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03,+-(((((((((((((((((((((((((  03,+14434((((((((((((((((((((( 0335(((((((((((((((((((((((((('8*3Ixix&& 8  '  & x)*++`FG8`0($ p,&"_]` L/ `Hh`V0^*^*>&` aI꽌ɪVɭ&Y&&Y&ސ 꽌ɪ\8`&&꽌ɪɖ'*&%&,E'зސ꽌ɪ`+*xS&xGx (`(8`I`B` ` `>J>J>VU)?`8'x0|&HhHh VY)'&Y)xꪽ)'      7:R1N1XO1:R15R15)2R15R15D2S1HXF1:S15S15S2S15S152P135R1HXF1P194R1HXF1M135S1N1XO1M194S1N1XO1Ă:1322227X7,107S18227X7,107R18:23:2P44121053V35:P194:P2V9e":V$"system":J$"overlap":K$"absolute value":M$"additive inverse":C(24576):L$"@I@":U$(4):U16384:Q16368:1}38:"@2D2H@Similarly, a ";:T$V$" of "I$:35:"@2D2H@is solved by finding the@2D2H@";:T$S$:35:"of the "I$".":52~6:X551:"@I16V2H@true@14H@true"L$:50{JC(5)1C(2):YC(5)1C(2):W0(V35YHJF)(V94YHJF)(M135YN1JO1)(M194YN1JO1):W2123:I16:"@138C"32J"H"13Y"V@$@133CB@";:121:"$":121::u1|Q$"inequality":W$"positive":P$"sentenc@"32Y"H"13J"V@$": Plot the "N$"s@D5H@for both.@I2D1H@The "O$"s H@"B$(I)::Q,0:K27173:89(^J10:"@5V28H@"10)"@"13G1"V2H@"16):171:B$(G)A$(1)B$(L)A$(5)B$(L)A$(1)B$(G)A$(5)J11:"@I13V18H@CORRECT"L$:96(_"@I13V18H@WRONG";L$",@D18H@these@D18H@are the@D18H@ones to@D18H@use."l)`I16:"@"12I"V2H"V2H@"16):89;'X"@13V@";:I16:"@2H@"B$(I)::Q,0:84V'Y"@I2H"12G"V@"B$(G)L$'ZK(U)128:K27K8K21K1390:GG(K8)(K21)(K21G1L)(K8G1L):G1(L1)G6'[G6G1(L1)'\K1394:"@13V@";:I16:ILII1:(]"@2:I120:DC(6):BC(6):T$B$(D):B$(D)B$(B):B$(B)T$::"@13V@";:I16:"@2H@"B$(I)::L1:Q,0|&T"@I2H"12L"V@"B$(L)L$&UK(U)128:K8K21K1385:LL(K21)(K8):L1L6&VL6L1'WK13ĺ"@5V28H@"(B$(L),7):G1(L1):Q,0:"@"13L10;Y"(V)(A)"X"F$%RA$(3)"X "D$" 0;Y"(V)A$"X"(44(((F$)1)))(((F$))):A$(4)"X < 0;Y"(V)A$"X"F$:A$(5)"X < 0;Y"(V)(A)"X"F$:A$(6)"X < 0;Y"(V)(A)"X"(44(((F$)1)))a&SA$(6)A$(6)(((F$))):I16:B$(I)A$(I):ice and @D27H@"(91)"RETURN] to@D27H@select it. @D27H@"(91)"ESC] will @D27H@cancel your%Q"@27H@choice. @D27H@"11)"@D27H@"11)L$:X10:Y83:O179:T155:110:Y99:110:F$(44(F))((F)):A$(A):A$(1)"X "D$" 0;Y"(V)A$"X"F$:A$(2)"X "D$" 155:110:"@4V1H@Plot the "E$"@D1H@for Y"(V)A;C$"X"C$(44(F))(F)".@2D1H@<1> Select two "P$"s@D5H@to plot for partial@D5H@"E$"s:@2D2H@RANGE;SENTENCE":7,52182,52X$P"@I7V27H@"11)"@D27H@Use the "(2)","(1)"@D27H@keys to @D27H@light your @D27H@chofor@D5H@the original "P$"@D5H@is the conjunction@D5H@(combination) of the@D5H@range-restricted@D5H@"P$" "N$"s.@I7V29H@Y"(V)A;C$"X"C$F$L$:111:5"NP195((E)0):31:AC(3):FC(3)1C(2):V60:GC(2):VV2(G1):F2V60#OX185:Y35:O269:T30H6V@Y"(V)"-"A"X"F$:39:"@I11V1H@<2> Plot the "N$"s@D5H@for both.":H10:I15:"@8V20H@X "D$" 0@D9H@X < 0@6V30H@Y"(V)"-"A"X"F$:107:"@5V30H@Y"(V)A"X"F$"@ID30H@Y"(V)"-"A"X"F$L$:AA:H15:I10"M107:AA:"@6V30H@Y"(V)"-"A"X"F$"@8D1H@<3> The "E$" 4A16 JF$(44(F))((F)):GC(2):V35:G1V94 K"@4V1H@Plot the "E$"@D1H@for Y"(V)A;C$"X"C$F$:39:"@7V1H@<1> Identify "P$"s@D5H@for the ranges @I@X "D$" 0@IRD5H@and X < 0.@I30H5V@Y"(V)A"X"F$"@ID30H@Y"(V)"-"A"X"F$:39:"@I9V9H@X < 0"!L"@G,I266,I:GG(I113)2::39:"@14V18H@X<0@D16H@Y"G$"-2X":2HI133153:97,I177,I::G226:I152641:G,I187,I:GG(.5(I113))::3:168:"@I5V27H@"11)"@D27H@ Y"G$C$"2X"C$" @D27H@"11)L$:39:5 IP441:120:168:FC(3)1C(2):AP:A269:T155:110:"@5V1H@In order to solve the@2D1H@"Q$" Y "G$" "C$"2X"C$" we@2D1H@really are solving these@2D1H@two "P$"s:":39:X10:Y107:O176:110:Y131:110:X94:Y107:110>G"@14V6H@X"D$"0@D5H@Y"G$"2X":1:I133153:12,I92,I::G227:I152641:$" of a@2D1H@variable X requires solving two@2D1H@"P$"s: one "P$" (valid for@2D1H@the domain X "D$" 0) uses the variable@2D1H@itself, while the second (valid for"E"@D1H@the domain X < 0) substitutes its@2D1H@additive inverse.":5F168:X185:Y35:O17H@ 1@23H@ 1@D17H@ 3@23H@ 3@D17H@ 5@23H@ 5@D17H@-1@I20H@ 1@23H@ 1@I12V31H@"H$:39:"@16V17H@-3@I20H@ 3@23H@ 3@I29H10V@"H$:39:"@17V17H@-5@I20H@ 5@23H@ 5@I8V27H@"H$:39:2:189,64227,107:3:5D"@4V1H@Solving an "R$" or "Q$"@2D1H@that contains the "KH$:39:"@13V17HI@ 3@23H@ 3@I35H10V@"H$:39B"@I14V17H@ 5@23H@ 5@8V37HI@"H$:39:1:266,64227,107:17:"@9V1H@But when X is @2D1H@negative, Y has@2D1H@the value of@2D1H@X's additive@2D1H@inverse, which@2D1H@is "W$".@9V20H@-X@11V17H@ 0@23H@ 0"C"@$" is in the@D1H@"R$": Y="C$"X"C$"@30H6V@Y="C$"X"C$:39:"@9V1H@Here, when X is@2D1H@"W$" (or@2D1H@zero), Y is@2D1H@equal to X."+AX115:Y67:O179:T155:110:Y83:110:Y67:X136:O157:110:"@9V18H@X@24H@Y@I17H11V@ 0@23H@ 0":39:"@17H@ 1@23H@ 1@I33H@"<@30H@>@16V7H@~@19H@~@31H@~@15V13H@4@25H@4@I18V6H@-4@31H@4"L$?52,12052,135:57,12391,123:98,123130,123:136,120136,136:141,123175,123:182,123214,123:220,120220,136:5@120:168:"@4V1H@Consider what a graph@D1H@looks like when an@D1H@"K5,123130,123:73,12073,135:136,120136,135:"@18V9HI@-3"L$:17:118:"@18V9H@ @2B@-3@5V2H@This means that the "K$"@2D2H@of any number Z is exactly the same@2D2H@as for its "M$", -Z."=>"@D2H@For example, "C$"4"C$" = "C$"-4"C$" = 4@15V8H@<@18H@>@20H@36,135:140,123161,123:193,123168,123:199,120199,136<17:118:"@18V28H@ @B@3@6V2H@The "K$" of a negative@2D2H@number Y is -Y. So the absolute@2D2H@value of -3 ("C$"-3"C$") is also 3.@14V12H@-(-3)@D11H@< 3 >@D10H@~@19H@~"=78,12398,123:10$" of any "W$"@2D2H@number X is X. For example, the@2D2H@"K$" of 3 ("C$"3"C$") is 3.":5:14,139259,139:I3124121:I,136I,144:I1,136I1,144::39;"@18V3H@";:N55:N1(N2))::39:"@I18V28H@3@I15V20H@<@23H@3@27H@>@D19H@~@28H@~":136,1201D1H@set, "(144)".":3:5262:U$"RUNAM4.5.1.1"E7M56,54,73,78^8P57,58,64,68,70,419"@6V1H@The ";:T$K$:35:"of a number is a@2D1H@"W$" value which indicates the@2D1H@number's ";:T$"distance":35:170:119:"@2H17V@<@34F@>@6V2H@The "KI641522:189,IG,I:GG1:G203G2034:39:"@30H6V@Y "G$" 2X-1":5:252,64210,152:252,66210,154:G152:I2112644:I,GI,152:GG8:G64G645:39:"@9V16H@The graphs@2D1H@of the two "I$"@2D1H@do not "J$", so the@2D1H@"E$" is the empty@22:191,66268,154:191,66268,154:189,64266,152:6:G107:I147:227I.5,G227I.8,G:GG1::3:17)3L1:R19:T4:B18:36:"@5V1H@Here is a graph of@2D1H@another "V$" of@2D1H@"I$".@5V30H@Y "(94)" 2X":121:1:245,64203,152:245,66203,154:G245:m@2D1H@together, the "J$"@2D1H@area is where both are@2D1H@true.@6V1H@"26):168:3:3138:"@14V8H@This is the@2D1H@"E$" for the@2D1H@"V$".":T7:B18:L26:R6:0:269,30269,155:36:168:1:204,152247,64:202,154245,66:5h23:168:189,64266,15 in the "V$" "J$".":5/168:3:"@5V5H@Here is the graph of:@30H@Y "G$" 2X":17:T4:B6:L2:R17:36:B18:L25:R7:36:168:3:"@6V5H@Here is the graph of:@30H@Y "G$" -X V017:T4:36:"@5V30H@Y "G$" 2X@D30H@Y "G$" -X@8V1H@When we plot both of theg the@2D2H@";:T$S$:35:"of the "R$"s.";:125."@6V1H@The "E$" for a "V$" of@2D1H@"I$" consists of all ordered@2D1H@pairs for which all of the@2D1H@"I$" in the "V$" are true.@2D1H@The graph of the "E$" for the@2D1H@"V$" is the area where the@2D1H@"I$"10H2D@<4> SAMPLE PROBLEM@10H3D@<0> Return To CONTENTS@6DI10H@WHICH (0-4) ??"L$:4:Kİ33:2:U$"RUNAM4.5"~*MK:P1:31:32+C355:M44,54,133,145,P45,46,47,41.-"@6V2H@We know that a ";:T$V$" of "R$"s":35:"@2D2H@is solved by findin1100$"@L@":XTB:X1:L1:R)::"@R@":N%XTB:X1:L1:R)::b&I1225:9::v'I1150:9::(35339:C(X)((1)X)1:(E)C0:78j)M0:P0:31:"@14H5V@LEARNING MODE@10H7V@<1> DISCUSSION@9V10H@<2> RULE@2D10H@<3> EXAMPLE@"13T"V@ ":XX1:X019:"@"32A(X)"H"13B(X)"V@ ":19? 20o A(X)O:B(X)T:A(1)A(0)B(1)B(0)X120: "@2H1V@"C"@5H@"P"@2HD@"M: 31051:34 !30976 ""@21V1HLI@"19)"@RI@":#Y(37)82:X(36)72:OX4(T$)7:TY12:T$" ";:8):(T)6T5(T)7 G80ĺ"@"32O"H"13T"V@$":30_ OO(G76)(G82):(O)6O5(O) Z,BA1,B1:C1,D1E1,G1:Z1,BA11,B1:C11,D1E11,G1:G78M330:(M4(E)5(E)6)X0āI0X1:"@"32A(I)"H"13B(I)"V@$":7 M4G27ĺ"@"32O"H"32O"H"13T"V@!":Q,0S 9:G(U)128:G85G68G80G76G82G78G2722:21n (E(C2M4))G7830 "@"32O"H"13T"V@ ":(T0O5O5)(O0T5T5)ē224O7,107T8231O7,107T8:227O7,103T8227O7,111T8 TT(G85)(G6YN@";, K(U):K12811:K155K20512:: Q,0:14D :13N 115r 33:K155İ2:U$"RUNALGEBRA 4"z 41 L$"@22V6H@Press SPACE BAR to Continue"L$:Q,0:9 K(U):9:K16018:"@I22V1H@"36)L$: T5:O5:X0X0 169:"@K(U)176:K0K44: L$"@3H21V@Press "(1)" Key to View the Next Page@D3H@Press "(2)" Key to View the Last Page"L$:Q,0 K(U)128:9:K21K86:K21PP1:32 K8PP1:33:P41 31:43 (U)155((E))ı Q,0:"@40X40627903:24577:124:C$(9):E36251:D$"^":G$"#":H$"$":O$"boundary line":I$"inequalities":S$"intersection":N$"solution":E$N$" set":R$"equation":40Q,0:0:1002:5:189,64266,152:G189:I641522:189,IG,I:GG1.7::3: 9:                  4"6%24576,C:C4Ĺ36251,5:I14:36266I,0::74O%2:(4)"RUNAM4.5.1"%C$(3)"ABSOLUTE VALUES":21,14042,14049,14842,15621,15614,14821,140:)ITIES TEST,IN TWO VARIABLES,OF LINEAR INEQUALITIES,INVOLVING ABSOLUTE VALUES,, e$K1R:12::$(36251)İ2:(4)"RUNAM4.5.1"$46$D0SDU:$U62SD60$U94SD35$U60SD62$U35SD94$%X34:2:B$"RUNALGEBRA#gX210:105#i"@"(X7)"H14V@<@7F@>@"((X31)7)"H10V@"(95)"@8DB@"(126):X1,115X60,115:X31,80X31,150:IX10X567:I,111I,119::I911448:X27,IX36,I::R$k GRAPH OF AN INEQUALITY,GRAPHS OF SYSTEMS,GRAPHING OPEN SENTENCES,INEQUAL6:I55.125:D64(U35U60)88:Q1AIG:Q15Q15I"`Q15Q15"a(Q15D64Q15D152)ē227I7,107Q18227I7,D"c:3:"e31,12031,138:60,3660,154:59,3659,154:X31:Y50:I13:X,YX,Y10:YY24::#fX,YO,YO,TX,TX,Y:07(ARG)8227W7,107(AWG)8:/!ZI12![C(I)B(3)1B(2):D(I)B(3)1B(2):((U35U60)D(I)AC(I)G)((U94U62)D(I)AC(I)G)94:91!^:O13:I12:"@"13D(I)"V"32C(I)"H@$"::41:I12:"@"13D(I)"V"32C(I)"H@ "::41:::"_60 SU2U62 TU3U35- UU4U943 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$!Y:227R7,1Y":U62U94E$"no":D$"N"PO25:I13:B1B1A(I)::B13Ĺ36267,(36267)1P21:T13:B19:L1:R26:37:T7:B18:L27:R38:37::33:200Q1000 RAB(3)1B(2):GB(3)1B(2):DB(3)1B(2):XB(3)1B(2):WB(3):YWX:UB(4):U1U1:E$"no":D$"N":U94U35E$"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$"d @D2H@for you. Complete the @D2H@graph by answering the "K"@2H@following questions @D2H@with Y (yes) or N (no):"A$:A195((36251)0)+LP1A1:32:82:87:"@29H7V@Y"(U)A"X+"G:88:"@13V1H@Should the graph include@D1H@the "R$"?":H21:V15:Iк"@13V16H@The graph@D6H@of the "N$" is@D6H@the partial plane on@D6H@the side of the@D6H@"R$" where@D6H@the points lie.":95:6;JX10:Y35:T99:O178:102:X185:O269:Y51:T155:102:"@5VI2H@The "R$" for @D2H@the "N$" on the @D2H@right has been plottet with a dotted@D6H@line).":0:88:3:U35U94İ88:72CG2:88H38:21:T7:B18:L2:R26:37:"@7V2H@<4> Find two solutions@D6H@for the "N$"@D6H@by picking a value@D6H@for X and plotting@D6H@two points for@D6H@which the "N$"@D6H@is true.":90~I5V2H@<2> Graph the boundary@D6H@line.":USDE88:38:21:T7:B18:L2:R26:37:"@7V2H@<3> If the "N$" is@D6H@of # or ^ form, then@D6H@include the boundary@D6H@line in the graph.@D6H@If it is of < or >"8F"@6H@form, then omit the@D6H@"R$" (show@D6H@i5:O269:102:T52:102:"@5V2H@GRAPH THE INEQUALITY "A$D"Y"(U)DA"X+"DG:500:"@7V29H@Y"(SD)A"X+"G+D"@I7V2H@<1> Transform the@D6H@"N$" into a@D6H@form which has Y@D6H@alone as one@D6H@member.":40:"@5V23H@"D"Y"(U)DA"X+"DG"@2D29H@Y"(SD)A"X+"G"@139:Y99:T147:102:"@13V2H@However, if the "N$" 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$" defines.":6eBP446:82:87:X10:Y35:T15, the@D2H@empty set.":21?T6:B10:L2:R38:37:T12:B19:L1:R39:37:"@6V2H@The edge(s) of the "N$"'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.'":D2H@open sentence 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 line@D17H@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<P61,46u=X10:Y43:O269:T92:102:"@6V2H@An "N$" in two variables is an@12V5H@Y#2X@33H@Y^2X":38:5:98,8063,152:98,8263,154:38:2:G152:I67102:I,GI,152:GG2::I102110:I,80I,152::38:6:231,80196,152:231,82196,154:38:1:G80:I2281921:I,80I,G:GG2::I1921851:I,80I,152::3:38;"@I13V17H@52::3:38:"@I6V27H@ The line @D27H@ Y=2X is @D27H@NOT included"A$: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:"@H6V@ The line @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.":39:103L85:259,80224,152:1:G80:I2562201:I,GI,80:GG2::I2202131:I,80I,1efines@2D1H@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"@I272D1H@ordered 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) dn To CONTENTS@I10H6D@WHICH (0-4) ??"A$:3:Kİ34:43I/MK:P1:32:33\1M50,60,66,74l2P51,57,46 3"@6V1H@An "N$" is similar to an@D1H@"S$": both are open sentences.":38:"@D1H@The ";:F$S$:36:"Y=2X defines@2D1H@a line consisting of all@ ALGEBRA Menu@2V7HI@"31)"@11H22V@WHICH (0-4) ??"A$:101:3:CK:32:24(C$(K))2:"@2VI@"C$(K)A$:C600z-33:C1815.M0:P0:32:"@14H5V@LEARNING MODE@10H7V@<1> DISCUSSION@9V10H@<2> RULE@2D10H@<3> EXAMPLE@10H2D@<4> SAMPLE PROBLEM@10H3D@<0> ReturP0:M0:C0:32:"@20H5V@CONTENTS@6V@":I14:"@10H@<"I"> "C$(I)"@D14H@"C$(I4):::"@7V4H@";:I13:(22)"@3DB@";::"@5V3H@";:I14:"<"I">@3D3B@";::X17:O45:Y38:T48:I14:102:YY24:TT24:j,5000:"@17V4H@"(31)"@3HD@<0>@10H@<0> Return To51:35 "30976. #"@21V1HLI@"19)"@RI@":l $Y(37)82:X(36)72:OX4(F$)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 +35399,0:(E)12889ĖH:V:"yes":306 H:V:"no"} (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$" ": "@2H1V@"C"@5H@"P"@2HD@"M: !310:17 1063 34:K155İ2:B$"RUN ALGEBRA 4"; 46 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& 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)155((36251))ı N,0:"@40X40YN@"; K(E):K12814:K155K205ı N,0:18 }24577:27903: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@        1)(A(1)1A(2)):C1(1)XY:D1(YT)(ZV):E1C1TA1VC<.h<(36251)6İ2:(4)"RUNAM4.5.1"<(36251)0129:50<:2:"ERROR"(222)" AT LINE # "(218)(219)256:@15H@, Y =":N4:H12:P18:27:G10:ATG112;;TA(9)B(1):VA(9):A(A(4)1):B1(A(4)1):EA(4)1:AEV2T0200:=<,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)-:"@I18V18H@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 =:13819L11ĺ"@17V18H@Try again":L1L11:135^9"@I16V18H@The answer is@I@X = @3B@"Tv9L10:"@31H18V@Y ="9H36:P19:N3:27:"@17V18H@"9):AVĺ"@17V18H@Correct":45:"@17V18H@"13):I1I11:142:L11ĺ"@17V18H@Try again":L1L11:139:H1B1TEV:"@13V3H@X + "A"Y = "G1"@D2H@"B1"X + "E"Y = "H1"@12V18H@Enter the value of@D18H@the "L$" shown,@D18H@then (RETURN) or ("(2)")@D18H@to erase.@16V31H@X = "9H36:N3:P17:27:"@17V18H@"9):ATĺ"@17V18H@Correct":45:"@17V18H@"13):I1I11low. First eliminate one of the @D2H@"L$"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$:K195((36251)0):O1K1:36:L10:2008G1TAV19H@"B1"X + "E"("V")";{6~"@30H@= "H1" @16V23H@"B1"X + "EV" @30H@= "H1"@D27H@"B1"X @30H@= "TB1"@D28HI@X @30H@= "T;V$:86T,VU,VU,WT,WT,V:q7T10:U269:V35:W92:127:W155:127:V92:T122:127:"@I5V2H@Solve the system of "Q$"s shown @D2H@be2:R38:40:"@8V2H@<4> Substitute@D2H@the Y value@D2H@";6}"into both@D2H@"Q$"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$:43:"@14V22H@"B1"X + "E"Y @30H@= "H1"@D43:"@24H13V@"B1"X + "E"Y = "H1"@D18H@"B1"("G1" + "A"Y)+"E"Y@32H@= "H1:23:"@17V2H@<3> Solve for Y":4385|"@22H15V@"B1G1"+"B1A"Y+"E"Y @32H@= "H1"@D22H@"B1G1" + "B1AE"Y @32H@= "H1"@D28H@"B1AE"Y @32H@= "H1B1G1"@D30HI@Y = "V"@I@":23:S8:B18:LG1TAV:H1B1TEV:"@5V23H@X - "A"Y = "G1"@D22H@"B1"X + "E"Y = "H1:43:"@8V2H@<1> Solve one@D2H@"Q$" for X@D2H@in terms of Y":43:"@21H9V@X - "A"Y = "G1"@D26H@X = "G1" + "A"Y":23i4{"@12V2H@<2> Substitute@D2H@the solution in@D2H@the other@D2H@"Q$:1H@<3> Solve second "Q$" 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:":2003zal equations in the system.@D5H@If your work was accurate the@D5H@resulting sentences will be@D5H@arithmetically true.":8P2t"@L4V1H10C@SUMMARY:@R143C7V1H@<1> Solve one "Q$" in terms of one@D5H@"L$".@2D1H@<2> Substitute solution into other@D5H@"Q$".@2D"@8V1H@<4> Substitute the value of the"0r"@5H@"L$" 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"{1s"@5H@origin1H@<2> Substitute 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:V10H@"G1" = "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$"."l/p43:"@12V= "EB1H1-l"@10V23H@Use Found Value@D23H@of X to find Y":43:"@13V2H@"A"("T") + "B1"("V") = "G1"@13V23H@Substitute X and@D23H@Y in the Other@D23H@Equation to see@D23H@if your answer@D23H@is correct.":43.m"@15V3H@"AT" + "B1V" = "G1:43:"@17s@2D23H@Combine Factors@2D23H@Isolate Variable@2D23H@Simplify":8 -k1:157,56157,144:3:"@138CL4V1H@NOW CHECK YOUR WORK@R143C7V2H@"A"X + "B1"Y = "G1"@D8H@Y = "E"X - "H1"@7V23H@Original System@D23H@of Equations":43:"@6H10V@Y = "E"("B1") - "H1"@D6H@Y 1"@2D1H@"A"X + "B1"("E"X - "H1")="G1"@2D1H@"A"X + "B1E"X - "B1H1"="G1"@2D8H@"AEB1"X - "B1H1"="G1"@2D13H@"AEB1"X="G1B1H1"@2D15HI@X = "T;V$A,j1:157,56157,152:3:"@7V23H@Original System@D23H@of Equations@2D23H@Substitution@2D23H@Simplify Factor"X - "H1"@I6V11H@"A"X + "B1"Y = "G1;V$*h5:143,112143,152:"@14V21H@"A"("T") + "B1"(Y) = "G1:44:"@16V22H@"AT" + "B1"(Y) = "G1:44:"@18V22H@"AT" + "B1"(@I@"V"@I@) = "G1:8+i"@4V1HL138C@IN SUMMARY:@R143C7V5H@"A"X + "B1"Y="G1"@D11H@Y="E"X - "H "H1:44:"@10V1H@Since we know now that X="T", we can@2D1H@substitute that value and solve for Y:":44&*g"@I18H8V@Y = "E" + "H1"@I14V7H@Y = "E"("T") - "H1:44:"@16V7H@Y = "ET" - "H1:44:"@18V7HI@Y@I@ = "ET"-"H1" = @I@"ETH1;V$:44:"@18H8V@Y = "E40:"@18V1H@Finally, we solve for this variable."(e"@13V18H@"AEB1"X@22H@= "G1B1H1:43:"@I15V20H@X@I@ = @I@"T;V$:5:115,99193,99193,131115,131115,99:43:8g)f"@4V1H@Here is our system of two "Q$"s:@2D11H@"A"X + "B1"Y = "G1"@2D18H@Y = "E" +D1H@on both sides of the equation.":43:0:91,6191,9011,9011,6191,612(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: the "Q$" further.":43:0:147,46147,7349,7349,46147,46:5:91,6191,9011,9011,6191,61I'c"@10V8HI@"AEB1"X@I15H@- "B1H1"@22H@= "G1:43:23:SS3:40:"@15V1H@Now we isolate the variable term by@2D1H@adding or subtracting";:" each other term@2 the factors so@2D1H@";%a"we can reduce the "Q$" 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:SS3:23:40^&b"@13V1H@Next, we add the common factors to@2D1H@simplify "B" @2D18H15C@ @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 multiply0#^"@I24H8V@"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$".":45:I1231m$_"@0C"I"H6V@"A"X + "B1"@15C"I1"H@"A"X + "B1;:46::"@0C6V12H@"A"X +Q$" tells us what the@2D1H@value of Y is, expressed in terms of@2D1H@X.";#]" Because these "Q$"s are a@2D1H@system, we already know that the value@2D1H@of Y is the @I@same@I@ in both "Q$"s.@6V12H@"A"X + "B1"@I@Y@I2D20HI@Y":44:23:S9:B19:L1:R39:4M89,110,118,129+!YO90,102,105,107,50y!ZTA(9):VA(9):A(A(4)1)B(1):B1(A(4)1):EA(4)1:AEV2T090I"\G1ATB1V:H1VET:"@4V1H@A system of two "Q$"s:@2D12H@"A"X + "B1"Y @22H@= "G1"@2D20H@Y = "E"X - "H1:43:"@10V1H@The second " to @D2H@find the value of the first variable@D2H@Then enter your answers below. "V$:O1F1:36:300 T145:H22:27:G11AVĺ"@18V33H@RIGHT":36265,(36265)1:86 U"@18V10H@WRONG. X="T" AND Y="V V23:S14:B18:L2:R38:40::37:400!X1)0)RT10:U269:V35:W99:127:W155:V107:127:"@5V2HI@Below is a system of simultaneous @D2H@linear "Q$"s. Use the addition @D2H@property of equality to eliminate @D2H@one "L$" so that you may solve "k S"@2H@for the other, and substitution by@D2H@substituting@D2H@them into the@D2H@second "Q$".@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((3625<5> Solve for X.@23H@X@30H@= "T:41:"@14V2H@<6> Check the@D2H@values by@D2H@substituting@D2H@them into the@D2H@first "Q$".@19H14V@"Y"("T")+"Z"("V")= "D1"@21HD@"YT"@24H@+"ZV"@29H@= "D1rO"@16V25H@"D1"@29H@= "D1:23:40:"@9V2H@<7> Check the@D2H@values61,83252,83:41:"@17V2H@<3> Solve for the@D2H@"L$" Y.@26H17V@Y = "VM23:S8:B18:L2:R38:40:"@8V2H@<4> In the first@D2H@"Q$", replace@D2H@Y by its value.@22H8V@"Y"X+"Z"(Y)= "D1"@D22H@"Y"X+"ZV"@30H@= "D1"@D22H@"Y"X@29H@ = "D1ZV:41N"@12V2H@@16H@"C1"X+"A1"Y="E1"@8V2H@<1> Multiply all@D2H@members of the@D2H@first "Q$"@D2H@by "X".@21H8V@"Y"X+"Z"Y="D1"@D21H@"C1"X+"A1"Y= "E1:41:"@21H8V@"YX"X+"ZX"Y= "D1XEL41:"@13V2H@<2> Add and@D2H@eliminate the@D2H@"L$" X.@13V25H@"ZXA1"Y = "D1XE1:1solution by@D6H@substituting the values of the@D6H@"L$"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"alue@D6H@of 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 "@5V2H@<3> 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 v"@6H@"L$"s appear on one side and@D6H@the constant is on the other.@2D2H@<2> Multiply both members of each@D6H@"Q$" by numbers that will@D6H@make the coefficients of one of@D6H@the "L$"s the same in@D6H@absolute value.":8ET10:U269:V35:W155:127:2H@is (1,-1).":8$BO67,69,71,50CT10: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$" into an@D6H@equivalent one in which the"D,131:23:S4:B19:L1:R38:40:"@5V2H@Now replace X for its value in either@2D2H@"Q$" 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@2Do "Q$"s below. By@2D2H@multiplying the first "Q$" 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@($3)"h@"@U13H@18X + 3Y = 15":41:"@17V13H@23X@22H@= 23@D15H@X@22H@= 1":91,131182will not eliminate "L$"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 twte 3 for"<"@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@":164,56262,112:252,41175,126:41:"@15V1H@We can subtract the@D1H@second from the first" ;"@17V1H@to get an "Q$"@D1H@that does not involve@D1H@X.@8V7H@3Y=9@D8H@Y=3":35,5977,59:5:161,83266,83:3:23:S11:B19:R22:L1:40:"@15V1H@Now substitu:9"@28H17V@";:F15:3:" ";::"@37H@X@5V25H@Y@17V24H@";:F1:3:T183:V51:I15:T,VT5,V:VV16::T171:V129:I17:T,VT,V5:TT14:]:"@5V5H@X+2Y=8@D5H@X-Y=-1@11V1H@The coefficients of@D1H@the "L$" X are@D1H@the same in both@D1H@"Q$"s.ivalent systems made up of the@2D2H@"Q$"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@";WHICH (0-4) ??"V$:K0:N4:5:Gİ38:2:(4)"RUNAM4.4"K3MG:O1:36:37W4C388j5M54,66,73,816O55,56,61,63,50m7"@5V2H@The Addition Property of Equality may@2D2H@be used for solving a system of@2D2H@linear "Q$"s in two "L$"s.@2D2H@Equ -Z250:144 .Z10:144d /35339:A(T)((1)T)1:B(T)1A(2):(36251)081:C(24576)72M0:O0:36:"@14H5V@LEARNING MODE@10H7V@<1> DISCUSSION@9V10H@<2> RULE@2D10H@<3> EXAMPLE@10H2D@<4> SAMPLE PROBLEM@10H3D@<0> Return To CONTENTS@I10H22V@8M #A$"":I0Q1:A$A$((256I))::A(A$):AQ27:H:" ";:H:A:m $"@2H1V@"C"@5H@"O"@2HD@"M:| %31051:39 &30976 '"@21V1HLI@"19)"@RI@": (JSB:L1:J1:RL):: )H170:144 *Z50:144 +Z150:144 ,Z100:144G16024:"@I22V1H@"36)V$:J H:P:N):Q0:I0N:256I,32::J1,0e HQ:V$((256Q))V$; G(ZZ):14:GXX29:J1,0:HQ:((256Q));:G14135:QQ(G149QN1)(G136Q0):GGXX:G47G58G45ĖHQ:(G);:256Q,G:QQ1:QN35 "26251))ı J1,0:"@40X40YN@";F G(ZZ):GXX16:G155G205ıU J1,0:20_ :19k 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::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)155((34 63900c24577:27903:XX128:V$"@I@":ZZ16384:J116368:Q$"equation":L$"variable":47r0:1002:"@G15C0KE@";:E(F):A(E10):BEA10:EE9:F0ĺ(12E)E)"@B@";(20A)"@B@"(B)"@ER@";: 14:G(ZZ):GXX5:J1,0:GG176                             AAI:BBN1IN2J-:AA213AA14:BB107BB8:CC213CC14:DD107DD8:ynV2:"++ERROR++ "(222)" AT LINE # "(218)(219)256:,6:V43:H2521611:H,V:H,V1:VV1:57::"@33H8V@Y=X+1":58:5:V135:,2:"RUNAM4.4.1":8,,43:2:(4)"RUNALGEBRA 4"-AA0:I33.5:IN1N25IN1N25CCI:DDN1IN2:AA0I19:X,YX5,Y:YY8::"@8V29H@Y@14V37H@X@29H17V@";:N44:N1NN2:"@2U@";+~3:"@UB@";::"@14V24H@";:N33:N0ĺ"@2F@";:NN1+3:"@F@";::+GRAPHIC METHOD,ADDITION AND SUBTRACTION,SUBSTITUTION METHOD,SOLVING SYSTEMS TEST5,:N1:"@2F@";F*{3:"@F@";::"@16V29H@";:N23:N0NN1:"@2U@";w*|3:"@2UB@";::X157:O269:Y35:T155:121:T+}"@8V30H@"(95)"@30H18V@"(126)"@13V23H@<@37H@>":162,107262,107:213,65213,149:X171:Y105:I17:X,YX,Y5:XX14::X211:Y75::X31:Y50:I13:X,YX,Y10:YY24::C)yX,YO,YO,TX,TX,Y: *z"@5V30H@"(95)"@30H18V@"(126)"@12V23H@<@37H@>":162,99262,99:213,41213,149:X211:Y51:I16:X,YX5,Y:YY16::X171:Y97:I17:X,YX,Y6:XX14::"@13V24H@";:N33:N0N4,(36264)1S(rT5:B18:L2:R19:23:46:L21:R38:46::42:(UU)Ĺ24576,C:200[(s64(t24576,C:C4ĹUU,4:I14:36263I,0::98(u200(vN5R(5)3:N6R(7)4:N1R(7)4:N3R(7)4:N1N3118:()x31,12031,138:60,3660,154:59,3659,154WRONG'o5:N1N3:N2N4:400:AA,BBCC,DD:3:"@5V21H@Enter coordinates@D21H@of intersection @D21H@( , )":H23:V8:MX3:47:A(1)A:H27:47:A(2)A:A(1)N5A(2)N6ĺ"@18V33H@RIGHT":ANAN1:113'p"@18V36H@ @25H@WRONG@F@("N5","N6") (qAN2Ĺ3626&k:"@5V21H@Graph first point@D21H@of Y="N3"X+"N4:27:R302S:Y13U:J1:"@27H5V@second @D21H@point of Y="N3"X+"N4:27:J0:H0:I1NE:X1(I)SY1(I)UJ1&lX1(I)(R30)2Y1(I)13YH1&m:H1J1ĺ"@18V33H@RIGHT":AN1:111'n"@18V33H@1"X+"N25)"@D2H@ Y = "N3"X+"N45)"@17V17H@ @D17H@ "X$%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:NENE1ow are two @D2H@linear "Q$"s,@D2H@the first of @D2H@which is already @D2H@graphed. Graph @D2H@two points of the8%g"@2H@second "Q$". @D2H@Use the keys: @D2H@D:Down L:Left @D2H@U:Up R:Right@D2H@ P:Plot @D2H@"17)"@D2H@ Y = "N,"N6")"(125)".":84#bP195((UU)0):P1P1:41#d118:J0:AN0:NE0:N2N6N5N1:N4N6N3N5:F0:I33:FF(N1IN2(N1IN2))::F2100:F0:I33:FF(N3IN4(N3IN4))::F2100x$fX10:O137:Y35:T155:121:X143:O269:121:"@5V2HI@Bel:"@8V2H@<4> Check that X="N5" and Y="N6"@2D6H@Y="N1"X+"N2"@21H@Y="N3"X+"N4"@2D6H@"N6"="N1"("N5")+"N2#a"@U21H@"N6"="N3"("N5")+"N4"@2D6H@"N6"="N6" (true)@21H@"N6"="N6" (true)@2D2H@The common solution is ("N5","N6").@D2H@The solution set is "(123)"("N5"V@$"::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@$@15C@":56:"@L13V6H@("N5","N6")@R@":23:R38:46e"`I15:0:154,51I161,51I:154,155I161,155I:3:L2:R21:46:X67:O108:Y83:T131:121:O87:121:O108:T99:121:T7:"@11V10H@X@13H@Y@8V2H@<2> Graph Y="N3"X+"N4:EW:E3E3!^JJN1:HHN2:N1N3:N2N4:400:N1JJ:N2HH:"@10H13V@";:I1E:X(I)"@13H@"Y(I)"@D10H@";::56:I1E:"@"30X(I)2"H"13Y(I)"25WW1:X(W)I:Y(W)N1IN26[:125:EW:E3E3\"@10H13V@";:I1E:X(I)"@13H@"Y(I)"@D10H@";::56:I1E:"@"30X(I)2"H"13Y(I)"V@$"::56:6:400:AA,BBCC,DD:W0:I33:N3IN45N3IN45WW1:X(W)I:Y(W)N3IN4 ]:3:23:T7:B18:F288YW0: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:N1IN25N1INthe graphs@D6H@to find the common solution.@D2H@<4> Check that the solution@D6H@satisfies both "Q$"s.":8uWP464XX10:O269:Y35:T51:121:T155:121:X157:O269:Y51:T155:121:118:N2N6N5N1:N4N6N3N5:F0:I33:FF(N1IN2(N1IN2))::phically:@2D2H@<1> Graph one "Q$" in a@D6H@coordinate plane.@D2H@<2> Graph the second "Q$" in a@D6H@coordinate plane using the sameiV"@6H@set of coordinate axes.@D2H@<3> Read the ordered number pair@D6H@associated with the point of@D6H@intersection of pair of linear "Q$"s can be@D2H@solved by graphing the "Q$"s in@D2H@the same coordinate plane system and@D2H@finding the coordinates of all@D2H@points common to the graphs.":8UX10:O269:Y35:T155:121:"@5V2H@To solve a pair of linear "Q$"s@D2H@graonsistent":45:"@2D2H@";:T$"equations":45:".":8DRP83,85,64SX10:O269:Y35:T75:121:Y83:T131:121:"@5V2H@The "S$" set of a system of two@D2H@linear "Q$"s is the intersection@D2H@of the "S$" sets of the@D2H@individual "Q$"s. T"@2D2H@A3H@Y=X+3":6:219,42161,112P219,43161,113:5:266,56182,152:266,55182,151:3:"@15V23H@Y=X-1":23:I113:2:5I:21)::"@5V2H@A set of linear@2D2H@"Q$"s that has4Q"@D2H@no "S$" set is@2D2H@called a ";:T$"system of":45:"@2D2H@";:T$"incn@2D2H@Y=X); they are@2D2H@";:T$"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.":"@6V2"lines have 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$"@2D2H@sets are notMN"@D2H@identical (for@2D2H@example, (0,2) is on@2D2H@Y=-X+2, but not oat (1,1), and their@2D2H@common point is@2D2H@called the ";:T$"point":45:"@2D2H@";:T$"of intersection":45:".L6:175,144259,48:175,145259,49:5:184,40260,120:184,41260,121:23:I114:2:5I:21)::"@5V2H@Because the two@2D2H@";:3Mthey@2D2H@coincide. This is@2D2H@called a ";:T$"system of":45:"@2D2H@";:T$"dependent "Q$"s":45:"@2D2H@The two "Q$"s'@2D2H@"S$" sets are@2D2H@identical.":8rK122:"@5V2H@The two lines have@2D2H@one point in common.@2D2H@The lines intersect@2D2H@S$" set of the":45:"@2D2H@";:T$"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; s.":8G"@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":457H"; the set of all "S$"s@2D2H@is called the ";:T$к"@5V2H@The 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$"@M0:P0:41:"@14H5V@LEARNING MODE@10H7V@<1> DISCUSSION@9V10H@<2> RULE@2D10H@<3> EXAMPLE@10H2D@<4> SAMPLE PROBLEM@10H3D@<0> Return To CONTENTS@I10H6D@WHICH (0-4) ??"X$:5:Kİ43:60AMK:P1:41:42CM68,82,87,98DP69,71,73,75,79,64E8:T48: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 (0-4) ??"X$:120:5:CK:41:24(C$(K))2:"@2VI@"C$(K)X$?C300:42:C111600:14::J;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:Y3'2QQ,0:HS:((256S));:K14155:3K136SSS1R4K149SMX1SS15KK128:(K47K58)K45ĖHS:(K);:256S,K:SS1:SMX556487I$"":I0S:I$I$((256I)):I:A(I$):8Q1150:14::9I12:14:: :I11D@"M: *31051:44 +30976= ,"@21V1HLI@"19)"@RI@":{ -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:K128498* !U1S0ĺ"@"302S"H14V@";:NS:3W "S1U1U0ĺ"@"13U"V29H@";:NU:3 #UU(K85)(K68):(U)5U4(U) %K80:"@"302S"H"13U"V@$": &SS(K76)(K82):(S)4S3(S) '6:AA,BBCC,DD:3:28 )"@2H1V@"C"@5H@"P"@2H2V1H@"36)X$: U0:S0:QQ,08 J1ĺ"@"R"H"Y"V@$"Y "@"302S"H"13U"V@!":QQ,0 K(ZZ)128:14:K030:K85K68K80K76K8230 "@"302S"H"13U"V@ ":U0S0ē210S14,107U8216S14,107U8:213S14,104U8213S14,112UQQ,0:"@40X40YN@";; K(ZZ):K12816:K155K205ıJ QQ,0:20T :19^ 129 43:K155İ2:"RUNALGEBRA 4" 64 35399,1:"@22V6H@Press SPACE BAR to Continue":35399,0:QQ,0:14 K(ZZ):14:K12824:QQ,0:K16024:"@I25:v 35399,1:"@3H21V@Press "(1)" Key to View the Next Page@D3H@Press "(2)" Key to View the Last Page":35399,0 K(ZZ)128:K09:14:QQ,0:K21K89:K21PP1:42 K8PP1:43:P64 41:67 D(1):(ZZ)155((UU))ı %Y24577:27903:ZZ16384:UU36251:QQ16368:Q$"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                          6:H3:42:F$:"<146:V99:106]<::3:"++ERROR++"(222)" AT LINE #"(218)(219)256>>{:(4)"RUN ALGEBRA 4"<<(36251)6125:(36251)091:490= A1A(9):B1A(9):C1A(9):DT10:U269:R35:V155:106k;D1A(5):E1A(5):C1A(5):G1A(5):A1G1D1E1:F1A(5):D1F1C1D1147:q;;H1A(3):D1A(3):E1A(3):C1A(3):G1H1E1:A1H1D1:F1C1E1:D1F1C1D1149:;210,91224,91:245,91266,91:<F$:S6:E12:X3(4)"RUNALGEBRA 4":48B:(36251)6132:(36251)0117:48:A1A(9):B1A(9):C1A(9):D1B1C1:E1A1C1:F1A1B1C1:D1B1D1A1143::A1A(9):B1A(9):C1A(9):D1B1C1:E1A1C1:F1A1B1C1::T10:U269:R35:V75:106:V155:106:;16V14H@A =":I1H1:135g9H23:O17:28:"@R@":I1Qĺ"@18V2H@RIGHT":36263,(36263)((36251)0):1389"@18V2H@WRONG.":24:38:M41399N123,128,123,128,123,128,123,128,123:249389N:(36251)6İ3:(4)"RUNAM4.4.1":(36251)İ3:ION@L14H16V@A = ":I1G1C1F1:1359149:"@I5V2H@A varies directly as B and inversely@D2H@as C.@D2H@A equals "A1" when B equals "D1" and@D2H@C equals "E1".@2D2H@If B equals "C1" and C equals "F1", what@D2H@is the value of A?@I14V2H@COMBINED VARIATION@L195((36251)0):37:152:147:151:K5:N2N4N6N8134"8"@I5V2H@A varies jointly as B and C.@2D2H@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 VARIAT@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@"F1:150:182,139196,139:217,139238,139:M4ı69?7Nı 5}95~N64851495146:"@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@";6"when B equals "C1" and C equals "F1".@11V2H@<1> Find the constant@D6H@B equals "C1" and C equals "F1".@3D2H@<1> Find the constant@D6H@of proportionality."5|"@3D2H@<2> Find the value of@D6H@A.@30H7U@A@35H@"A1"@26HD@K =@33H@=@D30H@BC@35H@"D1"$"E1"@28H2D@= "G1"@26H2D@A = KBC@D28H@= "G1"$"C1"$"F1"@D28H@= "G1C1F1:150:M43w"@17V3H@WRONG. ANSWER IS "E1:(36251)0120:24:38:114`3x24:38:N:(36251)0132:48m3yN3126v3z147U4{146:"@5V2H@A varies jointly as B and C. If@D2H@A equals "A1" when B equals "D1" and C@D2H@equals "E1", find the value of A when@D2H@I5V2H@Y varies inversely as X."12)"@D2H@Y is equal to "A1" when X equals "D1"."2)"@D2H@Find the value of Y when X equals"3)"@D2H@"B1"."34)"@6V35H@ @IL12V8H@Y = ";:K4:H16:O13:28:"@R@"2vQE1ĺ"@17V3H@RIGHT":36262,(36262)((36251)0):120=2H@<1> Find the constant@D6H@of proportionality.@14V2H@<2> Solve for Y when@D6H@XY = K.@10V27H@XY = K@D26H@("D1")("A1")= K@D28H@K = "F1"@2D27H@XY = K@D25H@("B1")Y = "F1"@D28H@Y = "E1:M4ı1t92uN195((36251)0):145:37:37:144:"nD1A(3):E1A(3):C1A(3):G1H1E1:A1H1D1:F1C1E1:C1D1E1G1C1E1C1G1D1E1D1G1110:h0ox0pN448:370q1440r145:"@5V2H@If Y varies inversely as X,@D2H@and if Y="A1" when X="D1",@D2H@find the value of Y when X = "B1"."1s"@10V(126)"@12V23H@<@37H@>":162,99263,99:213,41213,151:"@29H6V@";:F321:4:"@2DB@";:F1ĺ"@2D@";:FF1/lF:"@13V24H@";:F33:4:"@F@";:F1ĺ"@2F@";:FF1/m:T171:R98:I17:T,RT,R5:TT14::T211:R51:I16:T,RT5,R:RR16::b0"@36H@"E$.i"@17V15H@"D$"@22H@"E$"@29H@"E$"@34H@"E$"@36H@"D$"@R@":49,10763,107:168,107182,107:91,131119,131:140,131168,131:196,131210,131:231,131259,131:9.jT,RU,RU,VT,VT,R:m/kT157:U269:R35:V155:106:"@5V30H@"(95)"@18V30H@"D2H@non-zero constant k,@16V2H@Therefore@7H12V@kx@24H@zy@D3H@z =@10H@or zy = kx or" .h"@13V27H@= k@D8H@y@25H@x@15V13H@z y@20H@z y@28H@z@33H@x y@D18H@=@25H@or@31H@=@37H@.@D14H@x@21H@x@28H@z@33H@x y@G14H15V@"D$"@16H@"D$"@21H@"E$"@23H@"E$"@29H@"D$"@34H@"D$"@R@":84,9998,99:84,131112,131:133,131161,131:196,131210,131:231,131259,131:"@37H16V@.@13V12H@xy@R@":9S-g106:"@5V11H@COMBINED VARIATION@2D2H@occurs when a variable z varies@D2H@directly as a variable x and@D2H@inversely as another y. For a@nstant k,@11V12H@z@15HD@= k or z = kxy@15V2H@Therefore";+e" z@20H@z@28H@z@33H@x y@D17H@=@25H@or@31H@=@D12H@x y@19H@x y@28H@z x y@G15V13H@"D$"@21H@"E$"@29H@"D$"@34H@"D$"@36H@"D$"@17V13H@"D$"@15H@"D$"@20H@"E$"@22H@"E$"@29H@";r,fE$"@34H@"E$"@36H@"E$"@23H@"D$"@25H@"E$"@17V4H@"D$"@11H@"E$"@18H@"E$"@23H@"E$"@25H@"D$"@R@":9[*cN100,103,488+dT10:U269:R35:V155:106:"@12H5V@JOINT VARIATION@2D2H@occurs when a variable z varies@D2H@directly as the product of variables@D2H@x and y. For a non-zero coz@22H@x y@7H16V@=@14H@or@20H@=")a42,8356,83:77,83105,83:126,83154,83:14,13142,131:63,13191,131:119,131133,131:154,131182,131:"@G9V12H@"D$"@14H@"D$"@19H@"E$"@21H@"E$"@13H11V@"D$;I*b"@20H@"E$"@15V3H@"D$"@5H@"D$"@10H@"E$"@12H@"E$"@18H@"D$@kx@23H@zy@15V7H@y@24H@x"@(_42,11556,115:161,115175,115:9)`"@5V2H@Since k is a constant for all values@2D2H@of x, y, and z, then:@9V6H@zy z y z y@D2H@k=@9H@=@16H@=@11V7H@x@12H@x@19H@x@13V2H@So:@2D2H@z y z y@17H@z x y@2D3H@x@10H@x@17H@131:168,131196,131:"@10V23H@.@16V28H@.":9(^"@5V2H@";:B$"Combined variation":41:"is when a@2D2H@variable z varies directly as one@2D2H@variable x and inversely as another@2D2H@y. For a non-zero constant k,@3D2H@z = or zy = kx or@26H@= k.@13V6H@G@"E$"@R19H@z@G@"D$"@R24H@x@G@"D$"@R@y@G@"D$"@R17V2H@x@G@"D$"@R@y@G@"D$"@R9H@x@G@"E$"@R@y@G@"E$"@R19H@z@G@"E$"@R24H@x@G@"E$-']"@26H17V@y@G@"E$"@R7H16V@=@16H@or@22H@=":42,8356,83:77,83105,83:126,83154,83:14,13142,131:63,13191,131:133,131147,28,11542,115:9%["@5V2H@Since z = kxy, and k is a constant@2D2H@for all values of x, y, and z, then@2D6H@z@11H@z@G@"D$"@R18H@z@G@"E$"@RD2H@k =@9H@=@16H@=@D6H@xy x@G@"D$"@R@y@G@"D$"@R18H@x@G@"E$"@R@y@G@"E$"@R@"|&\"@13V2H@So:@15V3H@z@G@"D$"@R10H@z:9$Y"@5V2H@";:B$"Joint Variation":41:" is when a variable@2D2H@z varies directly as the product of@2D2H@variables x and y. When z varies@2D2H@jointly as x and y,@2D5H@z@D7H@= k, where k is a non-zero@D4H@xy@12H@constant,@2D2H@";%Z"or z = kxy.":D17H@base, and you can say@D17H@that the area of a@D17H@triangle";$X" varies@D17H@jointly with its base@D17H@and altitude.@11V7H@a@14V7H@b@16V4H@1@D2H@A=-ab or =-@D4H@2@11H@ab 2":78,13990,139:"@16V12H@A 1":28,11259,7284,11228,112:59,7259,1125> The graph of XY = K is a@D6H@hyperbola.":9C"UM86,99,121,132\"VN87,89,91,94,96,48B#W"@5V2H@The area A of a triangle depends upon@2D2H@its altitude a and its base b.@3D17H@The area is directly@D17H@proportional to the@D17H@altitude and to the@139196,139:217,139231,139:9!S106:"@6V2H@<3> If X is multiplied by a non-zero@D6H@number, Y must be divided by@D6H@that same number.@2D2H@<4> If X is divided by a non-zero@D6H@number, Y must be multiplied by@D6H@that same number.@2D2H@";."T"@D@ The product of X and Y is@D6H@constant: XY = K, where K is the@D6H@constant of proportionality.@2D2H@<2> The product of X@G@"D$"@R@ and Y@G@"D$"@R@, any" Q"@6H@pair of values for tD$"@R@ X@G@"E$"@R@Y@G@"D$"@R@":56,11584,115:105,115133,115:44:"@14V20H@or@23HU@X@G@"D$"@R@ Y@G@"E$"@R2D23H@X@G@"E$"@R@ Y@G@"D$"@R26HU@=":161,115175,115:196,115210,115:"@LE10H13V@$@15V10H@$@13V15H@$@15V15H@$@RE@":9ON80,83,48PT1043:" @6B@";::"@13V15H@X@G@"E$"@R@Y@G@"E$"@R@":24MI18:2:5I:37)::"@13V1H@ @5V2H@Since neither Y@G@"D$"@R@ nor X@G@"E$"@R@ is zero, you@2D2H@can divide both members by X@G@"E$"@R@Y@G@"D$"@R@ and@2D2H@get:@15V8H@X@G@"E$"@R@";N"Y@G@"$"@R@";K" = K":44:"@13V1H@SO:@11V8H@";:I12:"X@G@"D$"@R@Y@G@"D$"@R4B@";:43:" @D4B@";::"@13V8H@X@G@"D$"@R@Y@G@"D$"@R@ =@11V21H@";:I127L"X@G@"E$"@R@Y@G@"E$"@R4B@";:43:" @D4B@";::"@21H13V@";:I13:"X@G@"E$"@R@Y@G@"E$"@R4B@";:@ @D28H@ @27H16V@ ":0:168,112175,112:168,111175,111:168,110170,110:3:J"@5V2H@If (X@G@"D$"@R@,Y@G@"D$"@R@) and (X@G@"E$"@R@,Y@G@"E$"@R@) are ordered@2D2H@pairs of an inverse variation, then@3D8H@X@G@"D$"@R@Y@G@"D$"@R@ = K and X@G@"E$"@R@Y@G@"E1213,151:0:I14:32:19I:I)::"@14V33H@ @31H16V@ @DB@ @DB@ @13V35H@ @32H17V@ ":253,104253,112:254,104254,112:255,104255,112:252,104252,112:3:I16:6I:24(6I):I)::9NII16:6I:32:I)::"@14V24H@"4)"@D27H@ @D28H@ @D28H(I)(I10)10ĺ"@11V3H@ 2@D2B@ -@D2B@ "((I))"@L7H11V@"(I)"@R13V7H@ "G45::24:1:162,99263,99:"@5V2H@If K=0, then the @D2H@graph is along the@D2H@two axes."9)"@D2H@"9)"@L11V2H@"9)"@D2H@"9)"@R@"H212,41212,151:214,41214,151:213,46:(I)(I10)10ĺ"@11V3H@ 2@D2B@--@D2B@ "(I)"@7H11VL@"(I)"@R13V7H@ "E45::254,110:I31.1:TI:R2I:212T14,100R16:(I)(I10)10ĺ"@L11V3H@"T;C$"@R@ 2@D2B@--@D2B@ "I"@13V3H@ "JF45::I2.13.1:T2I:RI:212T14,100R16::73:"@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)JD45::I2.13.1:T2I:RI:212T14,100R1:TI:R2I:212T14,100R16:(I)(I10)10ĺ"@11V1HL@"(T)C$"@R11V7H@ 2@D2B@--@DB@"((I))"@13V3H@ "B45::I2.13.1:T2I:RI:212T14,100R16:(I)(I10)10ĺ"@11V1H@ 2 @D4B@ -- @D2B@"((I))"@11VL7H@"(I)"@R7H13V@ "C45::244,90:I31.1:TI:R2I:212T14,100R16:(I)(I10)10ĺ"@11V3HL@"I;C$" = 2@R8H@2@DB@-@DB@"I@45::I2.13.1:RI:T2I:212T14,100R16:(I)(I10)10ĺ"@11V3H@2 @D2B@- @D2B@"(I)"@11V7HL@"(R)"@R13V7H@ "jA45::170,110:I31.1))"@L4H11V@"C$(I)" = 1@R@"h>45::"@16V2H@This shape is called@2D2H@a ";:B$"hyperbola":41:".":9e?107:"@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"C$"Y = 2@R@":25I:R1I:212T14,100R16:(I)(I10)10ĺ"@L2H11V@"((T))C$"@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@--@DB@"((II31.1: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@"C$;R" = @R8H13V@ "<45::165,104:I31.1:TH@The graph of the@D2H@equation XY = K is@D2H@not a straight line.@D2H@The equation is not@D2H@linear.":107:"@L11V3H@X Y = 1@R18H11V@LET@D18H@K=1@11V5HL@"C$"@R@":45c:T171:R98:I17:T,RT,R5:TT14::T211:R51:I16:T,RT5,R:RR16::258,94:2D2H@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":418" to X, because@3D15H@Y = K $ .@23H15V@1@2DB@X":161,131168,131:99"@5V2tes of its ordered pairs is a@2D2H@non-zero constant. For any ordered@2D2H@pair (X,Y) of a function@3D2H@";6"XY = K, or Y = , where K is a@D20H@non-zero constant.@15V17H@K@17H17V@X":119,131126,131:97"@5V2H@As X increases, Y decreases so that@> RETURN TO CONTENTS@I10H22V@WHICH (0-4) ??"F$:J0:K4:6:Gİ39:47Y1MG:N1:37:38e2C485z3M52,79,112,1174N53,55,57,63,74,48k5"@5V2H@An ";:B$"inverse variation":41:" is a function@2D2H@in which the product of the@2D2H@coordina"@"W"K@"X)::% +Y1300::15:; ,Y12000::15:O -Y150::15:_ .35339:142v /3:(4)"RUNAM4.3"E0M0:N0:C(24576):37:"@14H5V@LEARNING MODE@10H7V@<1> DISCUSSION@9V10H@<2> RULE@2D10H@<3> EXAMPLE@10H2D@<4> SAMPLE PROBLEM@10H3D@<0 #29Y $A$"":I0P1(P6):A$A$((256I))::Q(A$):QP28:H:" ";:H:Q:y %"@2H1V@"C"@5H@"N"@2HD@"M: &31051:40 '30976 ("@21V1HLI@"19)"@RI@": )R(37)82:T(36)72:UT4(B$)7:VR12:B$" ";:106 *FSE:H:F:25 "@I22V1H@"36)F$:@ O:H:P0:I0K:256I,32::Z,0^ HP:F$((256P));:F$;w G(L):15:G12830 Z,0:HP:((256P));:G14136 G136PPP2 !G149PKPP2 "GG128:G47G58ĖHP:(G);:256P,G:PP2:PK36155((36251))ı( Z,0:"@40X40YN@";= G(L):G12817R G155G205ı` Z,0:21j :20t 137 39:G155İ3:(4)"RUN ALGEBRA 4" 48 "@22V6HI@Press SPACE BAR to Continue"F$:Z,0 G(L):15:G12825 Z,0:G160Z,0:GG176:GJGK6  "@3H21VI@Press "(1)" Key to View the Next Page@D3H@Press "(2)" Key to View the Last Page"F$ 15:G(L):G12810 Z,0:GG128:G21G810 G21NN1:38 G8NN1:39:N48 37:50 (L)њ4 153j24577:27903:A(D)((1)D)1:C$(19):D$(21):E$(22):L16384:Z16368:F$"@I@":46}0:1002:::"@G15C0KE@";:E(F):A(E10):BEA10:EE9:F0ĺ(12E)E)"@B@";(20A)"@B@"(B)"@ER@";:15:G(L):G1286                  B(1)B(3)115: 4tP441:100&4u74vP441:1004w32:"@18H16V@=@14V22H133C@3":33:"@14V22H@ @22H16V@3":33:H15:T15:R17:H:T:"2":33:H:T:" ":HH2::"@15C24H16V@"(19)"@133C@2@R15C@":6v15C@":6,109:B140:110:L126:110:L199:X137:110:L269:110:W75:B106:110:L199:110:63q30:"or the@2D2H@";:E$"constant of "M$:30:".":14,8321,83:63rK(G):10:K128114:4sB(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):z#     ʹ. ʹ6 ʹ' ʹ6 ʹ ʹ8ʹ խŠӳ 8ʹӠʹ'ʹ0ʹ-~<13a<?3>>?3?~6? ?#6?3080<0 1? 21 6g ~6?<'~8<<0x?p <011???016?0>>33?`?y$%GΩϩ  %%%GΩϩ  %%$$'к`&$%GΩϩ  %%%GΩϩ  %%$$м`  p6 6619 1991119 19910 99 9999999 9999997 100 10001001196 9669994 94499100 100010         214,36214,14867,14867,36:1:I7791:I,48I42,96::I7791:I,96I42,48::8I42,96::I7791:I,96I42,48::"::,84:175,91:183,90:185,97:"@R15C15H15V@2@20H@2@25H@2@L16H15V@=@21H@+@13H@"(97)"@18H@"(98)"@23H@"(99): X(222):X255Ħ :0:1002::::"++ERROR++";::" "(222):" AT LINE#"(219)256(218):R 66,35213,35213,14766,14766,35:67,363169,75171,77 178,67188,56189,56179,67:"@R27H6VG@"(33)"@I15C@":VT1012:"@"VT"V23H@ "::6:A80:I161189.9:I,A189,A:AA1::"@I@":3k 140,80161,80:126,112188,112:140,81160,81:126,111188,111:3:167,83:173,87:179,82:185,10H@All Rights Reserved@I@":1000 5:141,48196,72141,72141,48:142,49196,71142,71142,49:7:170,53168,55166,56165,58164,60163,64163,66164,68165,71166,73168,75170,77:171,53169,55167,56166,58165,60164,64164,66165,68166,71167,74,4277,4277,1884,1884,43,4276,4276,1883,1883,4:"@RI@":I24:I:2:38)::I2023:I:2:38)::"@L1V12H@ALGEBRA 4@R3V1H@VERSION 1.1@30H@01 MAY 84"! "@19V9H@Copyright @G@"(34)"@R@ 1983,1984@D11H@Peachtree Software@D13H@An MSA Company@22VT(205,255::::255:ZZ(0)::11`24577:27903:1012,0:3:(4)"BLOADEWS3":35339:5o0:1002::1002:51,0::::35328:35397,32:230,64:6:35339:"@R@":3:4:I121:36250I,0::I152:36298I,0::(4)"RUNALGEBRA 4"      0>I">p""""" *&""""""!"0`",,<>"""">@ppPpppp99c]~  6Ac>|>"" >`p`xx>0",,<>""""> R>>"@ """6&""""""!"" 0" 6"<& * Acx"" >``@xp0   <xA06 >>8>>>><""""<>""!"">>> 8 6 `>`@x`>> 8 D "LU :F`F`$L"%e%`$e($`%80%`$80$`'$L:{|0L_`F)׭F F)L i)`) qp`<) Ji L? Ji L? 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It is@D6H@called the@D6H@Y-axis.":99:36:"@16V2H@<3> They "Y$"@D6H@at the origin.":D17:"@I14V29H@ ";:37::W$:7k OV10:W269:U35:X155:98:"@5V2H@"D$" to the right@D2H@of the Y axis are@Dnd "U$" of P make@2D2H@up the ";:B$K$"s":34:400ILN77,79,81,83,43'MV10:W269:U35:X155:98:"@5V2H@To draw a "K$" plane:@2D2H@<1> Draw a "H$" "N$".@D6H@It is called the X-axis.":H30:O11:101:36:"@9V2H@<2> Draw a vertical "N$".@D6H@"Y$"ing :99:101:"@9V20H@";:B$S$:2:34:".":U95111:94,U:38::37J1:"@12V20H@The point where it@2D20H@meets the Y-axis@2D20H@is called the@2D20H@";:B$U$:34:".":V92711:V,91:38::3:73K99:101:V10:W129:U75:X155:98:"@5V2H@The "S$" a"@11V30H@The axes@2D20H@divide the plane@2D20H@into four regions@2D20H@called ";:B$"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@4D13H@$P":V10:W129:U75:X155:98he@D20H@"K$" plane.":7GV10: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$"origin":34:"@2D10H@"(30)"@U11H@ORIGIN@27HU@.@I14V9H@";:E17H" @BI@";:37::ph a "E$", one needs a@D2H@"H$" "N$".":V10:W129:U75:X155:98:H10:O11:6:101:36:600F".":36:3:"@20H10V@The vertical @4D20H@";:34:"@20H@Y":36:"@5U20H@Together they are@3D19H@"8)"@D19H@"9)"@D19H@"9)"@4U20H@the "K$"@D20H@axes and form t1)NNA(2)FLAG1:"@18V3H@RIGHT":HH1,(HH1)1:18:T12:B18:L2:R37:35:64?"@18V3H@WRONG. X = "OO" AND Y = "NN:FLAG0:18:(HH)1002:31:1000:35@:31:(HH)89:43B:31:43CM68,76,85,89DN69,71,73,75,43[E"@5V2H@To gra followed by the "E$" @D2H@for Y."30)W$:V10:W269:U35:X83:98:U91:X155:98=FLAG0:N195((HH)0):30:59:FLAG60:"@12V3H@If ("Y"X+"EE","Z"Y+"B") = ("F","D")@2D3H@Then X =":H13:O15:19:A(1)A:"@16V8H@Y = ":H13:O17:19:A(2)AP>OOA(:N14:30:59:1000:35::31:43u;YB(8)1:ZB(8)1:EEB(9):BB(9):OOB(9):NNB(9):FOOYEE:DNNZB:P<"@5V2HI@Find the values for X and Y that @D2H@will make the two "P$"s @D2H@given below equal. Then enter the @D2H@"E$" for X:X155:98:"@5V2H@In working with an "P$",@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$"s X,Y,W, and Z:@2D2H@(X,Y)=(W,Z) If and only if X=W, Y=Z.":7%";:I13:" @UB@";::"@17V20H10C@";:I15:" @UB@";6:"@10V138C@":I17:"@22H@ "::"@8V133C@":I19:"@24H@ "::"@15CI18V28H@1 2 3 4 5":197,72197,143259,143:198,72198,144:200,142256,75:77N56,438V10:W269:U35:X107:98:U116ate "P$"s in a@2D2H@"E$" of different ways.@19V5H@TABLE@15H@BAR GRAPH@28H@LINE GRAPH":V18:W87:U69:X139:98:U91:98:V52:98:"@10U4H@x f(x)@D@"65I15:"@4H@"I"@9H@"I21::I19:"@14H@"I"@12F@"I"@U@";::"@18V16H@1 2 3 4 5@I17V16H@ @17V18H5C@2H@in an ordered@D22H@pair is called@D22H@the @I@first@I2D22H@";3B$R$:34:" or@2D22H@";:B$K$".":34:"@6UI@";:I15:"@17H@"I::36:"@I9V@":I15:"@17H@"I"@I19H@"I21W$::"@I9V26H@second@I@ "E$"@I26H12V@second"W$:74"@5V2H@We may illustrthe domain is "(123)"1,2,3,4,5"(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$"s.":34:96:"@6U@":I15:"@16H@("I","2I1")"::36:"@9V22H@The @I@first@I@ "E$"@D2Gİ32:39,MG:N1:30:31+-C267>.M47,55,58,61Q/N48,50,52,43&0"@5V2H@Look at this ";:B$O$:34:" of@2D2H@five pairs of "E$"s:@3D16H@This is the function@D16H@f(x)=2x-1 for the@D16H@integers one through@D16H@five.@3D2H@Notice that RSE VARIATION@D2H@JOINT VARIATION@7V23H@";:D14:D(D)(HH8D)::1220y"@8V2H@GRAPHIC METHOD@D2H@ADD. AND SUB.@D2H@SUBSTITUTION@10V23H@";:D13:D(D)(HH12D):1zJ0:"@8V23H@";:K1D1:"@23H@"D(K)"@6F@"4D(K)::"@I138K8V34H@";:K1D1:D(K)EL. AND FUNC.@D2H@OPEN SENTENCES@7V23H@";:D14:D(D)(HHD)::122/w"@8V2H@GRAPHING EQUATIONS@D2H@SLOPE OF A LINE@D2H@SLOPE-INTERCEPT@D2H@EQUATION OF A LINE@D2H@";:D14:D(D)(HH4D)::122G0x"@8V2H@RATIO AND PROP.@D2H@DIRECT VARIATION@D2H@INVEU37:W268:X145:98:V11:W269:98:10,49228,49:10,110228,110:144,49144,110:143,49143,110:186,49186,110:13,39265,39:143:"@I5H1V@0@I5V8H@CONCEPT@21H@RIGHT WRONG"W$:(HH)118,119,120,121,2038,139D/v"@8V2H@ORDERED PAIRS@D2H@COORDINATES@D2H@RV)2V1:(24576)1039:(HH)0117::135:128:111:132`-qC0:109: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$"RUNAM4."G.uD16:A(D)0::135:V10:TEST,57,61,INTRODUCTION TO GRAPHING,58,77,LINEAR EQUATIONS,23,101,VARIATION,86,101,SOLVING SYSTEMS,86,117,INEQUALITIES,58,141,POSTTEST,mG(QQ):11:G128109:,n3:G$"RUN AM4.1.1",oD16:CA(D):VB(D):UC(D):131::;-pB(V)((1)V)1:C(143:219,U250,U:UU1::3:o+iH:U56:I133:209,U175,U:UU1::U94:I143:209,U175,U:UU1::3:+jH:I133:206,U175,U:219,U254,U:UUF::3:,kORDERED PAIRS,COORDINATES IN A PLANE,RELATIONS AND FUNCTIONS,OPEN SENTENCES,GRAPHING :H7:O4:"<@14F@>":(H8)72,(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:IV,UW,UW,XV,XV,U:V1,UV1,X:W1,UW1,X:)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@";:q*f4:" ";:$":37:E(]T10:B18:R23:L2:18:35:"@10V25H@"13)"@8D28H@"9):f(^31152,C:C5ĹHH,1:500:61o(_110(`V17:W45:U75:X123:98:V74:W101:98:"@9V@":I15:"@4H@"I"@2F@-->@2F@"2I1::(a31,12031,132:60,3660,154:59,3659,154:/)bt Point":E195((36251)0):N1E1:30:91::31:(36251)110:43'[YB(7)4:A1B(7)4:"@10V25H@PLOT ("Y","A1")":99:101:22:SYQA1ĺ"@I18V28H@RIGHT"W$:HH2,(HH2)1:93(\"@I18V28H@WRONG"W$"@"92S"H"14Q"V@x":R18:"@I"92Y"H"14A1"V@75:X155:98:H10:O11:"@I5V2H@Use the keys listed below to move @D2H@the point to the correct position of@D2H@the "P$" given below. Then @D2H@plot the point."21)"@I12V25H@U:Move Up"D'Z"@25H@D:Move Down@D25H@L:Move Left@D25H@R:Move Right@D25H@P:Plohe graph of@D2H@"A1" on the Y-axis@D2H@and draw a "H$"@D2H@line through it."%X6:162,99A18263,99A18:3:36:"@18V2H@Find where they "Y$".":F17:"@I"29Y2"H"12A1"V@ ";:37::T5:B6:L2:R38:18:35:T8:B18:35&YV10:W269:U35:X75:98:U)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 through it.":5L%W207Y14,68207Y14,133:3:36:"@13V2H@Find tthe "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$" plane.":7#UV10:W269:U35:X59:98:X155:98:N14:30:86::31:43$VYB(7the 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> There is exactly one point in@D6H@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@$":195,115210,115:36"R"@5U2H@The distance of a point@D2H@from YNEAR EQUATIONS@D2H@OPEN SENTENCES@10V23H@";:D13:D(D)(36266D)::122='(HH)6ĺ"@15V2H@YOU HAVE PASSED ALGEBRA 4"::126='=(222)255Ħ=3:::"++ERROR++ "(222)" AT LINE #"(218)(219)256" IN ALGEBRA 4":")@5D11H@"Y"X+"EE" = "F" AND "Z"Y+"B" = "D"@2D11H@"Y"X = "FEE"@25H@"Z"Y = "DB"@15V12H@X@16H@= "(FEE)Y"@26H@Y@30H@= "(DB)Z:18:T4:B19:L1:R35:<T12:B18:L2:R37:35:64@15H@<0>@8V9H@<1>@2D9H@<2>@3D4H@<3>@13H@<4>@2D13H@<5>@3D9H@<6>@8H2V@START@5V8H@MENU@6H@<@13H@>@9H3V@"(23)(24)"@10H9V@"Z$"@3D5H@"Z$"@14H@"Z$"@2D14H@"Z$"@3D10H@"Z$:6137:G$"READAM4.PROGRESS":D16:A(D)::1387137:G$"WRITEAM4.PROGR5570,60:71,5571,60569,3269,28:45,4349,43:91,4393,43:38,95101,95:38,9538,97:101,95101,97:73,9073,95:72,9072,95:38,135101,135101,130:38,13538,113:39,13539,113:73,13673,140:105,36126,36133,44126,52105,5298,44105,36:65> INEQUALITIES@D20H@<6> POSTTEST@D20H@<9> RESET MENU@D20H@<0> STOP":134w4C:WV29:XU12:98:VV1:WW1:98:3:556,1191,1198,1991,2756,2749,1956,11:49,4370,3291,4370,5649,43:9,3638,3638,489,489,36:37,3637,48:10,3610,48:70,@25HD@VOLUME "(96)"4@20HD@VER 1.1 01 MAY 84@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"I4140,7272,7:"@24H@EQUATIONS@D20H@<:3:G$"RUN AM4."L1N2D130:HH1D,0::GL:142:L139:3:G$"RUNAM4."L23:V3:U5:W276:X186:98:V4:W275:98:136,5136,186:136,34276,34:136,157276,157:137,5137,186:W$:D13:D1:"@20H@"19):D20:"@20H@"19):3"@26H1V@ALGEBRA3ĺ"@G@QR@R@";:JJ1n1{L((HH)):"@D34H@";::"@128K@":Jĺ"@14V2H@ARROWS SHOW AREAS OF WEAKNESS."W$:1251|800:10000:"@15V2H@YOU HAVE PASSED UNIT "(HH)" AND MAY @D2H@NOW GO ON TO "DE$1}JA(L)2(J1)32~136:18:140:HH,0:JGL1:142of the relation is a rule@D2H@to find the second ";H"element that must@2HD@be paired with the first element of@2HD@every ordered pair.@3D2H@A function is a relation in which@2HD@every member of the domain is paired@2HD@with one and only one member of 9H@>@13V4H@1@DB@2@DB@3@DB@4@13V13H@3@DB@5@DB@7@DB@9":49,10767,107xE49,11567,115:49,12367,123:67,13149,131:13FP71,553GX10:I269:Y35:J60:130:Y68:J108:130:129:"@5V2H@A set of ordered pairs is called a@D2H@relation.@3D2H@The rule ction":48:" is a special kind of@2D2H@relation.@2D4H@"(123)"(1,3),(2,5),(3,7),(4,9)"(125)"@11V3H@D@8F@R":128CD"@12V18H@Each element of the@D18H@domain is paired@D18H@with one and only@D18H@one element of the@D18H@range.@13V9H@>@14V9H@>@15V9H@>@16V@-1@D13H@1@D12H@-2@D13H@2"B128:"@13V9H@>@D9H@>@D9H@>@D9H@>":65,10749,11565,115:65,12349,13165,131:"@12V18H@Notice how each@D18H@number in the domain@D18H@is paired with one@D18H@or more numbers in@D18H@the range.":13qC"@5V2H@A ";:E$"fun@BD143C@5,@10V18H@";:X7:S8:7:"@FU@7":52:"@128C17V19H@ @BD143C@7":13A"@2H5V@The listing of ordered pairs is@2D2H@called a ";:E$"roster":48:" of the relation:@4H9V@"(123)"(1,1),(1,-1),(4,2),(4,-2)"(125)"@3H11V@D@3D4H@1@2D4H@4@12H11V@R@2D12HCDB@"X:"@22H16V@The set of@D22H@second elements@D22H@is the range.@6H18V@RANGE : ";J@(123)"@20H@"(125):"@6H10V@";:X3:S3:7:D5:9:S5:7:D4:"@U@";:9:"@15H17V128C@3@143CDB@3,@10V12H@";:S5:X5:7:"@14V12H@D@BD@";:S2:7:D5:9:"@128C17V17H@ >"@22H@ordered pairs is@D22H@the domain.@2H14V@"(123)"@8H@"(125)" : DOMAIN@10V4H@";:X1:S4:7:"@128C@"X"@2B143CU@"X;:52:"@128CB@ @143CDB@"X",@10V10H@";:X2:7T?"@U@";:E5:8:"@128C@ @143CDB@"X",@10V16H@";:X3:7:"@U@";:E9:8:"@128C@ @143,10:30009MK:P1:44:45*:C3101=;M60,70,73,84P<P61,65,67,55="@2H5V@A ";:E$"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"(31152):M0: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$:C10:D14:10:KĹ24576,10:46:3000 8KĹ24576 +$ ,"@2H1V@"C"@5H@"P"@2HD@"M:3 -31051:47> .30976[ /"@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:K14140 $K136SSS15 %K149SD11SS1u &KKG:(K47K58)K45ĖHS:(K);:256S,K:SS1:SD140} '33 (D$"":I0S1:D$D$((256I))::F1(D$):F10S032:H:" ";:H:D$: )K(W):KG41 *19:KKG:K89K7841 35399,1:"@22V6H@Press SPACE BAR to Continue":35399,0:Z,0:19V K(W):KG29j Z,0:K16029 "@I22V1H@"36)B$: V:H:D11);:S0:I0D1:256I,32::Z,0 !HS:B$((256S))B$; "K(W):19:KG34 #Z,0:HS:((256S));Last Page"B$@ 4000:Z,0:KKG:K21K814:K21PP1:45Z K8PP1:46:P55f 44:58 3:(W)155((36251))ı Z,0:"@40X40YN@"; K(W):KG21 K155K205ı Z,0:25 :24 135 46:K1553000:55CB@";: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 %27903:24577:135yZ16368: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"@               @replacement set @D7H@for X. Then find@D7H@the resulting Y@7HD@values.@10V26H@X Y =("K1"+"L1"X)@11V26H@";:I14:I6)K1L1I" @D26H@";:'sX179:I192:Y76:J124:130:I263:130:"@17V3H@<3> The solution set is,@D8H@"(123);:I14:"("I","K1L1I"),"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"@10V3H@<2> Replace X by each@D7H@member of the @D7HH@are solutions of the sentence, we@D2H@say that the solution set is the@D2H@empty set, "(16)".":13=%nP455:Y38:J66:X10:I270:130:J156:130:126:"@5V2H@Find the solution set of "H1"Y-"I1"X="J1"@D2H@when the replacement set for X is@D2H@"(123)" open@D2H@sentence in two variables is a@D2H@relation whose domain is the first@D2H@set of elements and whose range is"e$m"@9V2H@the second set of elements in the@D2H@ordered pairs satisfying the@D2H@sentence.@3D2H@If there are no ordered pairs which@D2tence. ";"j"The set of all solutions@2H2D@of the open sentence is called the@2D2H@";:E$"solution set":48:" over the replacement@2H2D@set of variables.":13"kP108,55w#lX10:I270:Y38:J100:130:Y108:J148:130:"@5V2H@The solution set of anyt 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@2D2H@and is said to ";:E$"satisfy":48:" the@2D2H@sen10,116 fP103,105,55 g"@2H5V@Look at this equation: 3X+4Y=25.@2H2D@We call equations or inequalities@2D2H@that involve two variables ";:E$"open":48:"@2H2D@";:E$"sentences in two variables":48:". To"S!h"@13V2H@solve such equations, you musA1]J,I:O0:A1K78O10^A0K89O1N_O1ĺ"@18V21H@RIGHT":97c`"@18V21H@WRONG"aU9O1Ĺ36254,(36254)1b28:T15:B18:L2:R19:49:T5:B18:L21:R38:49::45:(B1)6İ4:C$"RUNAM4.2"c(B1)116d55 eM102,107,1 )":V16:H6((B(1))):D13:32:U0:I172:B(I)B(1)F1B(I1)U1tZ:200:U1S1ĺ"@16V8H@RIGHT":U9:92["@8H16V@WRONG"\A0:"@17V2H@IS THE RELATION A@D2H@FUNCTION(Y/N)? "B$" "B$;:41:"@B@"(K):1000:I172:JI72:B(I)B(J)IJ9:"@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)"HY"@2H@elements of the @D2H@range."11)B$:I172:"@"29B(I)3"H"12B(I1)2"V@$"::"@15V3H@("B(1)", )("B(3)", "=S::"@17V2H@This relation is@D2H@"E$"a function.":13RTG19:(B1)G14bUP1G1:44V124:I172:B(I)0B(I1)0I8::86W:I172:KI272:B(I)B(K)B(I1)B(K1)I9:K9:K,I:86XK,I:131:X10:I136:Y35:J107:130:12@vertical line test@D2H@for the relation."Q1000:50:"@9V2H@If no vertical@D2H@line intersects@D2H@the graph of the@D2H@relation in more@D2H@";R"than one point,@D2H@the relation is a@D2H@function.":50:E$"":I172:KI272:B(I)B(K)E$"not1)))"H@"B(2)"@"4((B(3)))"F@"B(4)"@D"5((B(5)))"H@"B(6)"@"4((B(7)))"F@"B(8)B$:I282:"@29H"12B(I)2"V@$":*P"@18V2H@and the range on@D2H@the Y-axis.":28:I282:"@29H"12B(I)2"V@ ";::131:T4:B19:L2:R20:49:"@5V2H@Now we use the@D2HX-axis":I172:"@"29B(I)3"H12V@$":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)"@";uO8(A$(5))"H9V@"B(7)"@I8V"5((B(("B(3)","B(4)"),@D3H@("B(5)","B(6)"),("B(7)","B(8)")"(125)".":50:131:"@11V2H@The graph of the@D2H@relation looks";(M"@D2H@like this:":I172:"@133C"29B(I)3"H"12B(I1)2"V@$@15C@"::50:"@15V2H@The domain@D2H@elements are shown@D2H@along the the@D2H@range.":13IP455LJ124:I172:B(I)0B(I1)0I8::74K:K182:A$(K)(B(K))(B(K1))::S182:LS82:A$(S)A$(L)LSS9:L9:L,S:74vLL,S:"@5V2H@Look at this set@D2H@of ordered pairs:@2D2H@"(123)"("B(1)","B(2)"),MH:P1:21:22 #C36,66,683$M37,48,53,59L%P38,39,43,44,46,33@&"@5V2H@Every root of the "G$" X+Y=3 is@2D2H@an "H$" of "L$"s (X,Y). If@2H2D@the set "(6)" (real "L$"s) is the@2D2H@replacement set for X and Y, then we@2D2H@can find an infi2031,132:60,3660,154:59,3659,154:91:22:C2C4C566!M0:P0:21:"@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$:I4:6:Hİ23:30")"@2DB@";::"@6V3H@";:I15:"<"I">@2D3B@";::D19:R42:Q46:T56:I15:88:QQ16:TT16::"@10H16V@<0> Return To ALGEBRA Menu@2V7HI@"31)"@11H22V@WHICH (0-5) ?? "F$= "@16V4H@"(31)"@D3H@<0>":21,13242,13249,14042,14821,14814,14021,132:31,1(E)L:L L$"number":35339:I18:B$(I)::I15:C$(I)::(N)0104 P0:M0:C0:21:"@20H5V@CONTENTS@6V@":I12:"@10H@<"I">@2D3B@";::"@U@<3>@2D3B@";:I45:"<"I">@2D3B@";::"@7V14H@";:I18:B$(I)"@D14H@"; :"@7V4H@";:I14:(22(E):11:HL20:J,0:H16020:"@I22V1H@"36)F$:U "@2H1V@"C"@5H@"P"@2HD@"M:d 31051:24o 30976 "@21V1HLI@"19)"@RI@": Q(37)82:D(36)72:RD4(A$)7:TQ12:A$" ";:88 11:I175:28:: I1150:11:: 11:H H8PP1:23:P33# 21:35? (E)155((36251))ıV J,0:"@40X40YN@";z H(E):HL13:H155H205ı J,0:16 :15 97 23:H155İ3:E$"RUN ALGEBRA 4" 33 "@22V6HI@Press SPACE BAR to Continue"F$:J,05 H:G0ĺ(12F)F)"@B@";6 (20I)"@B@"(O)"@ER@";:f 11:H(E):HL6:J,0:HH176:H0HI6: "@3H21VI@Press "(1)" Key to View the Next Page@D3H@Press "(2)" Key to View the Last Page"F$ 11:H(E)L:H21H88:J,0:H21PP1:22 =. 9724577:27903:A(D)((1)D)1:D$"$":E$(4):E16384:J16368:F$"@I@":L128:G$"equation":H$"ordered pair":I$"coordinate":J$"Y-intercept":N36251:K$"straight line":290:1002: "@G15C0KE@";:F(G):I(F10):OFI10:FF9               ((B(3))):32:S0:I172:B(I)B(3)F1B(I1)S1<-:-I172:206B(I)21,40206B(I)21,150:207B(I)21,40207B(I)21,150::- 4:C$"RUNALGEBRA 4"-19:K(W):KG4000:N0ĺ"@3F@";:.,5:" ";::"@13V37H@X":,ORDERED PAIRS,COORDINATES IN A PLANE,RELATIONS AND FUNCTIONS,OPEN SENTENCES,GRAPHING TEST,(222)255Ħ,::4:"++ERROR++ "(222)" AT LINE # "(218)(219)256" IN AM4.1.1":4-H12((B(1)))155:130:"@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:N0ĺ"@2D@";:,5:"@B2D@";::"@13V23H@";:N22:3:B(4)A(6)3:B(6)A(6)3:B(8)A(6)3:B(1)B(3)124:w*~H1A(7)1:L1A(7)1:K1A(8):I1H1L1:J1H1K1:M1A(8)*I1Q:53::*X10:Y100:I45:J139:130:X73:I108:130*Y116:J156:130*X,YI,YI,JX,JX,Y:+X143:I269:Y35:J"@18V29H@";:F1Sĺ"RIGHT":B14,(B14)1:121U)x"@8H18V@WRONG, THE ANSWER IS "S)y28:I16:3:13I:36):I,P:(36251)1İ46:3000)z(B1)6İ45:4:C$"RUNAM4.2"){45:559*|B(1)A(5)3:B(3)A(5)3:B(5)A(5)3:B(7)A(5)3:B(2)A(6)irst component is the replacement@D2H@for X. The second component must be@D2H@the corresponding replacement for Y."B$:P1G1:44:1260)w"@L13V8H@"H1"Y-"I1"X="J1"@R16V2H@COMPLETE THE ORDERED PAIR: ("M1", )@4B@";:V17:H32((M1)):D13:32:SK1L1M1:;::"@B@"(125):13'tG195((B1)0):X10:I269:Y37:J89:130:Y100:J156:130:13,39265,39:B$:I16:3:5I:36)::"@5V2H@Below is an open sentence in X and"(u"@2H@Y. Complete the missing member in@D2H@the ordered pair beneath it. The@D2H@f graph of@D2H@Y=2X. It has a slope@D2H@of 2 and passes@D2H@through the origin.":245,40182,155:"@10V33H@Y=2X":27; G"@10V2H@Now look at the@D2H@graph of Y=2X+2. The@D2H@two lines have the@D2H@same slope but cross@D2H@the Y-axis at@D2H@different pointsI:3:18)::I113:5I:23:16):::22:C1933TA(N)2İ3:E$"RUNAM4.2.1"B31152,C:C5ĹN,2:I14:N41,0::36272,0:59C3:E$"RUNAM4.2.1"DM69,77,101,104EP70,74,75,33sF83:D157:R269:Q36:T156:88:"@5V2H@This is the21:86S=95:U161:"@16V2H@GRAPH "V"Y+("K"X)="W"@D2H@PLOT POINT 1";:D10:121>"@13H17V@2";:121:VE1KF1WVG1KH1Wĺ"@18V2H@RIGHT":N5,(N5)1:64?"@18V2H@WRONG"5@I1WVK2V:J1WVK2V:164,91I18249,91J18:19:I13:164:88:Q124:T156:88:Q36:D151:R270:88:F$:I19:5I:3:18):<"@5V2H@Graph the line of@D2H@the "G$" given@D2H@below by plotting@D2H@any two points@D2H@along the line.@2HD@Use the keys:@2H2D@U-UP D-DOWN L-LEFT@D2H@R-RIGHT P-PLOT"F$:P1C1:@with the three@D6H@solutions.@19H7V@(-1,"X")(0,"Y")(1,"Z")":85:15X:"@26H@"D$:26:15Y:"@29H@$":26:15Z:"@32H@"D$:26:"@13V2H@<4> Draw a@D6H@"K$"@D6H@through the@D6H@points which@D6H@were plotted.":94E;C195((N)0):D10:Q38:R144:T11@13V20H@X@3F@"B"-";8XBA1:YB:ZBA1:"("A")X@13V36H@Y@15V20H@";:I11:I"@23H@"B"-("A"("I"))@D20H@";::"@15V35H@"X"@D35H@"Y"@D35H@"Z:19:I112:3:7I:36)::A1BA2:B1BA2e9"@7V2H@<3> Plot the@D6H@points which@D6H@are associated@D6Hne member.@24H7V@"K"Y="W"-("V")X @D25H@Y="B"-("A")X":267"@4D2H@<2> Find three@D6H@solutions of@D6H@the "G$"@D6H@by picking@D6H@values for X.@24H7V@"K"Y="W"-("V")X @D25H@Y="B"-("A")X":D136:R261:Q99:T147:88:R157:D241:88:D136:R261:T116:88:"G$" has as its "I$"s@D6H@an "H$" of "L$"s which@D6H@satisfies the "G$".":75P433:D10:R270:Q36:T156:88:T52:88:95:U15386"@5V2H@GRAPH THE EQUATION "V"X+"K"Y="W".@7V2H@<1> Transform the@D6H@"G$" into@D6H@one which has@D6H@Y alone as@D6H@o$" in two@2HD@variables, X AND Y.":73D10:R270:Q36:T123:88:"@5V2H@<1> Every "H$" of "L$"s@D6H@which satisfies a linear@D6H@"G$" represents the@D6H@"I$"s of a point on the@D6H@graph of the "G$".@2H2D@<2> Every point on the graph of an"K4"@6H@"32:88:"@5V2H@The graph of any "G$" equivalent@2HD@to one of the form@2D4HL@AX + BY = C,@R28H9V@X"(19)(6)", Y"(19)(6)","%2"@11V2H@where A, B, and C are real "L$"s@D2H@with A and B not both zero is a@2HD@"K$". Such an "G$" is@D2H@called a linear "Gut it is good@2D2H@practice to plot a@2D2H@third point as a@2H2D@check on the first@2H2D@two.":D151:R270:Q36:T156:88/83:"@36H6V@"D$:26:"@10V32H@"D$:26:265,40168,152:26:"@34H8V@"D$:265,40168,152:70P49,51,33}1D10:R270:Q36:T1re A, B,@2H2D@and C are real "L$"s, with A and B@2H2D@not both zero. The graph of the@2H2D@"G$" will be a "K$", so"-"@D2H@it is called a ";:A$"linear "G$:25:".":7y."@5V2H@You need plot only@2H2D@two points to graph@2H2D@a linear "G$",@2H2D@b line is the set of all@2H2D@those points whose "I$"s@2H2D@satisfy the "G$". The line is@2H2D@called the ";:A$"graph of the "G$:25:".":7s,"@5V2H@A first degree "G$" in two@2D2H@variables, like X+Y=3, can be written@2H2D@in the form AX+BY+C=0, wheD$" @5V27HL@ @R27H@-1@DB@4"*26:"@31H5V@0@DB@3@I34H5V@1@DB@2@I13V31H@"D$:26:"@34H5V@1@DB@2@I37H5V@2@DB@1@33H14VI@"D$:26:"@37H5V@2@DB@1":168,79259,131:7+"@5V2H@Each root of the "G$" does give@2D2H@"I$"s of a point along this@2H2D@line. Thisq(D207:R228:88:D249:88:81:"@5V22H@X@1F@-2@F@-1";:I02:"@2F@"I;::"@6V22H@Y@14F@";:I15:I"@4B@";:)"@I5V24HL@ @R24H@-2@D25H@5@25H10VI@"D$:26:"@5V24HL@ @R24H@-2@D25H@5@27H5VLI@ @R27H@-1@D28H@4@I11V27H@"D$:26:"@5V31HI@0@DB@3@I29H12V@"nite "L$" of@2H2D@"H$"s that are solutions of":"@D2H@X+Y=3.":7'"@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@"K$".":D151:Q36:R270:T60:88:D165:R186:880:R150:Q38:T121:88:Q131:T156:88:D269:Q38:T156:8850i"@I5V2H@Below you are given@D2H@a linear "G$". @D2H@From its slope @D2H@and "J$", @D2H@graph two points on@D2H@the line. Use the @D2H@following keys. @D2H@"19)"@D2H@U-UP D-DOWN L-t on@D5H@the Y-axis whose@D5H@ordinate is the@D5H@"J$".@"11L1"V29H@$":27.g"@13V2H@<2>Use the slope to@D5H@find two other@D5H@points.":"@"11L1K1"V32H@$@"11(K1L1)"V26H@$":112:"@16V2H@<3>Draw a line@D5H@through these@D5H@three points.":7;/hD1T LINE # "(218)(219)256" IN AM4.2":G-c(N)6İ3:E$"RUNAM4.3.1"O-d33[-eP433J.f111:D150:R269:Q36:T156:88:D10:R150:88:T67:88:86:"@5V2H@Draw a line with a@D2H@slope of "K11" and a@D2H@"J$" of "L1".":27:"@9V2H@<1>Graph a poin5:D,QD,Q8:DD21::D202:Q83:I18:D,QD7,Q:QQ8::_,^164,115A18248,115B18:7,_KA(3)1A(2):AA(3)1A(2):BA(4)1A(2):VAK:WBK:U0:I11:BAI3BAI4U1,`:,a(222)255Ħ(-b::3:"++ERROR++ "(222)" AGRAPHING LINEAR EQUATIONS,SLOPE OF A LINE,SLOPE-INTERCEPT FORM,FINDING THE EQUATION OF A LINE,LINEAR EQUATIONS TEST +[I5:6:CH:21:24(C$(H))2:"@2VI@"C$(H)F$:Cİ23:3:E$"RUNALGEBRA 4"+\=,]"@8F@";:G1:4:G2:"@2F@";:4:D164:Q111:I1:Q51:I111:D,QD7,Q:QQ8::W*XD,QR,QR,TD,TD,Q:D1,QD1,T:R1,QR1,T:*Y GRAPHING LINEAR EQUATIONS,IN TWO VARIABLES,SLOPE OF A LINE,SLOPE INTERCEPT FORM OF,A LINEAR EQUATION,FINDING THE EQUATION OF,A LINE,LINEAR EQUATIONS TESTv+Z 1:4:" @30H@ ";:G1:4:" ";:G2:4:"@10V27H@";:G15:4:"@UB@";::"@16V27H@";:G52:4:"@UB@";::"@11V22H@<@37H@>@5V29H@"(95)"@17V29H@"(126)"@5V30H@Y@12V37H@X""*W206,40206,142:155,91263,91:D164:Q88:I15:D,QD,Q8:DD21::D202:I16:D,QD7,Q:QQ16::(U"@14V21H@<@15F@>@9V29H@"(95)"@18V29H@"(126)"@9V30H@Y@4D37H@X@13V27H@";:148,115262,115:206,72206,150:G14:4:"@UB@";::"@17V27H@";:G31:4:"@UB@";::"@15V23H@";:G2:4:93)V"@12V23H@";:G2:4:" ";:GH@"(126)"@12V23H@<@13F@>":213,40213,151:162,99262,99:"@31H5V@Y@13V24H@";:G31:4:" ";::"@32H13V@";(TG13:4:" ";::"@10V28H@";:G13:4:"@2UB@";::"@28H14V@";:G1:4:"@2DB@";:G2:4:D171:Q96:I17:D,QD,Q8:DD14::D210:Q511,123269,123:"@16V21H@";:G42:4:" ";::"@16V31H@";:G14:4:" ";:&R"@27H14V@";:G15:4:"@UB@";::"@27H18V@";:G31:4:"@UB@";::D150:Q119:I19:D,QD,Q8:DD14::D203:Q83:I19:D,QD7,Q:QQ8::o'S"@30H5V@"(95)"@18V30@D2H@the X-axis, when the "J$" is@D2H@b is@2D15HL138C@Y = b@R15C13V2H@An "G$" for a line parallel to@D2H@the Y-axis, when the X-intercept is"%P"@2H@a is@2D15HL138C@X = a@R15C@":7M&Q"@9V29H@"(95)"@29H19V@"(126)"@20H15V@<@17F@>":206,73206,159:146:88:"@5V2H@For all real "L$"s m and b, the@D2H@graph in the "I$" plane of the@D2H@"G$"@2D10HL133C@Y = mX + b@R15C12V2H@is the line whose slope is m and@D2H@"J$" is b.":7%OD10:R269:Q36:T91:88:T156:Q99:88:"@5V2H@An "G$" for a line parallel to#L"@17H16V@SLOPE@D3H@Y= 2X +0@28H@Y= 2X +2@D14H@Y-INTERCEPT@16V6H@"(126)"@31H@"(126):119,12945,12945,135:154,129220,129220,135:"@18V10H@"(95)"@35H@"(95):98,14973,14973,146:175,149248,149248,146:7#MP78,79,33$ND10:R269:Q36:T11@13V19H10C@Y=2X+2@D19H@Y=2"(19)"0+2@D19H@Y=2@R15C@":7"K"@5V2H@One way to describe a line is to@2D2H@write its "G$" in the form@2D2H@Y = mX+b, where m is the slope and b@2D2H@is the "J$". This is called@2D2H@the ";:A$"slope-intercept form":25:"."228,98220,98:228,100220,100:228,66220,66:228,68220,68:228,67220,67:3:77"J"@5V2H@To find the "J$" of a line,@2D2H@substitute 0 for X in the "G$" of@2D2H@the line. Then solve the resulting@2D2H@"G$" for Y.@2D2HL138C@Y=2X@D2H@Y=2"(19)"0@D2H@Y=0.@9V23H@Y=2X+2":229,40170,152:19:I113:3:4I:20): H"@5V2H@The ordinate of the@D2H@point where a line@D2H@crosses the Y-axis@D2H@is called the line's@2D2H@";:A$J$:25:".@17V32H@INTER-@D32H@CEPT":2:228,136228,67223,67:228,99220,99Q!I MODE@10H7V@<1> DISCUSSION@9V10H@<2> RULE@2D10H@<3> EXAMPLE@10H2D@<4> SAMPLE PROBLEM@10H3D@<0> Return To CONTENTS@I11H6D@WHICH (0-4) ??"B$:J0:U4:6:Kİ34:3:A$D$0MK:P1:32:331C4892M51,66,69,823P52,54,58,60,62,474"@130CG1)NA(A1)(NA(A1))DA(A1)(DA(A1))NNA(A1):DDA(A1):41f*:W0ND0ĢV:H:"@L@-":HH2+W(N)ĢV:H:"@L@"(W):HH2,NĢV1:H:"@R@"(N):H:"---":H:(D)-"@R@":.C(31152):(K1)281/M0:P0:32:"@14H5V@LEARNING2H1V@"C"@5H@"P"@2HD@"M:' !31051:352 "30976O #"@21V1HLI@"19)"@RI@": $U(37)82:J(36)72:OJ4(F$)7:TU12:F$" ";:105 %11:E175:39:: &11:A1115:39:: '11:K(B)128: (W((N)(D))(ND):NNWD:A1110A:B$((256X))B$; K(B)128:11:K029:I,0:HX:((256X));:K1331:XXD(K8X)D(K21X(U1)D):K47K58K45ĖHX:(K);:256X,K:XXD:XUD31 28 E$"":E0X1:E$E$((256E))::Y(E$):YX27:H:" ";:H:Y: "@1H@"36)B$:s K(B)128:K021:11:I,0:K85K68K82K76K8021:DU:JJ3(K82)3(K76):J36J24 J24J36 UU(K85)(K68):U7U17 U17U7 K80ı 103:21 V:H:U);:X0:E0UD:256E,32::I,0 HX32:49 (B)155((K1))ı5 I,0:"@40X40YN@"[ K(B):K12813:K155K205ıi I,0:16s :15} 107 34:K155İ3:A$C$ 47 "@22V6HI@Press SPACE BAR to Continue"B$:I,0 K(B):11:K12820:I,0:K16020:"@I22V"(E)"@ER@";:@ 11:K(B):K1286:I,0:KK176:KJKU6: "@3H21VI@Press "(1)" Key to View the Next Page@D3H@Press "(2)" Key to View the Last Page"B$ 11:K(B)128:K08:I,0:K21K88:K21PP1:33 K8PP1:34:P47 + 10727903:24577:K136251:E110:A(E)::I16368:B16384:A$(4):B$"@I@":C$"RUNALGEBRA 4":D$"RUNAM4.2":35339:B(J)((1)J)1:460:1002:"@G15C0KE@";:F(N):A(F10):EFA10:FF9:N0ĺ(12F)F)"@B@"; (20A)"@B@             0>>>>>>>55H80D1F1H1E1G1122V5"@E"29F13"H"11E1"V@$@E@":D1D11:G1E1:H1F1\5 LINEAR6INTERCE" ">"">4|11:J,0:H85H68H82H76H80123[4}"@"29F13"H"11E1"V@ @B@";:F1E187t4~E11F10GF1:44D1ĺ"@E"293H1"H"11G1"V@$@E@"4F1F1(H82)(H76):E1E1(H85)(H68):F12F124F12F124E15E155E15E11:165,Q1185,Q1185,O1&3u3:M327l3vI22.1:K1IL15K1IL15ē206I21,91(K1IL1)8:I23wI:I22.1:K1IL15K1IL15ē206I21,91(K1IL1)83xI:3yF12:E153z"@"29F13"H"11E1"V@!"3{H(E)L:H0123.q2p0:206,M1206,O1:1:227,N1227,M1205,M1205,O1185,O1:31:12L1K12:K1:34:12L1K12:K1:M31172qP1N1K18:Q1O1K18:P1133P1491152r37:12L1K1K12:K1:249,P1249,N1227,N12sQ1133Q1491173t28:12L1K1K12:K22:K1IL16K1IL16ĺ"@"11(K1IL1)"V"29I3"HE@$@E@"1m:112:19:I114:23:5I:16):I:I12:3:17I:19):I:P:22:(N)2İ3:E$"RUN AM4.2.1"1n992oK1(A(4)1)1A(2):L1A(4(K1))1A(2):M191L18:N1M1K18:O1M1K18:LEFT@D2H@R-RIGHT P-PLOT"F$:P195((N)0):860j111:21:SL1:"@17V2H@Y="K11"X+("L1")@D2H@GRAPH FIRST POINT";:D10:121:"@8H18V@SECOND POINT";:121:E1K1F1L1G1K1H1L1ĺ"@27H18V@RIGHT";:N7,(N7)1:1080k"@27H18V@WRONG";?1lIɺ" )-(Y";:N1:4:" ) @D2H@m=@D5H@(X";:N2:4:" )-(X";:N1:4N117:" )":35,91105,91:38:H5((C1)):N7:L1:10:JB1:104:"@10V6H@ @3B@"B1:38:H13((C1))((B1))((D1)):N13:L1:10:JE1:104W O"@12H10V@ @3B@"J:38:H4:V10:A4:JC1 a line is parallel@D2H@to the Y-axis, it has@D2H@no defined slope.":7LJ171:O269:U35:T155:105:101:100:U15B11576>M"@2H5V@The slope of the line@D2H@that passes through@D2H@the points@D2H@("C1","B1") and ("D1","E1") is@2D5H@(Y";:N2:4: line @D2H@is negative.":7J5:J11:O163:U35:T65:105:3:101:J171:O269:U36:T156:105:"@5V2H@If a line is parallel@D2H@to the X-axis, the@D2H@slope is zero.":5:175,59266,59GK37:2:J10:O164:U83:T115:105:248,40248,152:3:"@11V2H@If from@D2H@left to right along a@D2H@line that is falling,@D2H@Y decreases as X@D2H@increases.":J170:O269:U35:T156:105:101:Z.2((1)8):J182:U48:J,U:E183266:S182:G48:UUZ:U140ēE,U:S,GE2,UZ:SE:GUI:38:"@11V2H@The slope of the,@D2H@Y increases as X@D2H@increases.":J171:O269:U35:T155:105:101:Z.2((1)8):J183:U140:J,U:S182:G144:E183266:UUZ:U48ēE,U:S,GE2,UZ:SE:GUG:38:"@11V2H@The slope of the line@2HD@is positive.":7H"@2H5V@As a point movesge 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.":7EP70,72,74,76,76,76,76,47F"@5V2H@As a point moves from@D2H@left to right along a@D2H@line that is rising 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";D"@D6H@the point on the right.@D2H@<3> Find the vertical change, the@D6H@chan@X";:N2:4:"@11V4H@X";:4:"-X";:N1:4:37:"@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.":3:7BP67,47CJ10:O269:U35:T156:105:"@5V2H@To find the"@8V27H@(X";:N2:4:",Y";:4:")@9V27H@$":37:"@10V23H@M":143,99192,75:37@1:193,75193,99:"@9V29H@Y";:N2:4:"@29H12V@Y";:N1:4:3:"@10V2H@=@4H9V@Y";:N2:4:"-Y";:N1:4:28,8363,83:37:2:37:146,99193,99:"@20H13V@X";AN1:4:"@27HERENCE OF Y VALUES@2D12H@DIFFERENCE OF X VALUES":84,43238,43:37:J80:O270:U59:T123:105:"@12H11V@<@24F@>@3U24H@"(95)"@24H14V@"(126):172,64171,119:"@8V25H@Y@12V37H@X"N?85,91262,91:171,64171,119:"@12V12H@(X";:N1:4:",Y";:4:") $":37:;:N1:4:"(X";:4:",Y";:4:") and P";:N2:4:"(X";:4:",Y";:4=") is the@2D2H@quotient of the differences of the Y@2H2D@values of these points divided by the@2H2D@difference of the corresponding X@2H2D@values.":7>"@5V2H@SLOPE M = @U12H@DIFFis the accompanying@2H2D@change of abscissas, the slope of a@2H2D@line is:";"@16V13H@DIFFERENCE OF ORDINATES@13H2D@DIFFERENCE OF ABSCISSAS":91,139252,139:7C<"@5V2H@In general, the slope, m, of a line@2H2D@which passes through two points@2D2H@P"H@"(20)"@12V26H10C@20@5C15V26H@80@15C31H14V@=@U34H@1@15VB@4.@R@":185,115207,115:240,115250,115:7K:"@5V2H@Because the vertical change in moving@2H2D@from one point to another is the@2H2D@difference of the ordinates, and the@2D2H@horizontal change slope is the ratio:":37:"@16V19H@";:N5:4:"@UB@";:N1:4:N0:4:"@U2B@";:N1:4:N5:4:"@U2B@";:N2:4:N0:4:"@18V9H@RUN@4U21H@RISE"7H7:V18:D0:"@L@":Q1:J1204:1J:17Q:(20):H:V:D0HH5:578ND:4:N0:4f9109::"@11V17lope":36:",@2H2D@of a hill, you must determine the@2D2H@amount of vertical ";:F$"rise":36:" for the@2D2H@amount of horizontal ";:F$"run":36:".":76"@5V2H@For example, if a hill rises 20 feet@2D2H@over 80 feet of horizontal distance,@2H2D@its3V@";:E04:8E:"F"B$E)B$::"@143CR2H17V@S@UR@L@UR@O@UR@P@UR@E@4H18V@RUN@8H16V@RISE@14V13H@SLOPE = @U@VERTICAL CHANGE@2D21H@HORIZONTAL CHANGE@2D21H@RISE@D19H@=@D21H@RUN"5147,115266,115:147,147175,147:"@5V2H@To find the steepness, or ";:F$"s15@/r"@R8H17V@WRONG, Y = ("G1")X + ("I1")":C4İ19:130:127d/s19:33:P:(36251)2İ3:A$C$l/t47/uZ91B13:R220C13:D91E13:G220D13:R,ZR1,ZR1,Z1R,Z1R,Z:G,DG1,DG1,D1G,D1G,D:'0vQR:AZ:R,ZG,D:E150:RR(C1D1)ns of the equation."11)B$:125:32:S0:C(0)0/qC(1)0:C(2)0:"@11V2H@THE EQUATION THAT PASSES THROUGH@D2H@("D1","E1") AND ("C1","B1") IS:@2D8HL@Y =@22H@X+":H15:V15:U3:27:QY:U3:"@L@":H27:V15:27:QG1YI1ĺ"@R3H18V@RIGHT":36259,(36259)1:1J:17Q:"@128CL@"(20)"@143C@":QQ1:DD2:HH4:?-nP447U-o124:125:127:7..pP195((K1)2):D2:124:"@I5V2H@Find the equation of the line that @D2H@passes through the two points given @D2H@below. Then fill in the missing @D2H@portioHL:"@143C@";::-,iJ,UO,UO,TJ,TJ,U:c,j 2,3,5,7,11,13,17,19,-2,-3,-5,-7,-11,-13,-17,-19u,k(222)255Ħ,l::3:"ERROR "(222)" AT LINE "(218)(219)256"IN AM4.2.1":3-m2:21,136133,136:1:133,136133,104:3:133,10421,136:38:1:4:"@5V30H@";:N15155:Nĺ"@2D@";:+f4:"@2D"30(N0)"H@";::J192:E15:J,89J,95:JJ14::U43:E17:218,U223,U:UU16::+gE1A:H:V:J;:38:H:"@128C@"J;:"@143C@";:VV1:"@11V3H@="::,hEHNL:H:J;:38:H:"@128C@"J;:H1B(2):USJ:D1B(5)1B(2):E1B(5)1B(2):B1UD1:C1JD1:E115B115E115B115D1C1E1B1100:++e"@5V31H@"(95)"@31H18V@"(126)"@7U25H@<@37H@>":176,91261,91:220,41220,150:"@26H12V@";:N10:4:" ";:N5:4:"@3F@";:N5:4:" ";:N104:",Y";:4:"):")c"@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";:4:")@2D2H@<3> The equation is Y=mX+b":7n*dSB(5)1B(2):JB(5)=1+b@D7H@b=1":37:"@14V2H@So, the line's":"@15V2H@Y-intercept must be 1,@2HD@which it is.":7l(aP98,47)bJ10:O269:U36:T132:105:"@2H5V@To find the equation of the line@D2H@that passes through points (X";:N1:4:",Y";:4:")@D2H@and (X";:N2: we can find@D2H@the slope of the line.@2D2H@2-(-1) 3@2D2H@1-(-2) 3@9H17V@=@3F@=@L16V15H@1@R@":14,13956,139:77,13984,139:19_(`E116:3:4E:22)::"@5V2H@Point (1,2) must@D2H@satisfy the equation@2D7H@Y=mX+b":37:"@10V2H@So, 2=m(1)+b,@D7H@2^"@32H5V@Y@BD@(1,2)@37H12V@X@15V25H@(-2,-1)":220,88220,120:221,88221,140:37:248,46190,108:"@8V2H@A line can be drawn@D2H@through any two@D2H@points.":37:192,107234,107'_234,106234,58:"@10V34H@3@14V30H@3@12V2H@Using the slope@D2H@formula,1:4:"@3F@";:N1:4:" ";:N2:4:"@5V29H@";:N3:4:"@2D29H@";:N2:4:"@2D29H@";:N1:4:"@3D29H@";%]"@3D29H@";:N2:4:"@2D29H@";:N3:4:J217:U43:E17:J,UJ7,U:UU16::J192:U88:E15:J,UJ,U8:JJ14::"@33H7V@"(3)"@13V27H@"(3)&X47$YM90,97,110,112'$ZP91,47$["@2H5V@Look at the two points@D2H@graphed on the right.":J172:O269:U35:T156:105:"@11V25H@<@37H@>@31H5V@"(95)"@18V31H@"(126):220,41220,151:221,41220,151:176,91262,91h%\"@12V27H@";:N2:4:" ";:N:U3:27:Y084:AYh#UH15:V19:27:Y085:(B1E1)(C1D1)AYĺ"@18V18H@RIGHT":K16,(K16)1:87#V"@18V2H@WRONG@2U20H@"B1E1"@2D20H@"C1D1:140,140160,140#W19:E16:3:13E:21)::E114:5E:26:13)::101::33:(K1)2İ3:A$D$$below is@D2H@graphed on the right.@D2H@Find slope m of the@D2H@line.@I5V32H@Y@7D4F@X":P1F1:32:100:117:118#TD1:"@2H13V@What is the slope of@D2H@the line that passes@D2H@through ("D1","E1") and@D2H@("C1","B1")?@D11H@M= ":98,139119,139:V17:H15@=@R@":40:7)!QC5:32:(36272)1112!RD1:F195((K1)2):36272,1:J10:O164:U37:T89:105:U99:T155:105:U37:J171:O269:105:101:B$:E16:3:5E:21)::B$:14,39160,39p"S"@5V2HI@The line containing@D2H@the pair of points@D2H@given :103:14:N7:L1:104:"@12V6H@ @3B@"J:N17:H12((C1))((B1)):L1:JD1:10:104 !PH17:V11:A4:103:14:L1:H16:N13:104:"@12V12H@ @3B@"J:38:"@11V17H@ = @U@"B1E1"@2D20H@"C1D1:133,91154,91:118:N(B1E1):D(C1D1):H9:V16:"@L15V5H9H@"B1"@32HU@= "A1".":203,99217,99:6/"@5V2H@A ";:E$J$:30:"of one "H$" to another@2D2H@(not zero) is the quotient of the@2D2H@first "H$" divided by the second.@2D2H@The two "H$"s are called the@2D2H@";:E$"terms":30:"of the "J$".":60"@5V2n you compare the "H$"s "B1A1" and@2D2H@"B1", you say that "A1B1" is "A1" times as@2D2H@large as "B1". This comparison is made@2D2H@by computing the quotient'."@3D2H@You may also say that the two "H$"s@2D2H@have a "J$" of "A1" to 1.@6U29H@"B1A1"@2D2ISCUSSION@9V10H@<2> RULE@2D10H@<3> EXAMPLE@10H2D@<4> SAMPLE PROBLEM@10H3D@<0> RETURN TO CONTENTS@I10H6D@WHICH (0-4) ??"A$:Q4:5:Kİ28:36*MK:P1:26:27+C280:M44,58,64,74,P45,47,48,50,52,54,56,41-A1A(8)1:B1A(8)2:"@5V2H@Whe:59,3259,158:60,3260,158:31,12031,129:O0:Q5:5:CK:26:24(C$(K))2:"@2VI@"C$(K)A$:Cİ28:2:B$"RUNALGEBRA 4"'27:C5Ĺ36251,3:I14:36259I,0::74(C2İ2:24576,C:B$"RUNAM4.3.1")M0:P0:26:"@14H5V@LEARNING MODE@10H7V@<1> Dn to ALGEBRA Menu@2V7HI@"31)"@11H22V@WHICH (0-5) ??"A$"@7V4H@";:I14:(22)"@2DB@";::"@6V3H@";:I15:"<"I">@2D3B@";::X19:L42:W46:B56:I15:110:WW16:BB16:z&"@16V4H@"(31):21,13242,13249,14042,14821,14814,14021,132:"@17V3H@<0>"X4(E$)7:BW12:E$" ";:110B NSF:H:N:"@"Y"K@"Z)::R 33:33:33g !L1300::10:s "32:32 #35339:I15:C$(I): $C0:26:"@20H5V@CONTENTS@6V@":I15:"@10H@<"I"> "C$(I):I4ĺ"@14H11V@VARIATION"%:"@10H14V@<0> Retur21 D$"":I0U2:K$((256I)):II(K$" "K$"0"):D$D$((256I))::V(D$):VU21:H:" ";:H:"@133C@"V"@15C@": "@2H1V@"C"@5H@"P"@2HD@"M: 31051:29 "@15C@":30976 "@21V1HLI@"19)"@RI@": W(37)82:X(36)72:L019:"@I22V1H@"36)A$:F T:H:Q1);:U0:I0Q:256I,32::J,0d HU:A$((256U));:A$; 114:KK128:K022:J,0:HU:((256U));:K1325:UU2(K21UQ)2(K8U0) K47K58ĖHU:"@133C@"(K)"@15C@";:256U,K:UU2:UQ25 8:P41 26:43- (G)155((G1))ıD J,0:"@40X40YN@";j K(G):K12812:K155K205ıx J,0:15 :14 130 28:K155İ2:B$"RUN ALGEBRA 4" 41 "@22V6HI@Press SPACE BAR to Continue"A$:J,0 114:J,0:K162F)F)"@B@";, (20A)"@B@"(B)"@ER@";:S 10:114:J,0:KK176:K0KQ5: "@3H21VI@Press "(1)" Key to View the Next Page@D3H@Press "(2)" Key to View the Last Page"A$ 10:114:KK128:J,0:K21K87:K21PP1:27 K8PP1:2,24577:27903:A(E)((1)E)1:G16384:J16368:A$"@I@":G136251:H$"number":B$"":I$"proportion":J$"ratio":L$"variable":M$"variation":N$"directly":O$"equation":350:1002: "@G15C0KE@";:F(N):A(F10):BFA10:FF9:N0ĺ(1         !!! ! ! ! ! !!!!!!!!!""" " " " " """""""""quation in@D6H@slope-intercept@D6H@form.@23H10V@Y="G1"X+"I1"@15V2H@<4> Check the@D6H@equation by@D6H@using the other@D6H@point.@15V23H@"B1"="G1"("C1")+("I1")@23HD@"B1"="J1"+("I1")@23HD@"B1"="B1:3E19:10E:3:36):E:E69:3:E:36):E:@26H12V@"C1"-("D1")@11V35H@="G1:182,91238,912"@14V2H@<2> Find the@D6H@Y-intercept by@D6H@using one of@D6H@the points.@14V22H@b ="E1"-("G1")("D1")@24H15V@="E1"-("H1")@16V24H@="I1:19:E19:3:10E:36):E:J1G1C13"@10V2H@<3> Write the@D6H@e1B(2):C1B(9)1B(2):D1B(9)1B(2):E1G1D1I1:B1G1C1I1:H1G1D1:B1E1C1D1125e1~.2"@5V2H@Find the equation of the line that@D2H@passes through the points@D2H@("D1","E1") and ("C1","B1").@11V2H@<1> Find the slope: m = @U@"B1"-("E1")3:ZZ(B1E1)3:Z36Z156E50::121D0wR172R269E50::121P0xR,Z:0yG,D:E150:QQ(C1D1)3:AA(B1E1)3:A36A156E50::0zQ172Q269E50::0{Q,A:0|J10:O269:U36:T156:105:T75:105_1}G1B(4)1B(2):I1B(9)1:26:115:"@12V2H@FIND THE MISSING NUMBER":C(1)3:C(2)12:C(3)23:C(4)31:FA(4):I14:IFĂ:77"L"@L14V"C(I)"H@"B(I):"MT15:HC(F)1:Q4:"@10H14V@:@19H@=@29H@:":20:"@R@":VB(F)ĺ"@17H17V@RIGHT":G19,(G19)1:79c#N"@2H16V@WRONG, the !JX10:L269:W35:B83:110:W92:B155:110:"@I5V2H@An "O$" of two "J$"s is given @D2H@below. One of the four "H$"s is @D2H@missing. Fill in the missing "H$" @D2H@so that the "O$" will become a ":F195((G1)0)e"K"@9V2H@true "I$"."20)A$:P1F"@5V2H@Find the "J$" of "C1" "F$" 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$ I"@20H2D@"A1" "G$"@20H2D@"B1" "G$"@31HU@= "E1:140,123210,123:6D2H@<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."G"@10V25H@"C1"$"A1" = "E1"@25H3D@"B1"$X = "E1"@25H4D@X = = "D1"@29HU@"E1"@2D29H@"B1:203,139217,139:6 H"ounces":D1A(5):E1A(5):C1E1D1:B1D116:A1C116:72E0:12,59267,59:3:10,67269,67:C1A(5):B1A(9):A1A(5)B1:E1C1A1:D1E1B1:C1B169:"@5V2H@Find the value for X that will make"F"@2H@the "O$" "C1":X = "B1":"A1" a true@D2H@"I$".@3:61@P9X10:L269:W35:B59:110:B156:110SAP66,67,68,69,69,69,69,69,41BF$"nickels":G$"pennies":D1A(5):E1A(5):C1E1D1:B1D15:A1C15:72CF$"feet":G$"inches":D1A(5):E1A(4)1:C1E1D1:B1D112:A1C112:72:DF$"pounds":G$r b nor c equal@D2H@to zero, then their "J$" is:@2D17H@a:b:c@3D2H@If a:b = c:d, with neither b nor d?"@2H@equal to zero, then a$d = b$c.@2D2H@a and d are called the extremes.@D2H@b and c are called the means.@D2H@The "O$" itself is called a@D2H@"I$"."2DB@x@32H@bx@2U2B@ax@L7H9V@=@R12H@$@L14H@1@17H@=@23HR@$@28HL@=@R@":35,7542,75:70,7577,75=147,75154,75:224,75238,75:175,75182,75:6d>X10:L269:W35:B83:110:W91:B155:110:"@5V2H@In general, if a, b, and c are real@D2H@"H$"s with neithewith@D2H@neither b nor x equal to zero, then@6D2H@and a:b = ax:bx@2D2H@To find the "J$" of two quantities@D2H@of the same kind\<"@D2H@<1> Find measures in the same unit@D2H@<2> Divide the measures@8V5H@a@2DB@b@10H@b@2UB@a@21H@a@2DB@b@21H@b@2UB@a@25H@x@:T15:D12:T15ĖX:T2:" "9X:T:"@10C@1@15C@":33:TT2::H27:T15:D17:H:T:"@10C@6@15C@":33:H:T:" ":HH2::"@12H16V@"(19)"@10C@6@15C@":119:P59,62,41~;X10:L269:W35:B107:110:B156:110:"@5V2H@If a, b, and x are "H$"s extremes":30:", and the@2D2H@second and third "H$"s the ";:E$"means":30:".":68"@5V2H@In any true "I$", the product@2D2H@of the extremes is equal to the@2D2H@product of the means.@L3D10H10C@1@15C@:@133C@2 @15C@= @133C@3@15C@:@10C@6@15C@":32:X11@2D2H@states that two "J$"s are equal:@L11H3D@1:2 = 3:6@R@":32:"@9V2H@EXTREMES":73,75196,75:83,7583,81:196,75196,81:32:"@2D18H@MEANSS7111,97111,99118,99:162,99167,99167,97:32:"@D2H@The first and fourth "H$"s are@2D2H@called the ";:E$"0:"when both terms of@2D2H@the "J$" are whole "H$"s and when@2D2H@there is no whole "H$" other than@2D2H@one which is a common divisor of the5"@D2H@terms:@L2H2D@8:4@9H@=@12H@6:3@19H@=@22H@4:2@29H@=@32HI@2:1@RI@":66"@5V2H@A "I$" is an "O$" whichnvert the nickel to 5 pennies: 3"@15V2H@NICKEL@34H@PENNY@L19H@:";:32:"@132C13H@d@130C25H@d";:32:"@9H@";:K15:"d";::32:"@143C13H17V@5"(17)"@19H@:@25H@1"(17)"@R@":64"@5V2H@A "J$" is said to be expressed in@2D2H@";:E$"simplest form":3@"C1":"B1"@5U31H@"C1"@2D31H@"B1"@3D30H@"C1","B1:217,91231,91:X10:L126:W75:B106:1122"@5V2H@But both "H$"s must be expressed@2D2H@in the same units of measure. For@2D2H@example, you can't compare a nickel@2D2H@to a penny "N$", but you can@2D2H@coH@You may write a "J$" in a "H$" of@2D2H@different ways:@3D2H@USING A@D2H@DIVISION@D2H@SIGN@2D2H@USING A@D2H@RATIO@D2H@SIGN@6U20H@AS A@D20H@FRACTION@3D20H@AS AN@D20H@ORDERED@D20H@PAIR":B1A(6)1:B14B1648Y1C1(A(8)1)B1:"@11V13H@"C1(5)B1"@4D13H(2k18:27::(G1)İ2:B$"RUNAM4.3.1"02l41o2mD1A(10):C1A(9):A1A(9):B1D1C1:E1D1A1:C1A1109:2nX,WL,WL,BX,BX,W:2oRATIO AND PROPORTION,DIRECT VARIATION,INVERSE VARIATION,JOINT AND COMBINED,VARIATION TEST,S3p110:L73:110:W,"19)"@D2H@what is the value of y when x="A1"? @D2H@Enter your answer in the space"6)"@D2H@provided."27)"@6V37H@ "A$1i"@L6D4H@Y = ";:Q4:H13:T14:20:"@R@":VE1ĺ"@17V2H@RIGHT":36261,(36261)1:1072j"@14V19H@WRONG":C2107:18:27:100A1".@5U27H@y = kx@D27H@"B1" = k("C1")@D27H@k = "D1"@2D27H@y = kx@D27H@y = ("D1")("A1")@27HD@y = "E1:M4ıq0f60gP195((G1)0):26:X10:L269:W35:B83:110:W91:B155:110:109z1h"@I5V2H@y varies directly as x."13)"@D2H@If y="B1" when x="C1"ugh the origin.":6/dX10:L269:W35:B67:110:B139:110:M3109:"@5V2H@If y varies "N$" as x, and if@2HD@y="B1" when x="C1", find the value of y@D2H@when x="A1"."j0e"@10V2H@<1> Find the constant@D6H@of "I$"ality.@3D2H@<2> Solve for y when@D6H@x=" x is multiplied by a@D6H@"H$", y will be multiplied by@D6H@the same "H$".@2D2H@<4> When x is divided by a "H$",@D6H@y will be divided by the same@D6H@"H$"."/c"@D2H@<5> The graph of y=kx is a straight@D6H@line whose slope is k and which@D6H@passes thror the "L$"s, is@D6H@equal to the "J$" of y";:N2:3:" and x";:3:","-a"@6H@any other pair of values.@16V17H@y";:N1:3:" y";:N2:3:"@2D17H@x";:N1:3:" x";:N2:3:"@20H17V@=":119,139132,139:147,139161,139:6.bB155:110:"@6V2H@<3> WhenX10:L269:W35:B155:110:"@5V5H@PRINCIPLES OF PROPORTIONALITY@2D2H@<1> The "J$" y:x is constant.@D6H@That is, y=kx, where k is the@D6H@non-zero constant of@D6H@"I$"ality."F-`"@D2H@<2> The "J$" of y";:N1:3:" and x";:3:", any pair@D6H@of values foV30H@";:N14:3:" ";::X204:W51+]I16:X,WX4,W:WW8::X215:W98:I14:X,WX,W4:XX14::"@27H5V@y@9D36H@x":34:"@31H7V@k@8D9H@and having a@2D2H@slope equal to the constant of@2D2H@"I$"ality.":214,84235,84235,54:6+^P95,98,41,_ar@2D2H@direct "L$" with "(6)"@2D2H@as the domain and range@2D2H@is a straight line@2D2H@passing through the@2D2H@origin":X185:L269:W35:B123:110$+\"@5V29H@"(95)"@37H7D@>@5V28H@";:206,41206,99262,99:206,99245,41:N1422:3:"@DB@";::"@13 = @R@0. The "L$" k is the@3D2H@";:E$"constant of "I$"ality":113)Z"@5V2H@In a direct "M$", you can say@2D2H@that y varies "N$" as x, or@2D2H@y is "N$" "I$"al to x.@3D2H@y = kx, where k is a nonzero@2D2H@constant.":6*["@5V2H@The graph of a lineD2H@"H$" y of the range and the@2D2H@corresponding "H$" x of the domain@2D2H@is the same for all ordered pairs of":"@D2H@the function except (0,0).":6D)Y"@5V2H@This definition can be rewritten in@2D2H@mathematical terms:@9V2H@y@2D2H@x@4H10V@= k , x@G@2D2H@"L$" to the corresponding value@2D2H@of the other "L$" is constant,@2D2H@we say that one "L$" ";:E$"varies":30:"@2D2H@";:E$N$:30:"with the other.":6(X"@5V2H@A ";:E$"linear direct "M$:30:"is a@2D2H@function in which the ratio between a@2C1(C11)(C12):C1"@7F@"B1"@6V5H@"C1"@18V25H@"B1"@"3C1"H16V@"C1"@"14C1"V2H@"C1"@"4C12"H"14C1"V@"C1:"@"3C1"H"14C12"V@"C1:C13ı&V34:S9:F18:H3:Y0:Z9:31:'W"@5V2H@When two "L$"s are related so@2D2H@that the "J$" of one value of one@ is a ";:E$"constant":30:".@9V2H@12 8 4@2D3H@3 2 1@U@ ="%T"@10V5H@= =@U24H@SIDE PERIMETER@2D25H@1@7F@4@2D25H@2@7F@8@2D25H@3@7F@12@L15H5U@4@R@":14,8328,83:49,8356,83:77,8384,83:6&UX21:B120:LX(C12)8:WB(C12)8:110:26:10H@of 1 on the side.@24H8V@SIDE PERIMETER@18V2H@Its perimeter would be 4 .":C11:B14:85:C12:B18:85:C13:B112:85:18C%SS5:F19:Z37:H3:31:"@5V1H@Notice that the "J$" of the perimeter@D1H@to the side is always the same.@17V1H@We say that the "J$"product of the means must@D2H@equal the product of the extremes.@D2H@The correct answer is "B(F)"."#O18:K17:3:12K:36):::27:(G1)0103:41#PM81,94,118,103#QP82,87,88,89,90,91,41y$R"@5V2H@Suppose we had a square with a length@D2