@G15C0KE@";:F(G):I(F10):TTFI10:               @2D10HL@Y = mX + b@R12V2H@is the line whose slope is m and@D2H@"K$" is b.":9('dE10:V269:U36:X91:147:X156:U99:147:"@5V2H@An "Q$" for a line parallel to@D2H@the X-axis, when the "K$" is@D2H@b is@2D15HL@Y = b@R13V2H@An "Q$" for a line parallel to@19,12945,12945,135:154,129220,129220,135:"@18V10H@"(95)"@35H@"(95):98,14973,14973,146:175,149248,149248,146:9%bP99,100,48M&cE10:V269:U36:X116:147:"@5V2H@For all real "Y$"s m and b, the@D2H@graph in the "U$" plane of the@D2H@"Q$"ine is to@2D2H@write its "Q$" in the form@2D2H@Y = mX+b, where m is the slope and b@2D2H@is the "K$". This is called@2D2H@the ";:A$"slope-intercept form":40:"."|%a"@17H16V@SLOPE@D3H@Y= 2X +0@28H@Y= 2X +2@D14H@Y-INTERCEPT@16V6H@"(126)"@31H@"(126):1#^"@5V2H@To find the "K$" of a line,@2D2H@substitute 0 for X in the "Q$" of@2D2H@the line. Then solve the resulting@2D2H@"Q$" for Y.@2D2HL@Y=2X@D2H@Y=2"(19)"0@D2H@Y=0@13V19H@Y=2X+2@D19H@Y=2"(19)"0+2@D19H@Y=2@R@":9$`"@5V2H@One way to describe a late of the@D2H@point where a line@D2H@crosses the Y-axis@D2H@is called the line's@2D2H@";:A$K$:40:".@17V32H@INTER-@D32H@CEPT":2:228,136228,67223,67:228,99220,99#]228,98220,98:228,100220,100:228,66220,66:228,68220,68:228,67220,67:3:9245,40182,155:"@10V33H@Y=2X":42!["@10V2H@Now look at the@D2H@graph of Y=2X+2. The@D2H@two lines have the@D2H@same slope but cross@D2H@the Y-axis at@D2H@different points.@9V23H@Y=2X+2":229,40170,152:24:I113:3:4I:20):"\"@5V2H@The ordin31152,C:C5ĹHH,2:I14:HH41,0::36272,0:77I W3:E$"RUNAM4.2.1"^ XM89,98,102,107q YP90,94,96,48"!Z116:117:E157:V269:U36:X156:147:"@5V2H@This is the graph of@D2H@Y=2X. It has a slope@D2H@of 2 and passes@D2H@through the origin.":0)3:D1M1KN1E1N1ĺ"@18V2H@RIGHT":HH5,(HH5)1:84OS"@18V2H@WRONG"TP1E1D1K2D1:Q1E1D1K2D1:164,91P18249,91Q18:J0:CC0:24:I13:16I:3:18)::I113:5I:23:16):::37:L1948U(HH)2İ3:E$"RUNAM4.2.1"2 VD2H@R-RIGHT P-PLOT"F$:P1L1:36:120P157:F1180:"@16V2H@GRAPH "D1"Y+("K"X)="E1"@D2H@PLOT POINT 1";:E24:U7:122:28:E:U:"$":M112U:N1(E30)3:N0:D1M1KN1E1N1;RJE:CCU:"@13H17V@2";:E24:U7:122:28:E:U:"$":M112U:N1(E3((HH)0):E10:U38:V144:X114:147:U124:X156:147:U36:E151:V270:147:F$:I19:5I:3:18):*O"@5V2H@Graph the line of@D2H@the "Q$" given@D2H@below by plotting@D2H@any two points@D2H@along the line.@2HD@Use the keys:@2H2D@U-UP D-DOWN L-LEFT@re associated@D6H@with the three@D6H@solutions.@19H7V@(-1,"G1")(0,"H1")(1,"I1")":118:119:15G1:"@26H@"D$:41:15H1:"@29H@$"L41:15I1:"@32H@"D$:41:"@13V2H@<4> Draw a@D6H@"S$"@D6H@through the@D6H@points which@D6H@were plotted.":156gML195116:147:"@13V20H@X@3F@"B"-";JG1BA1:H1B:I1BA1:"("A")X@13V36H@Y@15V20H@";:I11:I"@23H@"B"-("A"("I"))@D20H@";::"@15V35H@"G1"@D35H@"H1"@D35H@"I1:24:I112:3:7I:36)::J1BA2:K1BA2K"@7V2H@<3> Plot the@D6H@points which@D6H@a6H@one member.@24H7V@"K"Y="E1"-("D1")X @D25H@Y="B"-("A")X":41I"@4D2H@<2> Find three@D6H@solutions of@D6H@the "Q$"@D6H@by picking@D6H@values for X.@24H7V@"K"Y="E1"-("D1")X @D25H@Y="B"-("A")X":E136:V261:U99:X147:147:V157:E241:147:E136:V261:X" has as its "U$"s@D6H@an "O$" of "Y$"s which@D6H@satisfies the "Q$".":9FP448:E10:V270:U36:X156:147:X52:147:157:F1170>H"@5V2H@GRAPH THE EQUATION "D1"X+"K"Y="E1".@7V2H@<1> Transform the@D6H@"Q$" into@D6H@one which has@D6H@Y alone as@Din two@2HD@variables, X AND Y.":9DE10:V270:U36:X123:147:"@5V2H@<1> Every "O$" of "Y$"s@D6H@which satisfies a linear@D6H@"Q$" represents the@D6H@"U$"s of a point on the@D6H@graph of the "Q$".@2H2D@<2> Every point on the graph of an"IE"@6H@"Q$147:"@5V2H@The graph of any "Q$" equivalent@2HD@to one of the form@2D4HL@AX + BY = C,@R28H9V@X"(19)(6)", Y"(19)(6)",""C"@11V2H@where A, B, and C are real "Y$"s@D2H@with A and B not both zero is a@2HD@"S$". Such an "Q$" is@D2H@called a linear "Q$" :(36);:M112U:N1(E30)3:N0:IE:XU:128:F1N:JE:CCU:E24:U7:"@8H18V@SECOND POINT";6E24:U7:122:28:IEXU153:6"@8F@";:G1:4:G2:"@2F@";:4:E164:U111:I15:E,UE,U8:EE21::E202:U83:I18:E,UE7,U:UU8::7164,115,SLOPE OF A LINE,SLOPE-INTERCEPT FORM,FINDING THE EQUATION OF A LINE,LINEAR EQUATIONS TEST 5I5:6:CH:36:24(C$(H))2:"@2VI@"C$(H)F$:Cİ38:3:E$"RUNALGEBRA 4"5\6SW:"@17V2H@Y="W"X+("O")@D2H@GRAPH FIRST POINT";:E24:U7:122:28:E:UW36W151I4::#4L,W::>4E,UV,UV,XE,XE,U:4 GRAPHING LINEAR EQUATIONS,IN TWO VARIABLES,SLOPE OF A LINE,SLOPE INTERCEPT FORM OF,A LINEAR EQUATION,FINDING THE EQUATION OF,A LINE,LINEAR EQUATIONS TEST]5 GRAPHING LINEAR EQUATIONS1383122:3"@E@":5331:R8:D0ĢR82c3D1:32:R81:"1":28:R8D1:"1":3L206:L,R:I14:LL21:RRD8:L155L268I4::3R36R151I4::3L,R::3L206:L,W:I14:LL21:WWD8:L155L268I4::40:+2M1ON10M1OSN1M1OSN1N1122"@E@":J30:CC12O:E7:U13:122:I14:JJ3:CCCCS:J22J36I4::1352CC6CC17I4::1352122:2J30:CC12O:122:I14:JJ3:CCCCS:J22J36I4::1383CC6CC17I4::@B@";:J0CC0124:J:CC:(36);:E:Uh1|T30ē206,48206,136:Q203:W51:I111:Q,WQ7,W:WW8:1}D12ē161,91252,91:Q164:W88:I15:Q,WQ,W8:QQ21:1~D13ĺ"@23H12V@";:G2:4:" ";:G1:4:"@5F@";:G1:4:" ";:G2:4:E:U2Z15:4:"@UB@";:G:"@16V27H@";:G52:4:"@UB@";:G:"@11V22H@<@37H@>@5V29H@"(95)"@17V29H@"(126):206,40206,142:155,91263,91:E164:U88:I150yE,UE,U8:EE21::E202:U51:I111:E,UE7,U:UU8::"@5V30H@Y@12V37H@X":(1z" ";:E:U:"!H@"(126):148,115262,115:206,72206,150:"@9V30H@Y@4D37H@X":/w"@13V27H@";:G14:4:"@UB@";::"@17V27H@";:G31:4:"@UB@";::"@15V23H@";:G2:4:1550x"@12V23H@";:G2:4:" ";:G1:4:" @30H@ ";:G1:4:" ";:G2:4:"@10V27H@";:G";:G31:4:" ";:G:"@32H13V@";:G13:4:" ";:G:"@10V28H@";:G13:4:"@2UB@";:G:"@28H14V@";:G1:4:"@2DB@";:G2:4:E171:U96:.uI17:E,UE,U8:EE14::E210:U51:I16:E,UE7,U:UU16::@/v"@14V21H@<@15F@>@9V29H@"(95)"@18V294V@";:G15:4:"@UB@";:G:"@27H18V@";:G31:4:"@UB@";:G:E150:U119:I19:E,UE,U8:EE14:I:E203:U83:I19:E,UE7,U:UU8:I:.t"@30H5V@"(95)"@18V30H@"(126)"@12V23H@<@13F@>":213,40213,151:162,99262,99:"@31H5V@Y@37H13V@Y@13V24H@I114:23:5I:16)::I12:3:17I:19):::37:(HH)2İ3:E$"RUNAM4.2.1"Z,q300,r"@9V29H@"(95)"@29H19V@"(126)"@20H15V@<@17F@>":206,73206,159:141,123269,123:-s"@16V21H@";:G42:4:" ";:G:"@16V31H@";:G14:4:" ";:G:"@27H1D2H@R-RIGHT P-PLOT"F$:P1L1:120:OA(4)1A(2):WA(5(O))1A(2):36+n152:E:U:(36);:M112U:N1(E30)3:N0:128:F11N1ĺ"@27H18V@RIGHT";:HH7,(HH7)1:112+o"@27H18V@WRONG";:132Q,pR91O8:DS:WR:141:144:C3:J0:24::X121:147:U131:X156:147:E269:U38:X156:147:"@I5V2H@Below you are given@D2H@a linear "Q$". @D2H@From its slope @D2H@and "K$", R+m"@9V2H@graph two points on@D2H@the line. Use the @D2H@following keys. @D2H@"19)"@D2H@U-UP D-DOWN L-LEFT@82:D0ĢR8)j183,RD82187,RD82187,RD82183,RD82183,RD82:185,RD8206,RD8206,RD8206,R227,R227,RD8:D1:42:"@16V2H@<3>Draw a line@D5H@through these@D5H@three points.":WR:141:144:9*kL195((HH)0):E10:V150:U38f "S".":42:"@9V2H@<1>Graph a point on@D5H@the Y-axis whose@D5H@ordinate is the@D5H@"K$".@"11S"V29H@$":R91S8 )i42:"@13V2H@<2>Use the slope to@D5H@find two other@D5H@points.":225,RD82229,RD82229,RD82225,RD82225,RD82:139:34:RD2H@the Y-axis, when the X-intercept is"K'e"@2H@a is@2D15HL@X = a@R@":9'fP448:DA(4)1A(2):SA(6(D))1A(2):E150:V269:U36:X156:147:E10:V150:147:X67:147:120:"@5V2H@Draw a line with a@D2H@slope of "D1" and a"q(h"@2H@"K$" oJ18248,115K18:9~7KA(3)1A(2):AA(3)1A(2):BA(4)1A(2):D1AK:E1BK:F10:I11:BAI3BAI4F117:7:3:"++ERROR++ "(222)" AT LINE # "(218)(219)256:7,(HH)6İ3:E$"RUNAM4.3.1"7.48-H12((B(1)))((B(3))):32:S0:I172:B(I)B(3)F1B(I1)S1O-:h- 4:C$"RUNALGEBRA 4"-19:K(W):KG4000:Eĺ"@2D@";:C,5:"@B2D@";::"@13V23H@";:N22:N0ĺ"@3F@";:b,5:" ";::"@13V37H@X":,ORDERED PAIRS,COORDINATES IN A PLANE,RELATIONS AND FUNCTIONS,OPEN SENTENCES,GRAPHING TEST,:4:"++ERROR++ "(222)" AT LINE # "(218)(219)256:G30+X,YI,YI,JX,JX,Y: ,X143:I269:Y35:J155:130:"@5V29H@"(95)"@18VB@"(126)"@12V21H@<@37H@>@DB@X@30H5V@Y":148,99263,99:206,41206,151:X164:Y96:I15:X,YX,Y8:XX21::X203:Y51:I16:X,YX7,Y:YY16::"@6V27H@";:N321:N03:B(3)A(5)3:B(5)A(5)3:B(7)A(5)3:B(2)A(6)3:B(4)A(6)3:B(6)A(6)3:B(8)A(6)3:B(1)B(3)124:*~H1A(7)1:L1A(7)1:K1A(8):I1H1L1:J1H1K1:M1A(8)*I1Q:53::*X10:Y100:I45:J139:130:X73:I108:130+Y116:J156:1", )@4B@";:V17:H32((M1)):D13:32:SK1L1M1:"@18V29H@";:F1Sĺ"RIGHT":B14,(B14)1:121)x"@8H18V@WRONG, THE ANSWER IS "S)y28:I16:3:13I:36):I,P:(36251)1İ46:3000)z(B1)6İ45:4:C$"RUNAM4.2"){45:55m*|B(1)A(5)member in@D2H@the ordered pair beneath it. The@D2H@first component is the replacement@D2H@for X. The second component must be@D2H@the corresponding replacement for Y."B$:P1G1:44:126d)w"@L13V8H@"H1"Y-"I1"X="J1"@R16V2H@COMPLETE THE ORDERED PAIR: ("M1tion set is,@D8H@"(123);:I14:"("I","K1L1I"),";::"@B@"(125):13'tG195((B1)0):X10:I269:Y37:J89:130:Y100:J156:130:13,39265,39:B$:I16:3:5I:36)::"@5V2H@Below is an open sentence in X and"(u"@2H@Y. Complete the missing 0V3H@<2> Replace X by each@D7H@member of the @D7H@replacement set @D7H@for X. Then find@D7H@the resulting Y@7HD@values.@10V26H@X Y =("K1"+"L1"X)@11V26H@";:I14:I6)K1L1I" @D26H@";:H'sX179:I192:Y76:J124:130:I263:130:"@17V3H@<3> The solu"@D2H@when the replacement set for X is@D2H@"(123)"1,2,3,4"(125)".@3D3H@<1> Transform the@D7H@equation into an"%p"@7H@equivalent one@D7H@that has Y alone@7HD@as one member.@10V27H@"H1"Y-"I1"X="J1"@D27H@"H1"Y="J1"+"I1"X@D28H@Y="K1"+"L1"X":28&r"@1entence.@3D2H@If there are no ordered pairs which@D2H@are solutions of the sentence, we@D2H@say that the solution set is the@D2H@empty set, "(16)".":13q%nP455:Y38:J66:X10:I270:130:J156:130:126:"@5V2H@Find the solution set of "H1"Y-"I1"X="J1130:Y108:J148:130:"@5V2H@The solution set of any open@D2H@sentence in two variables is a@D2H@relation whose domain is the first@D2H@set of elements and whose range is"$m"@9V2H@the second set of elements in the@D2H@ordered pairs satisfying the@D2H@s@and is said to ";:E$"satisfy":48:" the@2D2H@sentence. ";"j"The set of all solutions@2H2D@of the open sentence is called the@2D2H@";:E$"solution set":48:" over the replacement@2H2D@set of variables.":13"kP108,55#lX10:I270:Y38:J100:8:". To"!h"@13V2H@solve such equations, you must find@2H2D@all the ordered pairs of numbers@2H2D@that make the sentence true.":13="i"@5V2H@Each such ordered pair is called a@2H2D@";:E$"solution":48:" or ";:E$"root":48:" of the sentence@2D2HC$"RUNAM4.2" c(B1)116# d55: eM102,107,110,116L fP103,105,55 !g"@2H5V@Look at this equation: 3X+4Y=25.@2H2D@We call equations or inequalities@2D2H@that involve two variables ";:E$"open":48:"@2H2D@";:E$"sentences in two variables":4$;:41:"@B@"(K):I172:JI72:B(I)B(J)IJA1R]J,I:O0:A1K78O1d^A0K89O1_O1ĺ"@18V21H@RIGHT":97`"@18V21H@WRONG"aU9O1Ĺ36254,(36254)1 b28:T15:B18:L2:R19:49:T5:B18:L21:R38:49::45:(B1)6İ4:B(I)3"H"12B(I1)2"V@$"::"@15V3H@("B(1)", )("B(3)", )":V16:H6((B(1))):D13:32:U0:I172:B(I)B(1)F1B(I1)U1Z:200:U1S1ĺ"@16V8H@RIGHT":U9:92["@8H16V@WRONG"7\A0:"@17V2H@IS THE RELATION A@D2H@FUNCTION(Y/N)? "B$" "B:K9:K,I:86XK,I:131:X10:I136:Y35:J107:130:129:"@I5V2H@The graph to the @D2H@right describes a@D2H@relation. For @D2H@each element of @D2H@the domain, enter@D2H@one of the"7)"Y"@2H@elements of the @D2H@range."11)B$:I172:"@"29unction.":50:E$"":I172:KI272:B(I)B(K)E$"not "wS::"@17V2H@This relation is@D2H@"E$"a function.":13TG19:(B1)G14UP1G1:44V124:I172:B(I)0B(I1)0I8::86 W:I172:KI272:B(I)B(K)B(I1)B(K1)I9ine test@D2H@for the relation."QI172:206B(I)21,40206B(I)21,152:207B(I)21,40207B(I)21,152::50:"@9V2H@If no vertical@D2H@line intersects@D2H@the graph of the@D2H@function in more@D2H@";<R"than one point,@D2H@the relation is a@D2H@f)"@"4((B(3)))"F@"B(4)"@D"5((B(5)))"H@"B(6)"@"4((B(7)))"F@"B(8)B$:I282:"@29H"12B(I)2"V@$":P"@18V2H@and the range on@D2H@the Y-axis.":28:I282:"@29H"12B(I)2"V@ ";::131:T4:B19:L2:R20:49:"@5V2H@Now we use the@D2H@vertical l172:"@"29B(I)3"H12V@$":N"@I4H8V@"B(1)"@"8(A$(1))"H8V@"B(3)"@9V4H@"B(5)"@"8(A$(5))"H9V@"B(7):50:B$:I172:"@"29B(I)3"H12V@ "::131:"@4H8V@"B(1)"@"8(A$(1))"H8V@"B(3)"@9V4H@"B(5)"@";jO8(A$(5))"H9V@"B(7)"@I8V"5((B(1)))"H@"B(24)"),@D3H@("B(5)","B(6)"),("B(7)","B(8)")"(125)".":50:131:"@11V2H@The graph of the@D2H@relation looks";M"@D2H@like this:":I172:"@133C"29B(I)3"H"12B(I1)2"V@$@15C@"::50:"@15V2H@The domain@D2H@elements are shown@D2H@along the X-axis":Ige.":13IP455AJ124:I172:B(I)0B(I1)0I8::74K:K182:A$(K)(B(K))(B(K1))::S182:LS82:A$(S)A$(L)LSS9:L9:L,S:74kLL,S:"@5V2H@Look at this set@D2H@of ordered pairs:@2D2H@"(123)"("B(1)","B(2)"),("B(3)","B(tion is a rule@D2H@to find the second ";H"element that must@2HD@be paired with the first element of@2HD@every ordered pair.@3D2H@A function is a relation in which@2HD@every member of the domain is paired@2HD@with one and only one member of the@D2H@ran1@DB@2@DB@3@DB@4@13V13H@3@DB@5@DB@7@DB@9":49,10767,107mE49,11567,115:49,12367,123:67,13149,131:13zFP71,55(GX10:I269:Y35:J60:130:Y68:J108:130:129:"@5V2H@A set of ordered pairs is called a@D2H@relation.@3D2H@The rule of the rela" is a special kind of@2D2H@relation.@2D4H@"(123)"(1,3),(2,5),(3,7),(4,9)"(125)"@11V3H@D@8F@R":1288D"@12V18H@Each element of the@D18H@domain is paired@D18H@with one and only@D18H@one element of the@D18H@range.@13V9H@>@14V9H@>@15V9H@>@16V9H@>@13V4H@D12H@-2@D13H@2"B128:"@13V9H@>@D9H@>@D9H@>@D9H@>":65,10749,11565,115:65,12349,13165,131:"@12V18H@Notice how each@D18H@number in the domain@D18H@is paired with one@D18H@or more numbers in@D18H@the range.":13fC"@5V2H@A ";:E$"function":48:10V18H@";:X7:S8:7:"@FU@7":52:"@128C17V19H@ @BD143C@7":13A"@2H5V@The listing of ordered pairs is@2D2H@called a ";:E$"roster":48:" of the relation:@4H9V@"(123)"(1,1),(1,-1),(4,2),(4,-2)"(125)"@3H11V@D@3D4H@1@2D4H@4@12H11V@R@2D12H@-1@D13H@1@2H16V@The set of@D22H@second elements@D22H@is the range.@6H18V@RANGE : ";?@(123)"@20H@"(125):"@6H10V@";:X3:S3:7:D5:9:S5:7:D4:"@U@";:9:"@15H17V128C@3@143CDB@3,@10V12H@";:S5:X5:7:"@14V12H@D@BD@";:S2:7:D5:9:"@128C17V17H@ @BD143C@5,@@ordered pairs is@D22H@the domain.@2H14V@"(123)"@8H@"(125)" : DOMAIN@10V4H@";:X1:S4:7:"@128C@"X"@2B143CU@"X;:52:"@128CB@ @143CDB@"X",@10V10H@";:X2:7I?"@U@";:E5:8:"@128C@ @143CDB@"X",@10V16H@";:X3:7:"@U@";:E9:8:"@128C@ @143CDB@"X:"@29MK:P1:44:45:C31012;M60,70,73,84E<P61,65,67,55="@2H5V@A ";:E$"relation":48:" is any set of ordered@2D2H@pairs of elements:@2D2H@"(123)"(1,3),(2,5),(3,7)"(125):51:"@11V22H@The set of first@D22H@elements in the">"@22H:P0:44:(B1)1847M0:P0:44:"@14H5V@LEARNING MODE@10H7V@<1> DISCUSSION@9V10H@<2> RULE@2D10H@<3> EXAMPLE@10H2D@<4> SAMPLE PROBLEM@10H3D@<0> Return To CONTENTS@I10H6D@WHICH (0-4) ??"B$:C10:D14:10:KĹ24798,10:46:30008KĹ24798,10:3000@2H1V@"C"@5H@"P"@2HD@"M:( -31051:473 .30976P /"@21V1HLI@"19)"@RI@": 0Y(37)82:X(36)72:IX4(E$)7:JY12:E$" ";:130 1ITB:"@"L"H"I"V@"RL):: 2Q300:127 3Q75:127 4Q7:127 519:K(W)G:6C(31152):M0 $K136SSS1* %K149SD11SS1j &KKG:(K47K58)K45ĖHS:(K);:256S,K:SS1:SD140r '33 (D$"":I0S1:D$D$((256I))::F1(D$):F10S032:H:" ";:H:D$: )K(W):KG41 *19:KKG:K89K7841 + ,":"@22V6H@Press SPACE BAR to Continue":35399,0:Z,0:19K K(W):KG29_ Z,0:K16029w "@I22V1H@"36)B$: V:H:D11);:S0:I0D1:256I,32::Z,0 !HS:B$((256S))B$; "K(W):19:KG34 #Z,0:HS:((256S));:K14140Last Page"B$@ 4000:Z,0:KKG:K21K814:K21PP1:45Z K8PP1:46:P55f 44:58z 3:(W)155ı Z,0:"@40X40YN@"; K(W):KG21 K155K205ı Z,0:25 :24 135 46:K1553000:558 35399,1B@";:52:"@128C128K@"X"@143CDB@";::U J1E:"@128C@"X"@2B143C@"X"@B@";:52:: J1D:"@128C@"X"@143C@"X"@B@";:52:: 4000:Z,0:KK176:KC1KD110: "@3H21VI@Press "(1)" Key to View the Next Page@D3H@Press "(2)" Key to View the %27903:24800:135yZ16368:B$"@I@":C$(4):G128:W16384:B136251:35339:A(X)((1)X)1:(B1)084:540:1002:"@G15C0KE@";:F(N):A(F10):BFA10:FF9:N0ĺ(12F)F)"@B@";(20A)"@B@"(B)"@ER@";:& J1S:X"@           ZZ,0 "@"92S"H"14Q"V@!":ZZ,0:109:GG128:"@"92S"H"14Q"V@ ":Q0S0ē63S14,115Q870S14,115Q8:66S14,112Q866S14,120Q8 Q1S0ĺ"@"92S"H15V@";:FS:4 S1Q0ĺ"@"14Q"V7H@";:FQ:4 QQ(G85)(G68):(Q)4Q2:G155112:43z 35399,1:"@22V6H@Press SPACE BAR to Continue":35399,0:11:109:ZZ,0:G16018:"@I22V1H@"36)W$: O:H:K):P0:I0K:256I,32::ZZ,0 200 A$"":I0P1:A$A$((256I))::A(A$):A19:H:A$" ": Q0:S0:ess "(2)" Key to View the Last Page"W$` 11:109:ZZ,0:GG128:G21G88:G21NN1:31z G8NN1:32:N43 30:45 (QQ)155ı ZZ,0:"@40X40YN@" G(QQ):G12813:G155G205ı ZZ,0:16 :15 108 3$(I)::D16:B(D),C(D),A$(D)::1124 0:1002:} "@G15C0KE@";:D(F):A(D10):BDA10:DD9:F0ĺ(12D)D)"@B@"; (20A)"@B@"(B)"@ER@";: 11:109:ZZ,0:GG176:GJGK6:' "@I3H21V@Press "(1)" Key to View the Next Page@D3H@Pr4wZ$(22):E$"number":Y$"intersect":U$"ordinate":S$"abscissa":R$"component":D$"Distances":N$E$" line":108% H$"horizontal":W$"@I@":QQ16384:ZZ16368:G$(4):O$"correspondence":K$"coordinate":P$"ordered pair":HH36251:35339:I15:C              !!! ! ! ! ! !!!!!!!!!""" " " " " """""""""0H@the "K$" @D20H@axes and form the @D20H@"K$" plane.":7 GV10:W129:U75:X155:98:99:101:"@5V2H@Note that both "N$"s meet at@2D2H@their zero points. This "Y$"ion@2D20H@is called the@2D20H@";:B$"origin":34:"@2D10H@"(30)"@U11H@ORIGIN@27HU@.@I14VV26H@But to graph@D2H@an "P$", two "N$"s@D2H@are needed.":99:36:"@10V20H@The "H$"@D20H@"N$" is@D20H@called the9F"@D20H@";:B$"X-axis":34:".":36:"@20H10V@The vertical @4D20H@Y":36:"@5U20H@Together they are@3D19H@"8)"@D19H@"9)"@D19H@"9)"@4U2V3H@WRONG. X = "V" AND Y = "UK@18:T12:B18:L2:R37:NEİ35::31:43^A(HH)İ31:89lB:31:43CM68,76,85,89DN69,71,73,75,43qE"@5V2H@To graph a "E$", one needs a@D2H@"H$" "N$".":V10:W129:U75:X155:98:H10:O11:101:36:"@61E:30:YB(8)1:ZB(8)1:AB(9):BB(9):VB(9):UB(9):FVYA:DUZB:"@12V3H@If ("Y"X+"A","Z"Y+"B") = ("F","D")@2D3H@Then X =":H13:O15:19:A(1)A:"@16V8H@Y = ":H13:O17:19:A(2)A>VA(1)UA(2)ĺ"@18V3H@RIGHT":HH1,(HH1)1:64?"@1835::31:43<"@5V2HI@Find the values for X and Y that @D2H@will make the two "P$"s @D2H@given below equal. Then enter the @D2H@"E$" for X followed by the "E$" @D2H@for Y."30)W$:E95((HH)0):V10:W269:U35:X83:98=U91:X155:98:N1:AB(9):BB(9):EB(9)YA:DB(9)ZB:"@L5V2H@IF@8V2H@THEN@R9H6V@("Y"X+"A","Z"Y+"B") = ("E","D")@5D11H@"Y"X+"A" = "E" AND "Z"Y+"B" = "D"@2D11H@"Y"X = "EA"@25H@"Z"Y = "DB ;"@15V12H@X@16H@= "(EA)Y"@26H@Y@30H@= "(DB)Z:18:T4:B19:L1:R35:th an "P$",@D3H@(X,Y):@2D4H@X is called the first "R$",@D17H@or first "K$".@2D4H@Y is called the second "R$",@D17H@or second "K$"."9"@3D2H@For all real "E$"s X,Y,W, and Z:@2D2H@(X,Y)=(W,Z) If and only if X=W, Y=Z.":7:N14:30:YB(8)1:ZB(8)0C@";:I15:" @UB@";6:"@10V138C@":I17:"@22H@ "::"@8V133C@":I19:"@24H@ "::"@15CI18V28H@1 2 3 4 5":197,72197,143259,143:198,72198,144:200,142256,75:77N56,438V10:W269:U35:X107:98:U116:X155:98:"@5V2H@In working wierent ways.@19V5H@TABLE@15H@BAR GRAPH@28H@LINE GRAPH":V18:W87:U69:X139:98:U91:98:V52:98:"@10U4H@x f(x)@D@"5I15:"@4H@"I"@9H@"I21::I19:"@14H@"I"@12F@"I"@U@";::"@18V16H@1 2 3 4 5@I17V16H@ @17V18H5C@";:I13:" @UB@";::"@17V20H1lled@D22H@the @I@first@I2D22H@";3B$R$:34:" or@2D22H@";:B$K$".":34:"@6UI@";:I15:"@17H@"I::36:"@I9V@":I15:"@17H@"I"@I19H@"I21W$::"@I9V26H@second@I@ "E$"@I26H12V@second"W$:7v4"@5V2H@We may illustrate "P$"s in a@2D2H@"E$" of diff(125)E1"@D2H@and the range is "(123)"1,3,5,7,9"(125)".":96:7 2"@5V2H@We may also write the "O$"@2D2H@as a list of ";:B$P$"s.":34:96:"@6U@":I15:"@16H@("I","2I1")"::36:"@9V22H@The @I@first@I@ "E$"@D22H@in an ordered@D22H@pair is ca -C267.M47,55,58,601/N48,50,52,430"@5V2H@Look at this ";:B$O$:34:" of@2D2H@five pairs of "E$"s:@3D16H@This is the function@D16H@f(x)=2x-1 for the@D16H@integers one through@D16H@five.@3D2H@Notice that the domain is "(123)"1,2,3,4,5"VI@"C$(G)W$:Cİ32:112(*31:C294+M0:N0:30:"@14H5V@LEARNING MODE@10H7V@<1> DISCUSSION@9V10H@<2> RULE@2D10H@<3> EXAMPLE@10H2D@<4> SAMPLE PROBLEM@10H3D@<0> Return To CONTENTS@I10H6D@WHICH (0-4) ??"W$:J0:K4:6:Gİ32:39,MG:N1:30:31"<"I">@2D3B@";::V19:W42:U46:X56:I15:98:UU16:XX16:)"@16V4H@"(31)"@3HD@<0>@10H3U@<0> Return To ALGEBRA Menu@2V7HI@"31)"@11H22V@WHICH (0-5) ??"W$:21,13242,13249,14042,14821,14814,14021,132:97:J0:K5:6:CG:30:24(C$(G))2:"@2"L"H"I"V@"RL)::($I1300:11::=%I1200:11::Q&I150::11:'24798,0:N0:M0:C0:30:"@20H5V@CONTENTS@6V@":I15:"@10H@<"I"> "C$(I):I4ĺ"@29H10V@IN TWO@D14H@VARIABLES"?(:"@7V4H@";:I14:Z$"@2DB@";::"@6V3H@";:I15:3(Q)* G80ĺ"@"92S"H"14Q"V@$":R SS(G76)(G82):(S)4S3(S)Z 23z "@2H1V@"C"@5H@"N"@2HD@"M: 31051:33 30976 !"@21V1HLI@"19)"@RI@": "U(37)82:V(36)72:WV4(B$)7:XU12:B$" ";:3:98#ITB:"@3:S,iH:U56:I133:209,U175,U:UU1::U94:I143:209,U175,U:UU1::3:,jH:I133:206,U175,U:219,U254,U:UUF::3:k-kORDERED PAIRS,COORDINATES IN A PLANE,RELATIONS AND FUNCTIONS,OPEN SENTENCES,GRAPHING TEST,57,61,INTRODUCTION TO G72,(O3)83(H7)73,(O3)83:F(H7)73(H7)7914:F,(O3)8F,(O4)8::+g"@6V30H@"(95)"Y@30H17V@"(126)"@11V24H@<@36H@>@D37H@X":213,50213,142:169,91256,91:,hH:U56:I133:219,U250,U:UU1::U94:I143:219,U250,U:UU1::UW,UW,XV,XV,U:*cH:O:(95)"@8DB@"(126):H74,O86H74,(O8)82:F(O1)85(O7)828:(H1)7,FH7,F::O7:H2:F33:F0ĺ"@U@";:*d4:"@UB@";::*eH6:O5:F33:F0ĺ"@2F@";:U+f4:" ";::H7:O4:"<@14F@>":(H8)4A1"V@$":37:@)]T10:B18:R23:L2:18:35:"@10V25H@"13):d)^31152,C:C5Ĺ36251,1:500:60m)_110)`V17:W45:U75:X123:98:V74:W101:98:"@9V@":I15:"@4H@"I"@2F@-->@2F@"2I1::)a31,12031,132:60,3660,154:59,3659,154:*bV,ft@D25H@R:Move Right@D25H@P:Plot Point":E195((36251)0):N1E1:30:91::31:(36251)110:43([YB(7)4:A1B(7)4:"@10V25H@PLOT ("Y","A1")":99:101:22:SYQA1ĺ"@12V18H@RIGHT":36253,(36253)1:93)\"@12V18H@WRONG":R18:"@I"92Y"H"1'YV10:W269:U35:X75:98:U75:X155:98:H10:O11:"@I5V2H@Use the keys listed below to move @D2H@the point to the correct position of@D2H@the "P$" given below. Then @D2H@plot the point."21)"@I12V25H@U:Move Up"c(Z"@25H@D:Move Down@D25H@L:Move Le14,133:3:36:"@13V2H@Find the graph of@D2H@"A1" on the Y-axis@D2H@and draw a "H$"@D2H@line through it."&X6:162,99A18263,99A18:3:36:"@18V2H@Find where they "Y$".":F119:"@I"29Y2"H"12A1"V@ ";:37::T5:B6:L2:R38:18:35:T8:B18:35:30:86::31:43%VYB(7)4:A1B(7)4:Y0A1086:H30:O9:99:101:"@5V2H@To locate the graph of the ordered@D2H@pair ("Y","A1"):@2D2H@Find the graph of@D2H@"Y" on the X-axis and@D2H@draw a vertical@D2H@line throught it.":5j&W207Y14,68207Y is exactly one point in@D6H@the "K$" plane paired@D6H@with each "P$" of real@D6H@"E$"s.@3D2H@<2> There is exactly one ordered@D6H@pair of real "E$"s paired"$T"@6H@with each point on the@D6H@"K$" plane.":7%UV10:W269:U35:X59:98:X155:98:N14distance of a point@D2H@from the X-axis is@D2H@called the Y-"K$"@D2H@or "U$".@12V26H@Y":192,93192,115:36:"@2D2H@They make up the@D2H@"K$"s of the@D2H@point and are written@D2H@as an "P$".@16V25H@(X,Y)":7$SV10:W269:U35:X155:98:"@6V2H@<1> There6:"@128K16V2H@"D$" downward are@2HD@called negative.":H2:U94:F1:106:7"Q103:V10:W269:U35:X155:98:"@5V2H@The distance of a point@D2H@from the Y-axis is@D2H@called the X-"K$"@D2H@or "S$".@7D28H@X@14V27H@$":195,115210,115:36#R"@5U2H@The right@D2H@of the Y axis are@D2H@called positive.":103:H1:104:36:H0:104:"@2H9V@"D$" to the left@D2H@are called negative.":H2:105K"PH0:36:105:"@12V2H@"D$" upward from@D2H@the X-axis are called@D2H@positive.":H1:F1:U88:106:36:H0:U88:10aw a vertical "N$".@D6H@"Y$"ing it@D6H@at right angles@D6H@at the zero" N"@6H@point. It is@D6H@called the@D6H@Y-axis.":99:36:"@16V2H@<3> They "Y$"@D6H@at the origin.":D119:"@I14V29H@ ";:37::7!OV10:W269:U35:X155:98:"@5V2H@"D$" to the 75:X155:98:"@5V2H@The "S$" and "U$" of P make@2D2H@up the ";:B$K$"s":34:400iLN77,79,81,83,43G MV10:W269:U35:X155:98:"@5V2H@To draw a "K$" plane:@2D2H@<1> Draw a "H$" "N$".@D6H@It is called the X-axis.":H30:O11:101:36:"@9V2H@<2> Dr2D20H@";:B$S$:34:".@11V13H@$P":V10:W129:U75:X155:98:99:101:2:U95111:94,U:38::37J"@12V20H@The point where it@2D20H@meets the Y-axis@2D20H@is called the@2D20H@";:B$U$:34:".":1:V92711:V,91:38::3:7SK99:101:V10:W129:U9H@";:E119H" @BI@";:37::"@11V30H@The axes@2D20H@divide the plane@2D20H@into four regions@2D20H@called ";:B$"quadrants":34:".@10V5H@II@14H@I@18V4H@III@13H@IV":7aI"@5V2H@From point P, the point where it@2D2H@meets the X-axis is called the@OST TEST"<"y<"@8V2H@TWO VARIABLES@D2H@LINEAR EQUATIONS@D2H@OPEN SENTENCES@10V23H@";:D13:D(D)(36266D)::122<'(HH)6ĺ"@15V2H@YOU HAVE PASSED ALGEBRA 4"::126<'YZ:PP1:PK1ı;200;" of P. The@2D20H@"K$"s are@2D20H@written as an@2D20H@"P$", with@2D20H@the "S$" first.@11V13H@$P@13H10V@(2,3)":1:71,9191,91:3:94,9494,111:7;I1414(G5):36251I,0:: < DE$"UNIT "((HH)):(HH)5DE$"THE P49185,110:228,49228,110:227,49227,110:"@5V2HI@"36):D14:3:14D:36)::D18:34:6D:5):::HP:W$((256P))W$;:109:ZZ,0:HP:((256P));:G141ı:G136PPP1:G149PKPP1;GG128:G47G58ĖHP:(G);:256P,G(HH)6ĹHH,0:11299:3:V3:U5:W276:X185:98:V4:W275:98:3,25276,25:3,165276,165:45,545,25:46,546,25:7,167272,167:W$:D12:D1:"@7H@"32):D21:"@1H@"38)::24(A$(G))2:"@1V@"A$(G)"@I1H@C0 P0@D1H@M0":49,7272,7:j:185,A(D)::138*83:G$"OPENAM4.PROGRESS":B8G$"CLOSE":35339:8"@8V2H@INTRO. TO GRAPHING@D2H@LINEAR EQUATIONS@D2H@VARIATION@D2H@SOLVING SYSTEMS@D2H@INEQUALITIES":D(1)(36254):D(2)(36258):D(3)(36263):D(4)(36265):D(5)(36269):D6:122915H@<0>@8V9H@<1>@2D9H@<2>@3D4H@<3>@13H@<4>@2D13H@<5>@3D9H@<6>@8H2V@START@5V8H@MENU@6H@<@13H@>@9H3V@"(23)(24)"@10H9V@"Z$"@3D5H@"Z$"@14H@"Z$"@2D14H@"Z$"@3D10H@"Z$:7137:G$"READAM4.PROGRESS":D16:A(D)::138 8137:G$"WRITEAM4.PROGRESS":D16:5571,60669,3269,28:45,4349,43:91,4393,43:38,95101,95:38,9538,97:101,95101,97:73,9073,95:72,9072,95:38,135101,135101,130:38,13538,113:39,13539,113:73,13673,140:105,36126,36133,44126,52105,5298,44105,36:7"@5V2H@<9>@S@D20H@<6> POST TEST@D20H@<9> RESET MENU@D20H@<0> STOP":134j5C:WV29:XU12:98:VV1:WW1:98:3:656,1191,1198,1991,2756,2749,1956,11:49,4370,3291,4370,5649,43:9,3638,3638,489,489,36:37,3637,48:10,3610,48:70,5570,60:71,ME "(96)"4@20HD@VER "B$" "C$"@21H21V@WHICH (0-9) ??@I25H5V@CONTENTS@20H7V@<1> INTRODUCTION TO@24HD@GRAPHING@D20H@<2> LINEAR@D24H@EQUATIONS@20HD@<3> VARIATION@D20H@<4> SOLVING SYSTEMS@24HD@OF LINEAR"<5140,7272,7:"@24H@EQUATIONS@D20H@<5> INEQUALITIEN AM4."L1D3D130:HH1D,0::GL:142:L139:3:G$"RUNAM4."L33:V3:U5:W276:X186:98:V4:W275:98:136,5136,186:136,34276,34:136,157276,157:137,5137,186:W$:D13:D1:"@20H@"19):D20:"@20H@"19):4"@26H1V@ALGEBRA@25HD@VOLU<--";:JJ1d2{L((HH)):"@D34H@";::"@128K@":Jĺ"@14V2H@ARROWS SHOW AREAS OF WEAKNESS."W$:1252|800:10000:"@15V2H@YOU HAVE PASSED UNIT "(HH)" AND MAY @D2H@NOW GO ON TO "DE$2}JA(L)2(J1)3 3~136:18:140:HH,0:JGL1:142:3:G$"RUARIATION@D2H@JOINT VARIATION@7V23H@";:D14:D(D)(HH8D)::1221y"@8V2H@GRAPHIC METHOD@D2H@ADD. AND SUB.@D2H@SUBSTITUTION@10V23H@";:D13:D(D)(HH12D): 2zJ0:"@8V23H@";:K1D1:"@23H@"D(K)"@6F@"4D(K)::"@I138K8V34H@";:K1D1:D(K)3ĺ"ND FUNC.@D2H@OPEN SENTENCES@7V23H@";:D14:D(D)(HHD)::1220w"@8V2H@GRAPHING EQUATIONS@D2H@SLOPE OF A LINE@D2H@SLOPE-INTERCEPT@D2H@EQUATION OF A LINE@D2H@";:D14:D(D)(HH4D)::122B1x"@8V2H@RATIO AND PROP.@D2H@DIRECT VARIATION@D2H@INVERSE VW268:X145:98:V11:W269:98:10,49228,49:10,110228,110:144,49144,110:143,49143,110:186,49186,110:13,39265,39:143:"@I5H1V@0@I5V8H@CONCEPT@21H@RIGHT WRONG"W$:(HH)118,119,120,121,2038,139?0v"@8V2H@ORDERED PAIRS@D2H@COORDINATES@D2H@REL. AN 83":(24798)1039:(HH)0117::135:128:111:132[.q109:ZZ,0:GG176:Gĉ:::.rG9A(1)6:D26:A(D)3::111:136:113.sG1G6113:142:G6ĹHH,6:500:31152,6:G1.1.tG139:3:G$"RUNAM4."G/uD16:A(D)0::135:V10:U37:RAPHING,58,77,LINEAR EQUATIONS,23,101,VARIATION,86,101,SOLVING SYSTEMS,86,117,INEQUALITIES,58,141,POST TEST-mG(QQ):11:G128109:-n3:G$"RUN AM4.1.1"-oD16:CA(D):VB(D):UC(D):131::8.pB(V)((1)V)1:C(V)2V1:B$"1.0":C$"10 JA