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# Introductory Algebra MTH 65

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This 14 page Class Notes was uploaded by August Feeney on Monday October 19, 2015. The Class Notes belongs to MTH 65 at Portland Community College taught by Staff in Fall. Since its upload, it has received 43 views. For similar materials see /class/224638/mth-65-portland-community-college in Math at Portland Community College.

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Date Created: 10/19/15

MTH 65 Mr Simonds class solving systems graphically Key Concepts Systems of two linear equations with two unknowns Solving systems graphically Types of solution sets to systems of linear equations MartinGave sections and practice problems 41 1 27 odd 31 43 odd Definition A solution to a system of two linear equations with two unknowns is an ordered pair that makes each of the equations true Examples Decide whether or not the given ordered pairs are solutions to the system of equations 2x 3 y 7 3x 7 2y 7 9 Is 571 0 solution to the system 3 X q 7 x 3 39 J 7 2m J I 4 1134r3l l17 5 p1 9 41E 39 N 0 Is 713 0 solution to the system 3 39 1 7 2 30quot 0 q c5 2 Page 1 of 6 1x 3 7 9L 1 1 30 7 36S cum W1 7 q 7 j rm 5 7 3 c l w MTH 65 Mr Simonds class solving systems graphically Solve each sysTem of equaTions by graphing The Two linear equaTions and finding The poinTs They have in common Make sure ThaT you check your souTion before sTaTing your conclusion 3x 7 5 Solve The sysTem y7x3 yxS39 i35 x43 m Show m 3 Il il n 39 o j 39P 5 03 CWk 513 z I szi 3 7lt 437 I zm g I 2 4 Figure1 ILo SJ Hquot 4 Ha 53449 5 CL Solve The sysTem y 3 2xy77 1quot 1 397 3 J x j I JK7 K 39 27 M J m 1 3 l L0 391 324 go 3 i Uquot Elna L Ndl FIgure2 Sl 9 mun 3 quotquot me cum sown 5 f Y2 3 v mu 2 i3 I a 3 Lam 3 LLPI I 1x 7 zca win 7 quot Page 2 of 6 fka SI J39kM 4quot 44quot Sjsh jjvl MTH 65 Mr Simonds class solving systems graphically 2x7y9 Solve The system x 3 3911 j 7 X 4 4 H 7 S I IDj q 3 X CHE LB 14 1 j q 2L3 a 7W 3 is k939 quotMm yu v 3 1 LI Soluxh 5 YL J JJ Fv 7 C 31 37 Solve The system 4x 6y 242 y7 x71 LIXLCJ3JI ks39 X l x M i J 0 Y 4L0I 0 El LDM39S Lula Parhdv rutd39l Figure4 ch We bu f 21 6 2 an 2 3 Q a x 3 f Ii 11 77c j f Jv qlt j zs b L Vxghf L quotx4 l j Page3of6 4 1 73 3 I Pgrud fr SjShM h quot0 Julv gnn5 CH amp 139 115V L 1 333 1 V MTH 65 Mr Simonds class solving systems graphically I L S 5 1 02 I 4 quot p FRAd IJ Lh bl unclailulj om 3quotquot quot5 Sm 39 y 3x 7 2 Solve The system 6x 7 2y 4 y Ijvi X is 75 74 73 72 71 l I 6391 r 2 lllj l 0y 1 ti Figures 239 L L GE il 391 rd 3x quot LI Types of solutions sets to systems of two linear equatio with two unknowns y gt4 A system can have exactly one solution In this case the system is called 6 consistent and the equations are called Independent ThIs happe when the two equations graph to lines that intersect at a single point A system can have an in nite numberquot of solutions In this case the system is called consistent and the equations are called dependent x ThIs happens when the two equatIons graph to the same lIn A system can have no solution In this case the system is called inconsistent and the equations are called independent This happens x when the two equations graph to parallel lines Page 4 of 6 MTH 65 Mr Simonds class solving systems graphically Examples For each sys rern wriTe bo rh equaTions in sope I39Ilfercepf form and decide wi rhou r graphing wheTher The sys rem is consisTenT or inconsis ren r and wheTher The equaTions are dependenT or independent qxquot 13 q y2x72 4x2y74 le 4amp1 u L L 1xL j 3 ll 1 g 39 1 ltu sn3 UMIIJ 39 39qcoaIUJHI rhplw 12 v r utl MIJ 5 fL Jjjl39tw I I NJ 4quot ltiva l n5 Art iqdz und V 2x73y8 1x 3j Y jx h Y 3x72y8 I j lx7 j Ll JltY39 f M i I 3 a 3 4 L 1 x 3 3X 3 J L 1quot 531 5 3 fr gbguue ryx l hj r u ixquot 7t A I L 39 39 til I 1 lmquot H39k ll grltn soreJ v l39 MJ39 J kc 652 f Sh L S LURIJI39quot 0 a UL lttA out 39 na z P ltq lw a ie5ofe MTH 65 Mr Simonds class solving systems graphically 4 y 7 x 2 Find the solution to the system of equations 3 y 4 x 2y 7 3 3 AlCult 39 I39 clear y L jrtpl u391 X J I jrlw 1 B Jv rk XLLHJ uh fvld 4 l4 1 ki l39 4 I Ll r Figure9 um zl Ad 56quot 9 Hm m 1 4 L 64 UJ JAlt M IW o 01quot NLL J I Lg JabN 39 Page 6 of 6 MTH 65 Mr Simonds class Key Concepts More factoring Trinomials where the leading coefficient isn t 1 Binomials some special formulas Factor problems thatjust keep on keeping on MartinGave sections and practice 63 1 67 odd 64 1 71 odd 65 1 99 odd Trinomials where the leading coefficient isn t 1 ax2 b x c a at 1 As we should when factoring any expression the first thing we should look for is a GFC other than 1 If there is such a beast yank it out Make sure that you check to see if the resultant trinomial factor is factorable Examples Factor each expression completely 4x2y 20xy24y 9t5 90m 216x2 14x2 y5 56x2 y 420x2 y3 Page 1 of 8 MTH 65 Mr Simonds class ax2bxc ail IT is frequenle The case ThaT The GCF is indeed 1 When This is The case one TacTic you can Tr39y is called guess and check Think of a pair39 of numbers ThaT mulTiply To a and a differenT pair39 of numbers ThaT mulTiply To a Wr39iTe down possible facTor39izaTions and check To see if They FOIL To The original expression Hmmmmmmmmm Use guess and check To facTor39 6x2 25x 25 Include a lisT of all The possible facTor39izaTions Use guess and check To facTor39 31 2 61 8 Include a lisT of all The possible facTor39izaTions Page 2 of 8 MTH 65 Mr Simonds class ax2bxc ail Plan b Find if iT exisTs a pair of numbers h and k whose producT is ac and whose sum is b u If such a pair exisTs rewriTe The polynomial as ax2 hx kx c and facTor by grouping If no such pair exisTs The polynomial is prime assuming of course ThaT you didn39T forgeT To sTarT The process by looking for The GCF of all of The Terms FacTor by grouping 8x2 14x 15 FacTor by grouping 36x2 48xy 15 yz FacTor by grouping 6x2 15 x 12 Page 3 of 8 MTH 65 Mr Simonds class Factor x10 25 y4 Factor 8x3 y Factor 1251 12 27x9 Page 4 of 8 Binomials some special formulas Binomials some special formulas aZ b2aba b a3 b3a bazabb2 a3b3aba2 abb2 Factor factor factor 36pZ q2 36pzq2 Page 5 of 8 125 y3 MTH 65 Mr Simonds class A special fact that you need to incorporate into your reality Unless a and b sharea common factor other than 1 a2 b2 is prime 8x3 27y3 MTH 65 Mr Simonds class Factor x2 7x 12 and x2 7xy 12y2 Factor 1 2 101 24 and x6 10x3 24 Factor x2 29x 28 and am 29615 b3 28 b6 Page 6 of 8 MTH 65 Mr Simonds class Factor problems thatjust keep on keeping on Factor factor factor factor factor factor Factor x8 y8 Factor x6 y2 y8 Factor 8x4 10x2 3 Page 7 of 8 MTH 65 Mr Simonds class A Factor Plan 1 Always begin by facToring ouT The GCF of The Terms 2 If There39s a binomial lurking abouT see if iT fiTs one of The special form a2 b2 a ba b a3 b3 a ba2 ab b2 613 b3 a ba2 abb2 3 If There39s a Trinomial hanging around If The leading coefficienT is land The Trinomial is facTorable Then 2 x bxcxhxk where hkc and hkb If The leading coefficienT isn39T 1 you could Try guess and checkquot or you could Try facToring by grouping afTer finding a pair of numbers h and k whose producT is ac and whose sum is b and wriTing39 ax2bxcax2hxkxc 4 If There39s a fourTermed polynomial in The room Try To facTor by grouping 5 In all of The above circumsTances check your answer by multiplying check your answer by multiplying check your answer by multiplying check your answer by multiplying check your answer by multiplying 6 Remember prime happens Be open To The possibiliTy buT make sure ThaT you39ve considered and checked all viable facTorizaTions before concluding ThaT The polynomial is prime Page 8 of 8

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