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|00000650| 00 20 20 20 20 20 11 31 | 34 20 20 20 20 34 28 34 |.     .1|4    4(4|
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|00000920| 20 6f 66 20 11 33 78 11 | 31 2e 20 20 54 68 75 73 | of .3x.|1.  Thus|
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|00000940| 61 6e 20 69 64 65 6e 74 | 69 74 79 2e 0d 0a 00 00 |an ident|ity.....|
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|00000c20| 6f 77 73 2e 12 30 0d 0a | 00 0d 0b 00 20 20 20 20 |ows..0..|....    |
|00000c30| 20 32 33 11 33 78 20 11 | 31 2d 20 34 20 3d 20 34 | 23.3x .|1- 4 = 4|
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|00000c60| 0a 00 20 20 20 20 20 20 | 20 20 20 11 31 32 33 11 |..      |   .123.|
|00000c70| 33 78 20 11 31 3d 20 34 | 36 20 20 20 20 20 20 20 |3x .1= 4|6       |
|00000c80| 20 20 20 11 32 12 31 41 | 64 64 20 34 20 74 6f 20 |   .2.1A|dd 4 to |
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|00000ca0| 20 20 20 20 20 20 20 20 | 20 20 11 33 78 20 11 31 |        |  .3x .1|
|00000cb0| 3d 20 32 20 20 20 20 20 | 20 20 20 20 20 20 11 32 |= 2     |      .2|
|00000cc0| 12 31 44 69 76 69 64 65 | 20 62 6f 74 68 20 73 69 |.1Divide| both si|
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|00000d10| 68 65 20 65 71 75 61 74 | 69 6f 6e 20 61 73 20 66 |he equat|ion as f|
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|00000d30| 20 20 20 20 32 33 11 33 | 78 20 11 31 2d 20 34 20 |    23.3|x .1- 4 |
|00000d40| 3d 20 34 32 20 20 20 20 | 20 20 20 20 20 20 11 32 |= 42    |      .2|
|00000d50| 12 31 47 69 76 65 6e 20 | 65 71 75 61 74 69 6f 6e |.1Given |equation|
|00000d60| 12 30 0d 0a 00 20 20 20 | 20 20 20 20 20 20 11 31 |.0...   |      .1|
|00000d70| 32 33 11 33 78 20 11 31 | 3d 20 34 36 20 20 20 20 |23.3x .1|= 46    |
|00000d80| 20 20 20 20 20 20 11 32 | 12 31 41 64 64 20 34 20 |      .2|.1Add 4 |
|00000d90| 74 6f 20 62 6f 74 68 20 | 73 69 64 65 73 12 30 0d |to both |sides.0.|
|00000da0| 0a 00 20 20 20 20 20 20 | 20 20 20 20 20 11 33 78 |..      |     .3x|
|00000db0| 20 11 31 3d 20 32 20 20 | 20 20 20 20 20 20 20 20 | .1= 2  |        |
|00000dc0| 20 11 32 12 31 44 69 76 | 69 64 65 20 62 6f 74 68 | .2.1Div|ide both|
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|00000e00| 72 6f 6e 67 2e 20 20 57 | 65 20 73 6f 6c 76 65 20 |rong.  W|e solve |
|00000e10| 74 68 65 20 65 71 75 61 | 74 69 6f 6e 20 61 73 20 |the equa|tion as |
|00000e20| 66 6f 6c 6c 6f 77 73 2e | 11 32 12 30 0d 0a 00 0d |follows.|.2.0....|
|00000e30| 0b 00 20 20 20 20 11 31 | 30 2e 33 11 33 78 20 11 |..    .1|0.3.3x .|
|00000e40| 31 2b 20 31 2e 35 20 3d | 20 38 2e 34 20 20 20 20 |1+ 1.5 =| 8.4    |
|00000e50| 20 20 20 20 20 20 20 11 | 32 12 31 47 69 76 65 6e |       .|2.1Given|
|00000e60| 20 65 71 75 61 74 69 6f | 6e 12 30 0d 0a 00 20 20 | equatio|n.0...  |
|00000e70| 20 20 20 20 20 11 33 33 | 78 20 11 31 2b 20 31 35 |     .33|x .1+ 15|
|00000e80| 20 3d 20 38 34 20 20 20 | 20 20 20 20 20 20 20 20 | = 84   |        |
|00000e90| 20 11 32 12 31 4d 75 6c | 74 69 70 6c 79 20 62 6f | .2.1Mul|tiply bo|
|00000ea0| 74 68 20 73 69 64 65 73 | 20 62 79 20 31 30 12 30 |th sides| by 10.0|
|00000eb0| 0d 0a 00 20 20 20 20 20 | 20 20 20 20 20 20 20 11 |...     |       .|
|00000ec0| 31 33 11 33 78 20 11 31 | 3d 20 36 39 20 20 20 20 |13.3x .1|= 69    |
|00000ed0| 20 20 20 20 20 20 20 20 | 11 32 12 31 53 75 62 74 |        |.2.1Subt|
|00000ee0| 72 61 63 74 20 31 35 20 | 66 72 6f 6d 20 62 6f 74 |ract 15 |from bot|
|00000ef0| 68 20 73 69 64 65 73 12 | 30 0d 0a 00 20 20 20 20 |h sides.|0...    |
|00000f00| 20 20 20 20 20 20 20 20 | 20 11 33 78 20 11 31 3d |        | .3x .1=|
|00000f10| 20 32 33 20 20 20 20 20 | 20 20 20 20 20 20 20 11 | 23     |       .|
|00000f20| 32 12 31 53 6f 6c 75 74 | 69 6f 6e 12 30 0d 0a 00 |2.1Solut|ion.0...|
|00000f30| 00 17 4f 07 0d 0b 00 12 | 30 57 72 6f 6e 67 2e 20 |..O.....|0Wrong. |
|00000f40| 20 57 65 20 73 6f 6c 76 | 65 20 74 68 65 20 65 71 | We solv|e the eq|
|00000f50| 75 61 74 69 6f 6e 20 61 | 73 20 66 6f 6c 6c 6f 77 |uation a|s follow|
|00000f60| 73 2e 11 32 12 30 0d 0a | 00 0d 0b 00 20 20 20 20 |s..2.0..|....    |
|00000f70| 11 31 30 2e 33 11 33 78 | 20 11 31 2b 20 31 2e 35 |.10.3.3x| .1+ 1.5|
|00000f80| 20 3d 20 38 2e 34 20 20 | 20 20 20 20 20 20 20 20 | = 8.4  |        |
|00000f90| 20 11 32 12 31 47 69 76 | 65 6e 20 65 71 75 61 74 | .2.1Giv|en equat|
|00000fa0| 69 6f 6e 12 30 0d 0a 00 | 20 20 20 20 20 20 20 11 |ion.0...|       .|
|00000fb0| 33 33 78 20 11 31 2b 20 | 31 35 20 3d 20 38 34 20 |33x .1+ |15 = 84 |
|00000fc0| 20 20 20 20 20 20 20 20 | 20 20 20 11 32 12 31 4d |        |   .2.1M|
|00000fd0| 75 6c 74 69 70 6c 79 20 | 62 6f 74 68 20 73 69 64 |ultiply |both sid|
|00000fe0| 65 73 20 62 79 20 31 30 | 12 30 0d 0a 00 20 20 20 |es by 10|.0...   |
|00000ff0| 20 20 20 20 20 20 20 20 | 20 11 31 33 11 33 78 20 |        | .13.3x |
|00001000| 11 31 3d 20 36 39 20 20 | 20 20 20 20 20 20 20 20 |.1= 69  |        |
|00001010| 20 20 11 32 12 31 53 75 | 62 74 72 61 63 74 20 31 |  .2.1Su|btract 1|
|00001020| 35 20 66 72 6f 6d 20 62 | 6f 74 68 20 73 69 64 65 |5 from b|oth side|
|00001030| 73 12 30 0d 0a 00 20 20 | 20 20 20 20 20 20 20 20 |s.0...  |        |
|00001040| 20 20 20 11 33 78 20 11 | 31 3d 20 32 33 20 20 20 |   .3x .|1= 23   |
|00001050| 20 20 20 20 20 20 20 20 | 20 11 32 12 31 53 6f 6c |        | .2.1Sol|
|00001060| 75 74 69 6f 6e 12 30 0d | 0a 00 00 17 4f 03 0d 0b |ution.0.|....O...|
|00001070| 00 52 69 67 68 74 2e 20 | 20 54 68 65 20 73 6f 6c |.Right. | The sol|
|00001080| 75 74 69 6f 6e 20 74 6f | 20 74 68 65 20 67 69 76 |ution to| the giv|
|00001090| 65 6e 20 65 71 75 61 74 | 69 6f 6e 20 69 73 20 11 |en equat|ion is .|
|000010a0| 33 78 20 11 31 3d 20 32 | 33 2e 0d 0a 00 00 17 4f |3x .1= 2|3......O|
|000010b0| 07 0d 0b 00 12 30 57 72 | 6f 6e 67 2e 20 20 57 65 |.....0Wr|ong.  We|
|000010c0| 20 73 6f 6c 76 65 20 74 | 68 65 20 65 71 75 61 74 | solve t|he equat|
|000010d0| 69 6f 6e 20 61 73 20 66 | 6f 6c 6c 6f 77 73 2e 11 |ion as f|ollows..|
|000010e0| 32 12 30 0d 0a 00 0d 0b | 00 20 20 20 20 11 31 30 |2.0.....|.    .10|
|000010f0| 2e 33 11 33 78 20 11 31 | 2b 20 31 2e 35 20 3d 20 |.3.3x .1|+ 1.5 = |
|00001100| 38 2e 34 20 20 20 20 20 | 20 20 20 20 20 20 11 32 |8.4     |      .2|
|00001110| 12 31 47 69 76 65 6e 20 | 65 71 75 61 74 69 6f 6e |.1Given |equation|
|00001120| 12 30 0d 0a 00 20 20 20 | 20 20 20 20 11 33 33 78 |.0...   |    .33x|
|00001130| 20 11 31 2b 20 31 35 20 | 3d 20 38 34 20 20 20 20 | .1+ 15 |= 84    |
|00001140| 20 20 20 20 20 20 20 20 | 11 32 12 31 4d 75 6c 74 |        |.2.1Mult|
|00001150| 69 70 6c 79 20 62 6f 74 | 68 20 73 69 64 65 73 20 |iply bot|h sides |
|00001160| 62 79 20 31 30 12 30 0d | 0a 00 20 20 20 20 20 20 |by 10.0.|..      |
|00001170| 20 20 20 20 20 20 11 31 | 33 11 33 78 20 11 31 3d |      .1|3.3x .1=|
|00001180| 20 36 39 20 20 20 20 20 | 20 20 20 20 20 20 20 11 | 69     |       .|
|00001190| 32 12 31 53 75 62 74 72 | 61 63 74 20 31 35 20 66 |2.1Subtr|act 15 f|
|000011a0| 72 6f 6d 20 62 6f 74 68 | 20 73 69 64 65 73 12 30 |rom both| sides.0|
|000011b0| 0d 0a 00 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |...     |        |
|000011c0| 11 33 78 20 11 31 3d 20 | 32 33 20 20 20 20 20 20 |.3x .1= |23      |
|000011d0| 20 20 20 20 20 20 11 32 | 12 31 53 6f 6c 75 74 69 |      .2|.1Soluti|
|000011e0| 6f 6e 12 30 0d 0a 00 00 | 17 4f 0a 0d 0b 00 57 72 |on.0....|.O....Wr|
|000011f0| 6f 6e 67 2e 20 20 54 68 | 65 20 6c 65 61 73 74 20 |ong.  Th|e least |
|00001200| 63 6f 6d 6d 6f 6e 20 64 | 65 6e 6f 6d 69 6e 61 74 |common d|enominat|
|00001210| 6f 72 20 69 73 20 11 33 | 78 11 31 28 11 33 78 20 |or is .3|x.1(.3x |
|00001220| 11 31 2d 20 32 29 2c 20 | 73 6f 20 61 66 74 65 72 |.1- 2), |so after|
|00001230| 20 6d 75 6c 74 69 70 6c | 79 69 6e 67 0d 0a 00 62 | multipl|ying...b|
|00001240| 6f 74 68 20 73 69 64 65 | 73 20 62 79 20 11 33 78 |oth side|s by .3x|
|00001250| 11 31 28 11 33 78 20 11 | 31 2d 20 32 29 2c 20 77 |.1(.3x .|1- 2), w|
|00001260| 65 20 6f 62 74 61 69 6e | 20 74 68 65 20 66 6f 6c |e obtain| the fol|
|00001270| 6c 6f 77 69 6e 67 2e 12 | 30 0d 0a 00 0d 0b 00 20 |lowing..|0...... |
|00001280| 20 20 20 31 30 20 2b 20 | 34 28 11 33 78 20 11 31 |   10 + |4(.3x .1|
|00001290| 2d 20 32 29 20 3d 20 35 | 11 33 78 11 31 2c 20 11 |- 2) = 5|.3x.1, .|
|000012a0| 33 78 20 11 34 3d 20 11 | 31 30 2c 20 11 33 78 20 |3x .4= .|10, .3x |
|000012b0| 11 34 3d 20 11 31 32 20 | 20 20 20 20 20 11 32 12 |.4= .12 |     .2.|
|000012c0| 31 4d 75 6c 74 69 70 6c | 79 20 67 69 76 65 6e 20 |1Multipl|y given |
|000012d0| 65 71 75 61 74 69 6f 6e | 20 62 79 20 78 28 78 2d |equation| by x(x-|
|000012e0| 32 29 12 30 0d 0a 00 20 | 20 20 20 20 20 11 31 31 |2).0... |     .11|
|000012f0| 30 20 2b 20 34 11 33 78 | 20 11 31 2d 20 38 20 3d |0 + 4.3x| .1- 8 =|
|00001300| 20 35 11 33 78 20 20 20 | 20 20 20 20 20 20 20 20 | 5.3x   |        |
|00001310| 20 20 20 20 20 20 20 20 | 20 11 32 12 31 44 69 73 |        | .2.1Dis|
|00001320| 74 72 69 62 75 74 69 76 | 65 20 50 72 6f 70 65 72 |tributiv|e Proper|
|00001330| 74 79 12 30 0d 0a 00 20 | 20 20 20 20 20 20 20 20 |ty.0... |        |
|00001340| 20 20 20 20 20 20 20 11 | 31 32 20 3d 20 11 33 78 |       .|12 = .3x|
|00001350| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00001360| 20 20 20 20 20 11 32 12 | 31 54 72 69 61 6c 20 53 |     .2.|1Trial S|
|00001370| 6f 6c 75 74 69 6f 6e 12 | 30 0d 0a 00 0d 0b 00 11 |olution.|0.......|
|00001380| 31 12 30 57 65 20 6b 6e | 6f 77 20 74 68 61 74 20 |1.0We kn|ow that |
|00001390| 69 6e 20 74 68 65 20 6f | 72 69 67 69 6e 61 6c 20 |in the o|riginal |
|000013a0| 65 71 75 61 74 69 6f 6e | 20 11 33 78 20 11 31 3d |equation| .3x .1=|
|000013b0| 20 32 20 79 69 65 6c 64 | 73 20 61 20 64 65 6e 6f | 2 yield|s a deno|
|000013c0| 6d 69 6e 61 74 6f 72 20 | 6f 66 20 7a 65 72 6f 2e |minator |of zero.|
|000013d0| 0d 0a 00 54 68 75 73 2c | 20 11 33 78 20 11 31 3d |...Thus,| .3x .1=|
|000013e0| 20 32 20 69 73 20 61 6e | 20 65 78 74 72 61 6e 65 | 2 is an| extrane|
|000013f0| 6f 75 73 20 73 6f 6c 75 | 74 69 6f 6e 20 61 6e 64 |ous solu|tion and|
|00001400| 20 74 68 65 20 65 71 75 | 61 74 69 6f 6e 20 68 61 | the equ|ation ha|
|00001410| 73 20 6e 6f 20 73 6f 6c | 75 74 69 6f 6e 2e 0d 0a |s no sol|ution...|
|00001420| 00 00 17 4f 0a 0d 0b 00 | 57 72 6f 6e 67 2e 20 20 |...O....|Wrong.  |
|00001430| 54 68 65 20 6c 65 61 73 | 74 20 63 6f 6d 6d 6f 6e |The leas|t common|
|00001440| 20 64 65 6e 6f 6d 69 6e | 61 74 6f 72 20 69 73 20 | denomin|ator is |
|00001450| 11 33 78 11 31 28 11 33 | 78 20 11 31 2d 20 32 29 |.3x.1(.3|x .1- 2)|
|00001460| 2c 20 73 6f 20 61 66 74 | 65 72 20 6d 75 6c 74 69 |, so aft|er multi|
|00001470| 70 6c 79 69 6e 67 0d 0a | 00 62 6f 74 68 20 73 69 |plying..|.both si|
|00001480| 64 65 73 20 62 79 20 11 | 33 78 11 31 28 11 33 78 |des by .|3x.1(.3x|
|00001490| 20 11 31 2d 20 32 29 2c | 20 77 65 20 6f 62 74 61 | .1- 2),| we obta|
|000014a0| 69 6e 20 74 68 65 20 66 | 6f 6c 6c 6f 77 69 6e 67 |in the f|ollowing|
|000014b0| 2e 12 30 0d 0a 00 0d 0b | 00 20 20 20 20 31 30 20 |..0.....|.    10 |
|000014c0| 2b 20 34 28 11 33 78 20 | 11 31 2d 20 32 29 20 3d |+ 4(.3x |.1- 2) =|
|000014d0| 20 35 11 33 78 11 31 2c | 20 11 33 78 20 11 34 3d | 5.3x.1,| .3x .4=|
|000014e0| 20 11 31 30 2c 20 11 33 | 78 20 11 34 3d 20 11 31 | .10, .3|x .4= .1|
|000014f0| 32 20 20 20 20 20 20 11 | 32 12 31 4d 75 6c 74 69 |2      .|2.1Multi|
|00001500| 70 6c 79 20 67 69 76 65 | 6e 20 65 71 75 61 74 69 |ply give|n equati|
|00001510| 6f 6e 20 62 79 20 78 28 | 78 2d 32 29 12 30 0d 0a |on by x(|x-2).0..|
|00001520| 00 20 20 20 20 20 20 11 | 31 31 30 20 2b 20 34 11 |.      .|110 + 4.|
|00001530| 33 78 20 11 31 2d 20 38 | 20 3d 20 35 11 33 78 20 |3x .1- 8| = 5.3x |
|00001540| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00001550| 20 20 20 11 32 12 31 44 | 69 73 74 72 69 62 75 74 |   .2.1D|istribut|
|00001560| 69 76 65 20 50 72 6f 70 | 65 72 74 79 12 30 0d 0a |ive Prop|erty.0..|
|00001570| 00 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |.       |        |
|00001580| 20 11 31 32 20 3d 20 11 | 33 78 20 20 20 20 20 20 | .12 = .|3x      |
|00001590| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 11 |        |       .|
|000015a0| 32 12 31 54 72 69 61 6c | 20 53 6f 6c 75 74 69 6f |2.1Trial| Solutio|
|000015b0| 6e 12 30 0d 0a 00 0d 0b | 00 11 31 12 30 57 65 20 |n.0.....|..1.0We |
|000015c0| 6b 6e 6f 77 20 74 68 61 | 74 20 69 6e 20 74 68 65 |know tha|t in the|
|000015d0| 20 6f 72 69 67 69 6e 61 | 6c 20 65 71 75 61 74 69 | origina|l equati|
|000015e0| 6f 6e 20 11 33 78 20 11 | 31 3d 20 32 20 79 69 65 |on .3x .|1= 2 yie|
|000015f0| 6c 64 73 20 61 20 64 65 | 6e 6f 6d 69 6e 61 74 6f |lds a de|nominato|
|00001600| 72 20 6f 66 20 7a 65 72 | 6f 2e 0d 0a 00 54 68 75 |r of zer|o....Thu|
|00001610| 73 2c 20 11 33 78 20 11 | 31 3d 20 32 20 69 73 20 |s, .3x .|1= 2 is |
|00001620| 61 6e 20 65 78 74 72 61 | 6e 65 6f 75 73 20 73 6f |an extra|neous so|
|00001630| 6c 75 74 69 6f 6e 20 61 | 6e 64 20 74 68 65 20 65 |lution a|nd the e|
|00001640| 71 75 61 74 69 6f 6e 20 | 68 61 73 20 6e 6f 20 73 |quation |has no s|
|00001650| 6f 6c 75 74 69 6f 6e 2e | 0d 0a 00 00 17 4f 0a 0d |olution.|.....O..|
|00001660| 0b 00 57 72 6f 6e 67 2e | 20 20 54 68 65 20 6c 65 |..Wrong.|  The le|
|00001670| 61 73 74 20 63 6f 6d 6d | 6f 6e 20 64 65 6e 6f 6d |ast comm|on denom|
|00001680| 69 6e 61 74 6f 72 20 69 | 73 20 11 33 78 11 31 28 |inator i|s .3x.1(|
|00001690| 11 33 78 20 11 31 2d 20 | 32 29 2c 20 73 6f 20 61 |.3x .1- |2), so a|
|000016a0| 66 74 65 72 20 6d 75 6c | 74 69 70 6c 79 69 6e 67 |fter mul|tiplying|
|000016b0| 0d 0a 00 62 6f 74 68 20 | 73 69 64 65 73 20 62 79 |...both |sides by|
|000016c0| 20 11 33 78 11 31 28 11 | 33 78 20 11 31 2d 20 32 | .3x.1(.|3x .1- 2|
|000016d0| 29 2c 20 77 65 20 6f 62 | 74 61 69 6e 20 74 68 65 |), we ob|tain the|
|000016e0| 20 66 6f 6c 6c 6f 77 69 | 6e 67 2e 12 30 0d 0a 00 | followi|ng..0...|
|000016f0| 0d 0b 00 20 20 20 20 31 | 30 20 2b 20 34 28 11 33 |...    1|0 + 4(.3|
|00001700| 78 20 11 31 2d 20 32 29 | 20 3d 20 35 11 33 78 11 |x .1- 2)| = 5.3x.|
|00001710| 31 2c 20 11 33 78 20 11 | 34 3d 20 11 31 30 2c 20 |1, .3x .|4= .10, |
|00001720| 11 33 78 20 11 34 3d 20 | 11 31 32 20 20 20 20 20 |.3x .4= |.12     |
|00001730| 20 11 32 12 31 4d 75 6c | 74 69 70 6c 79 20 67 69 | .2.1Mul|tiply gi|
|00001740| 76 65 6e 20 65 71 75 61 | 74 69 6f 6e 20 62 79 20 |ven equa|tion by |
|00001750| 78 28 78 2d 32 29 12 30 | 0d 0a 00 20 20 20 20 20 |x(x-2).0|...     |
|00001760| 20 11 31 31 30 20 2b 20 | 34 11 33 78 20 11 31 2d | .110 + |4.3x .1-|
|00001770| 20 38 20 3d 20 35 11 33 | 78 20 20 20 20 20 20 20 | 8 = 5.3|x       |
|00001780| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 11 32 12 |        |     .2.|
|00001790| 31 44 69 73 74 72 69 62 | 75 74 69 76 65 20 50 72 |1Distrib|utive Pr|
|000017a0| 6f 70 65 72 74 79 12 30 | 0d 0a 00 20 20 20 20 20 |operty.0|...     |
|000017b0| 20 20 20 20 20 20 20 20 | 20 20 20 11 31 32 20 3d |        |   .12 =|
|000017c0| 20 11 33 78 20 20 20 20 | 20 20 20 20 20 20 20 20 | .3x    |        |
|000017d0| 20 20 20 20 20 20 20 20 | 20 11 32 12 31 54 72 69 |        | .2.1Tri|
|000017e0| 61 6c 20 53 6f 6c 75 74 | 69 6f 6e 12 30 0d 0a 00 |al Solut|ion.0...|
|000017f0| 0d 0b 00 11 31 12 30 57 | 65 20 6b 6e 6f 77 20 74 |....1.0W|e know t|
|00001800| 68 61 74 20 69 6e 20 74 | 68 65 20 6f 72 69 67 69 |hat in t|he origi|
|00001810| 6e 61 6c 20 65 71 75 61 | 74 69 6f 6e 20 11 33 78 |nal equa|tion .3x|
|00001820| 20 11 31 3d 20 32 20 79 | 69 65 6c 64 73 20 61 20 | .1= 2 y|ields a |
|00001830| 64 65 6e 6f 6d 69 6e 61 | 74 6f 72 20 6f 66 20 7a |denomina|tor of z|
|00001840| 65 72 6f 2e 0d 0a 00 54 | 68 75 73 2c 20 11 33 78 |ero....T|hus, .3x|
|00001850| 20 11 31 3d 20 32 20 69 | 73 20 61 6e 20 65 78 74 | .1= 2 i|s an ext|
|00001860| 72 61 6e 65 6f 75 73 20 | 73 6f 6c 75 74 69 6f 6e |raneous |solution|
|00001870| 20 61 6e 64 20 74 68 65 | 20 65 71 75 61 74 69 6f | and the| equatio|
|00001880| 6e 20 68 61 73 20 6e 6f | 20 73 6f 6c 75 74 69 6f |n has no| solutio|
|00001890| 6e 2e 0d 0a 00 00 17 4f | 03 0d 0b 00 47 6f 6f 64 |n......O|....Good|
|000018a0| 20 6a 6f 62 2e 20 20 54 | 68 65 20 67 69 76 65 6e | job.  T|he given|
|000018b0| 20 65 71 75 61 74 69 6f | 6e 20 68 61 73 20 6e 6f | equatio|n has no|
|000018c0| 20 73 6f 6c 75 74 69 6f | 6e 2e 0d 0a 00 00 17 4f | solutio|n......O|
|000018d0| 08 0d 0b 00 57 72 6f 6e | 67 2e 20 20 57 65 20 73 |....Wron|g.  We s|
|000018e0| 6f 6c 76 65 20 74 68 65 | 20 65 71 75 61 74 69 6f |olve the| equatio|
|000018f0| 6e 20 61 73 20 66 6f 6c | 6c 6f 77 73 2e 12 30 0d |n as fol|lows..0.|
|00001900| 0a 00 0d 0b 00 20 20 20 | 20 20 33 28 11 33 78 20 |.....   |  3(.3x |
|00001910| 11 31 2b 20 32 29 20 2d | 20 39 20 3d 20 34 28 11 |.1+ 2) -| 9 = 4(.|
|00001920| 33 78 20 11 31 2d 20 33 | 29 20 20 20 20 20 20 20 |3x .1- 3|)       |
|00001930| 20 20 20 11 32 12 31 47 | 69 76 65 6e 20 65 71 75 |   .2.1G|iven equ|
|00001940| 61 74 69 6f 6e 12 30 0d | 0a 00 20 20 20 20 20 20 |ation.0.|..      |
|00001950| 20 11 31 33 11 33 78 20 | 11 31 2b 20 36 20 2d 20 | .13.3x |.1+ 6 - |
|00001960| 39 20 3d 20 34 11 33 78 | 20 11 31 2d 20 31 32 20 |9 = 4.3x| .1- 12 |
|00001970| 20 20 20 20 20 20 20 20 | 20 20 11 32 12 31 52 65 |        |  .2.1Re|
|00001980| 6d 6f 76 65 20 70 61 72 | 65 6e 74 68 65 73 65 73 |move par|entheses|
|00001990| 12 30 0d 0a 00 20 20 20 | 20 20 20 20 20 20 20 20 |.0...   |        |
|000019a0| 11 31 33 11 33 78 20 11 | 31 2d 20 33 20 3d 20 34 |.13.3x .|1- 3 = 4|
|000019b0| 11 33 78 20 11 31 2d 20 | 31 32 20 20 20 20 20 20 |.3x .1- |12      |
|000019c0| 20 20 20 20 20 11 32 12 | 31 53 69 6d 70 6c 69 66 |     .2.|1Simplif|
|000019d0| 79 12 30 0d 0a 00 20 20 | 20 20 20 20 20 20 20 20 |y.0...  |        |
|000019e0| 20 20 20 20 20 11 31 2d | 11 33 78 20 11 31 3d 20 |     .1-|.3x .1= |
|000019f0| 2d 39 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |-9      |        |
|00001a00| 20 20 11 32 12 31 41 64 | 64 20 33 20 61 6e 64 20 |  .2.1Ad|d 3 and |
|00001a10| 73 75 62 74 72 61 63 74 | 20 34 78 12 30 0d 0a 00 |subtract| 4x.0...|
|00001a20| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00001a30| 11 33 78 20 11 31 3d 20 | 39 20 20 20 20 20 20 20 |.3x .1= |9       |
|00001a40| 20 20 20 20 20 20 20 20 | 20 20 11 32 12 31 44 69 |        |  .2.1Di|
|00001a50| 76 69 64 65 20 62 79 20 | 2d 31 12 30 0d 0a 00 00 |vide by |-1.0....|
|00001a60| 17 4f 08 0d 0b 00 57 72 | 6f 6e 67 2e 20 20 57 65 |.O....Wr|ong.  We|
|00001a70| 20 73 6f 6c 76 65 20 74 | 68 65 20 65 71 75 61 74 | solve t|he equat|
|00001a80| 69 6f 6e 20 61 73 20 66 | 6f 6c 6c 6f 77 73 2e 12 |ion as f|ollows..|
|00001a90| 30 0d 0a 00 0d 0b 00 20 | 20 20 20 20 33 28 11 33 |0...... |    3(.3|
|00001aa0| 78 20 11 31 2b 20 32 29 | 20 2d 20 39 20 3d 20 34 |x .1+ 2)| - 9 = 4|
|00001ab0| 28 11 33 78 20 11 31 2d | 20 33 29 20 20 20 20 20 |(.3x .1-| 3)     |
|00001ac0| 20 20 20 20 20 11 32 12 | 31 47 69 76 65 6e 20 65 |     .2.|1Given e|
|00001ad0| 71 75 61 74 69 6f 6e 12 | 30 0d 0a 00 20 20 20 20 |quation.|0...    |
|00001ae0| 20 20 20 11 31 33 11 33 | 78 20 11 31 2b 20 36 20 |   .13.3|x .1+ 6 |
|00001af0| 2d 20 39 20 3d 20 34 11 | 33 78 20 11 31 2d 20 31 |- 9 = 4.|3x .1- 1|
|00001b00| 32 20 20 20 20 20 20 20 | 20 20 20 20 11 32 12 31 |2       |    .2.1|
|00001b10| 52 65 6d 6f 76 65 20 70 | 61 72 65 6e 74 68 65 73 |Remove p|arenthes|
|00001b20| 65 73 12 30 0d 0a 00 20 | 20 20 20 20 20 20 20 20 |es.0... |        |
|00001b30| 20 20 11 31 33 11 33 78 | 20 11 31 2d 20 33 20 3d |  .13.3x| .1- 3 =|
|00001b40| 20 34 11 33 78 20 11 31 | 2d 20 31 32 20 20 20 20 | 4.3x .1|- 12    |
|00001b50| 20 20 20 20 20 20 20 11 | 32 12 31 53 69 6d 70 6c |       .|2.1Simpl|
|00001b60| 69 66 79 12 30 0d 0a 00 | 20 20 20 20 20 20 20 20 |ify.0...|        |
|00001b70| 20 20 20 20 20 20 20 11 | 31 2d 11 33 78 20 11 31 |       .|1-.3x .1|
|00001b80| 3d 20 2d 39 20 20 20 20 | 20 20 20 20 20 20 20 20 |= -9    |        |
|00001b90| 20 20 20 20 11 32 12 31 | 41 64 64 20 33 20 61 6e |    .2.1|Add 3 an|
|00001ba0| 64 20 73 75 62 74 72 61 | 63 74 20 34 78 12 30 0d |d subtra|ct 4x.0.|
|00001bb0| 0a 00 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |..      |        |
|00001bc0| 20 20 11 33 78 20 11 31 | 3d 20 39 20 20 20 20 20 |  .3x .1|= 9     |
|00001bd0| 20 20 20 20 20 20 20 20 | 20 20 20 20 11 32 12 31 |        |    .2.1|
|00001be0| 44 69 76 69 64 65 20 62 | 79 20 2d 31 12 30 0d 0a |Divide b|y -1.0..|
|00001bf0| 00 00 17 4f 03 0d 0b 00 | 47 6f 6f 64 2e 20 20 54 |...O....|Good.  T|
|00001c00| 72 79 20 63 68 65 63 6b | 69 6e 67 20 74 68 69 73 |ry check|ing this|
|00001c10| 20 61 6e 73 77 65 72 20 | 69 6e 20 74 68 65 20 6f | answer |in the o|
|00001c20| 72 69 67 69 6e 61 6c 20 | 65 71 75 61 74 69 6f 6e |riginal |equation|
|00001c30| 2e 0d 0a 00 00 17 4f 04 | 0d 0b 00 57 72 6f 6e 67 |......O.|...Wrong|
|00001c40| 2e 20 20 54 68 69 73 20 | 65 71 75 61 74 69 6f 6e |.  This |equation|
|00001c50| 20 63 61 6e 20 62 65 20 | 73 6f 6c 76 65 64 20 66 | can be |solved f|
|00001c60| 6f 72 20 11 33 78 11 31 | 2e 20 20 52 65 76 69 65 |or .3x.1|.  Revie|
|00001c70| 77 20 47 75 69 64 65 64 | 20 45 78 61 6d 70 6c 65 |w Guided| Example|
|00001c80| 20 32 20 61 6e 64 20 0d | 0a 00 74 72 79 20 61 67 | 2 and .|..try ag|
|00001c90| 61 69 6e 2e 0d 0a 00 00 | 17 4f 08 0d 0b 00 57 72 |ain.....|.O....Wr|
|00001ca0| 6f 6e 67 2e 20 20 53 69 | 6e 63 65 20 74 68 65 20 |ong.  Si|nce the |
|00001cb0| 64 65 6e 6f 6d 69 6e 61 | 74 6f 72 73 20 61 72 65 |denomina|tors are|
|00001cc0| 20 36 20 61 6e 64 20 33 | 2c 20 74 68 65 20 6c 65 | 6 and 3|, the le|
|00001cd0| 61 73 74 20 63 6f 6d 6d | 6f 6e 20 64 65 6e 6f 6d |ast comm|on denom|
|00001ce0| 69 6e 61 74 6f 72 20 0d | 0a 00 28 4c 43 44 29 20 |inator .|..(LCD) |
|00001cf0| 69 73 20 36 2e 20 20 54 | 68 75 73 2c 20 77 65 20 |is 6.  T|hus, we |
|00001d00| 73 6f 6c 76 65 20 74 68 | 65 20 65 71 75 61 74 69 |solve th|e equati|
|00001d10| 6f 6e 20 61 73 20 66 6f | 6c 6c 6f 77 73 2e 12 30 |on as fo|llows..0|
|00001d20| 0d 0a 00 20 20 20 20 20 | 20 20 20 37 11 33 78 20 |...     |   7.3x |
|00001d30| 20 20 20 20 20 11 31 31 | 20 20 20 20 20 20 20 20 |     .11|        |
|00001d40| 20 20 20 20 20 32 0d 0b | 00 20 20 20 20 20 28 36 |     2..|.     (6|
|00001d50| 29 11 34 32 32 20 11 31 | 2b 20 28 36 29 11 34 32 |).422 .1|+ (6).42|
|00001d60| 20 11 31 3d 20 28 36 29 | 11 33 78 20 11 31 2d 20 | .1= (6)|.3x .1- |
|00001d70| 28 36 29 11 34 32 20 20 | 20 20 20 20 20 20 20 20 |(6).42  |        |
|00001d80| 20 11 32 12 31 4d 75 6c | 74 69 70 6c 79 20 67 69 | .2.1Mul|tiply gi|
|00001d90| 76 65 6e 20 65 71 75 61 | 74 69 6f 6e 20 62 79 20 |ven equa|tion by |
|00001da0| 4c 43 44 12 30 0d 0b 00 | 20 20 20 20 20 20 20 20 |LCD.0...|        |
|00001db0| 20 11 31 36 20 20 20 20 | 20 20 33 20 20 20 20 20 | .16    |  3     |
|00001dc0| 20 20 20 20 20 20 20 20 | 33 0d 0a 00 20 20 20 20 |        |3...    |
|00001dd0| 20 20 20 20 20 20 20 37 | 11 33 78 20 11 31 2b 20 |       7|.3x .1+ |
|00001de0| 32 20 3d 20 36 11 33 78 | 20 11 31 2d 20 34 20 20 |2 = 6.3x| .1- 4  |
|00001df0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 11 32 |        |      .2|
|00001e00| 12 31 52 65 64 75 63 65 | 20 61 6e 64 20 6d 75 6c |.1Reduce| and mul|
|00001e10| 74 69 70 6c 79 12 30 0d | 0a 00 0d 0b 00 20 20 20 |tiply.0.|.....   |
|00001e20| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 11 33 78 |        |     .3x|
|00001e30| 20 11 31 3d 20 2d 36 20 | 20 20 20 20 20 20 20 20 | .1= -6 |        |
|00001e40| 20 20 20 20 20 20 20 20 | 20 20 20 11 32 12 31 53 |        |   .2.1S|
|00001e50| 75 62 74 72 61 63 74 20 | 32 20 61 6e 64 20 36 78 |ubtract |2 and 6x|
|00001e60| 12 30 0d 0a 00 00 17 4f | 03 0d 0b 00 52 69 67 68 |.0.....O|....Righ|
|00001e70| 74 2e 20 20 43 68 65 63 | 6b 20 74 68 69 73 20 73 |t.  Chec|k this s|
|00001e80| 6f 6c 75 74 69 6f 6e 20 | 69 6e 20 74 68 65 20 6f |olution |in the o|
|00001e90| 72 69 67 69 6e 61 6c 20 | 65 71 75 61 74 69 6f 6e |riginal |equation|
|00001ea0| 2e 0d 0a 00 00 17 4f 08 | 0d 0b 00 57 72 6f 6e 67 |......O.|...Wrong|
|00001eb0| 2e 20 20 53 69 6e 63 65 | 20 74 68 65 20 64 65 6e |.  Since| the den|
|00001ec0| 6f 6d 69 6e 61 74 6f 72 | 73 20 61 72 65 20 36 20 |ominator|s are 6 |
|00001ed0| 61 6e 64 20 33 2c 20 74 | 68 65 20 6c 65 61 73 74 |and 3, t|he least|
|00001ee0| 20 63 6f 6d 6d 6f 6e 20 | 64 65 6e 6f 6d 69 6e 61 | common |denomina|
|00001ef0| 74 6f 72 20 0d 0a 00 28 | 4c 43 44 29 20 69 73 20 |tor ...(|LCD) is |
|00001f00| 36 2e 20 20 54 68 75 73 | 2c 20 77 65 20 73 6f 6c |6.  Thus|, we sol|
|00001f10| 76 65 20 74 68 65 20 65 | 71 75 61 74 69 6f 6e 20 |ve the e|quation |
|00001f20| 61 73 20 66 6f 6c 6c 6f | 77 73 2e 12 30 0d 0a 00 |as follo|ws..0...|
|00001f30| 20 20 20 20 20 20 20 20 | 37 11 33 78 20 20 20 20 |        |7.3x    |
|00001f40| 20 20 11 31 31 20 20 20 | 20 20 20 20 20 20 20 20 |  .11   |        |
|00001f50| 20 20 32 0d 0b 00 20 20 | 20 20 20 28 36 29 11 34 |  2...  |   (6).4|
|00001f60| 32 32 20 11 31 2b 20 28 | 36 29 11 34 32 20 11 31 |22 .1+ (|6).42 .1|
|00001f70| 3d 20 28 36 29 11 33 78 | 20 11 31 2d 20 28 36 29 |= (6).3x| .1- (6)|
|00001f80| 11 34 32 20 20 20 20 20 | 20 20 20 20 20 20 11 32 |.42     |      .2|
|00001f90| 12 31 4d 75 6c 74 69 70 | 6c 79 20 67 69 76 65 6e |.1Multip|ly given|
|00001fa0| 20 65 71 75 61 74 69 6f | 6e 20 62 79 20 4c 43 44 | equatio|n by LCD|
|00001fb0| 12 30 0d 0b 00 20 20 20 | 20 20 20 20 20 20 11 31 |.0...   |      .1|
|00001fc0| 36 20 20 20 20 20 20 33 | 20 20 20 20 20 20 20 20 |6      3|        |
|00001fd0| 20 20 20 20 20 33 0d 0a | 00 20 20 20 20 20 20 20 |     3..|.       |
|00001fe0| 20 20 20 20 37 11 33 78 | 20 11 31 2b 20 32 20 3d |    7.3x| .1+ 2 =|
|00001ff0| 20 36 11 33 78 20 11 31 | 2d 20 34 20 20 20 20 20 | 6.3x .1|- 4     |
|00002000| 20 20 20 20 20 20 20 20 | 20 20 20 11 32 12 31 52 |        |   .2.1R|
|00002010| 65 64 75 63 65 20 61 6e | 64 20 6d 75 6c 74 69 70 |educe an|d multip|
|00002020| 6c 79 12 30 0d 0a 00 0d | 0b 00 20 20 20 20 20 20 |ly.0....|..      |
|00002030| 20 20 20 20 20 20 20 20 | 20 20 11 33 78 20 11 31 |        |  .3x .1|
|00002040| 3d 20 2d 36 20 20 20 20 | 20 20 20 20 20 20 20 20 |= -6    |        |
|00002050| 20 20 20 20 20 20 20 20 | 11 32 12 31 53 75 62 74 |        |.2.1Subt|
|00002060| 72 61 63 74 20 32 20 61 | 6e 64 20 36 78 12 30 0d |ract 2 a|nd 6x.0.|
|00002070| 0a 00 00 17 4f 08 0d 0b | 00 57 72 6f 6e 67 2e 20 |....O...|.Wrong. |
|00002080| 20 53 69 6e 63 65 20 74 | 68 65 20 64 65 6e 6f 6d | Since t|he denom|
|00002090| 69 6e 61 74 6f 72 73 20 | 61 72 65 20 36 20 61 6e |inators |are 6 an|
|000020a0| 64 20 33 2c 20 74 68 65 | 20 6c 65 61 73 74 20 63 |d 3, the| least c|
|000020b0| 6f 6d 6d 6f 6e 20 64 65 | 6e 6f 6d 69 6e 61 74 6f |ommon de|nominato|
|000020c0| 72 20 0d 0a 00 28 4c 43 | 44 29 20 69 73 20 36 2e |r ...(LC|D) is 6.|
|000020d0| 20 20 54 68 75 73 2c 20 | 77 65 20 73 6f 6c 76 65 |  Thus, |we solve|
|000020e0| 20 74 68 65 20 65 71 75 | 61 74 69 6f 6e 20 61 73 | the equ|ation as|
|000020f0| 20 66 6f 6c 6c 6f 77 73 | 2e 12 30 0d 0a 00 20 20 | follows|..0...  |
|00002100| 20 20 20 20 20 20 37 11 | 33 78 20 20 20 20 20 20 |      7.|3x      |
|00002110| 11 31 31 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |.11     |        |
|00002120| 32 0d 0b 00 20 20 20 20 | 20 28 36 29 11 34 32 32 |2...    | (6).422|
|00002130| 20 11 31 2b 20 28 36 29 | 11 34 32 20 11 31 3d 20 | .1+ (6)|.42 .1= |
|00002140| 28 36 29 11 33 78 20 11 | 31 2d 20 28 36 29 11 34 |(6).3x .|1- (6).4|
|00002150| 32 20 20 20 20 20 20 20 | 20 20 20 20 11 32 12 31 |2       |    .2.1|
|00002160| 4d 75 6c 74 69 70 6c 79 | 20 67 69 76 65 6e 20 65 |Multiply| given e|
|00002170| 71 75 61 74 69 6f 6e 20 | 62 79 20 4c 43 44 12 30 |quation |by LCD.0|
|00002180| 0d 0b 00 20 20 20 20 20 | 20 20 20 20 11 31 36 20 |...     |    .16 |
|00002190| 20 20 20 20 20 33 20 20 | 20 20 20 20 20 20 20 20 |     3  |        |
|000021a0| 20 20 20 33 0d 0a 00 20 | 20 20 20 20 20 20 20 20 |   3... |        |
|000021b0| 20 20 37 11 33 78 20 11 | 31 2b 20 32 20 3d 20 36 |  7.3x .|1+ 2 = 6|
|000021c0| 11 33 78 20 11 31 2d 20 | 34 20 20 20 20 20 20 20 |.3x .1- |4       |
|000021d0| 20 20 20 20 20 20 20 20 | 20 11 32 12 31 52 65 64 |        | .2.1Red|
|000021e0| 75 63 65 20 61 6e 64 20 | 6d 75 6c 74 69 70 6c 79 |uce and |multiply|
|000021f0| 12 30 0d 0a 00 0d 0b 00 | 20 20 20 20 20 20 20 20 |.0......|        |
|00002200| 20 20 20 20 20 20 20 20 | 11 33 78 20 11 31 3d 20 |        |.3x .1= |
|00002210| 2d 36 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |-6      |        |
|00002220| 20 20 20 20 20 20 11 32 | 12 31 53 75 62 74 72 61 |      .2|.1Subtra|
|00002230| 63 74 20 32 20 61 6e 64 | 20 36 78 12 30 0d 0a 00 |ct 2 and| 6x.0...|
|00002240| 00 17 4f 09 0d 0b 00 57 | 72 6f 6e 67 2e 20 20 57 |..O....W|rong.  W|
|00002250| 65 20 73 6f 6c 76 65 20 | 66 6f 72 20 11 33 78 20 |e solve |for .3x |
|00002260| 11 31 61 73 20 66 6f 6c | 6c 6f 77 73 2e 20 12 30 |.1as fol|lows. .0|
|00002270| 0d 0a 00 0d 0b 00 20 20 | 20 20 32 11 33 78 20 11 |......  |  2.3x .|
|00002280| 31 2b 20 34 20 3d 20 34 | 28 11 33 78 20 11 31 2d |1+ 4 = 4|(.3x .1-|
|00002290| 20 31 29 20 2d 20 32 11 | 33 78 20 20 20 20 20 20 | 1) - 2.|3x      |
|000022a0| 20 20 20 11 32 12 31 47 | 69 76 65 6e 20 65 71 75 |   .2.1G|iven equ|
|000022b0| 61 74 69 6f 6e 12 30 0d | 0a 00 20 20 20 20 11 31 |ation.0.|..    .1|
|000022c0| 32 11 33 78 20 11 31 2b | 20 34 20 3d 20 34 11 33 |2.3x .1+| 4 = 4.3|
|000022d0| 78 20 11 31 2d 20 34 20 | 2d 20 32 11 33 78 20 20 |x .1- 4 |- 2.3x  |
|000022e0| 20 20 20 20 20 20 20 20 | 20 11 32 12 31 52 65 6d |        | .2.1Rem|
|000022f0| 6f 76 65 20 70 61 72 65 | 6e 74 68 65 73 65 73 12 |ove pare|ntheses.|
|00002300| 30 0d 0a 00 20 20 20 20 | 11 31 32 11 33 78 20 11 |0...    |.12.3x .|
|00002310| 31 2b 20 34 20 3d 20 32 | 11 33 78 20 11 31 2d 20 |1+ 4 = 2|.3x .1- |
|00002320| 34 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |4       |        |
|00002330| 20 11 32 12 31 43 6f 6d | 62 69 6e 65 20 6c 69 6b | .2.1Com|bine lik|
|00002340| 65 20 74 65 72 6d 73 12 | 30 0d 0a 00 20 20 20 20 |e terms.|0...    |
|00002350| 20 20 20 20 20 11 31 34 | 20 3d 20 2d 34 20 20 20 |     .14| = -4   |
|00002360| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00002370| 20 11 32 12 31 53 75 62 | 74 72 61 63 74 20 32 78 | .2.1Sub|tract 2x|
|00002380| 12 30 0d 0a 00 11 31 53 | 69 6e 63 65 20 74 68 69 |.0....1S|ince thi|
|00002390| 73 20 6c 61 73 74 20 73 | 74 65 70 20 70 72 6f 64 |s last s|tep prod|
|000023a0| 75 63 65 73 20 61 20 63 | 6f 6e 74 72 61 64 69 63 |uces a c|ontradic|
|000023b0| 74 69 6f 6e 2c 20 77 65 | 20 73 61 79 20 74 68 61 |tion, we| say tha|
|000023c0| 74 20 74 68 65 20 67 69 | 76 65 6e 20 0d 0a 00 65 |t the gi|ven ...e|
|000023d0| 71 75 61 74 69 6f 6e 20 | 68 61 73 20 6e 6f 20 73 |quation |has no s|
|000023e0| 6f 6c 75 74 69 6f 6e 2e | 0d 0a 00 00 17 4f 09 0d |olution.|.....O..|
|000023f0| 0b 00 57 72 6f 6e 67 2e | 20 20 57 65 20 73 6f 6c |..Wrong.|  We sol|
|00002400| 76 65 20 66 6f 72 20 11 | 33 78 20 11 31 61 73 20 |ve for .|3x .1as |
|00002410| 66 6f 6c 6c 6f 77 73 2e | 20 12 30 0d 0a 00 0d 0b |follows.| .0.....|
|00002420| 00 20 20 20 20 32 11 33 | 78 20 11 31 2b 20 34 20 |.    2.3|x .1+ 4 |
|00002430| 3d 20 34 28 11 33 78 20 | 11 31 2d 20 31 29 20 2d |= 4(.3x |.1- 1) -|
|00002440| 20 32 11 33 78 20 20 20 | 20 20 20 20 20 20 11 32 | 2.3x   |      .2|
|00002450| 12 31 47 69 76 65 6e 20 | 65 71 75 61 74 69 6f 6e |.1Given |equation|
|00002460| 12 30 0d 0a 00 20 20 20 | 20 11 31 32 11 33 78 20 |.0...   | .12.3x |
|00002470| 11 31 2b 20 34 20 3d 20 | 34 11 33 78 20 11 31 2d |.1+ 4 = |4.3x .1-|
|00002480| 20 34 20 2d 20 32 11 33 | 78 20 20 20 20 20 20 20 | 4 - 2.3|x       |
|00002490| 20 20 20 20 11 32 12 31 | 52 65 6d 6f 76 65 20 70 |    .2.1|Remove p|
|000024a0| 61 72 65 6e 74 68 65 73 | 65 73 12 30 0d 0a 00 20 |arenthes|es.0... |
|000024b0| 20 20 20 11 31 32 11 33 | 78 20 11 31 2b 20 34 20 |   .12.3|x .1+ 4 |
|000024c0| 3d 20 32 11 33 78 20 11 | 31 2d 20 34 20 20 20 20 |= 2.3x .|1- 4    |
|000024d0| 20 20 20 20 20 20 20 20 | 20 20 20 20 11 32 12 31 |        |    .2.1|
|000024e0| 43 6f 6d 62 69 6e 65 20 | 6c 69 6b 65 20 74 65 72 |Combine |like ter|
|000024f0| 6d 73 12 30 0d 0a 00 20 | 20 20 20 20 20 20 20 20 |ms.0... |        |
|00002500| 11 31 34 20 3d 20 2d 34 | 20 20 20 20 20 20 20 20 |.14 = -4|        |
|00002510| 20 20 20 20 20 20 20 20 | 20 20 20 20 11 32 12 31 |        |    .2.1|
|00002520| 53 75 62 74 72 61 63 74 | 20 32 78 12 30 0d 0a 00 |Subtract| 2x.0...|
|00002530| 11 31 53 69 6e 63 65 20 | 74 68 69 73 20 6c 61 73 |.1Since |this las|
|00002540| 74 20 73 74 65 70 20 70 | 72 6f 64 75 63 65 73 20 |t step p|roduces |
|00002550| 61 20 63 6f 6e 74 72 61 | 64 69 63 74 69 6f 6e 2c |a contra|diction,|
|00002560| 20 77 65 20 73 61 79 20 | 74 68 61 74 20 74 68 65 | we say |that the|
|00002570| 20 67 69 76 65 6e 20 0d | 0a 00 65 71 75 61 74 69 | given .|..equati|
|00002580| 6f 6e 20 68 61 73 20 6e | 6f 20 73 6f 6c 75 74 69 |on has n|o soluti|
|00002590| 6f 6e 2e 0d 0a 00 00 17 | 4f 09 0d 0b 00 57 72 6f |on......|O....Wro|
|000025a0| 6e 67 2e 20 20 57 65 20 | 73 6f 6c 76 65 20 66 6f |ng.  We |solve fo|
|000025b0| 72 20 11 33 78 20 11 31 | 61 73 20 66 6f 6c 6c 6f |r .3x .1|as follo|
|000025c0| 77 73 2e 20 12 30 0d 0a | 00 0d 0b 00 20 20 20 20 |ws. .0..|....    |
|000025d0| 32 11 33 78 20 11 31 2b | 20 34 20 3d 20 34 28 11 |2.3x .1+| 4 = 4(.|
|000025e0| 33 78 20 11 31 2d 20 31 | 29 20 2d 20 32 11 33 78 |3x .1- 1|) - 2.3x|
|000025f0| 20 20 20 20 20 20 20 20 | 20 11 32 12 31 47 69 76 |        | .2.1Giv|
|00002600| 65 6e 20 65 71 75 61 74 | 69 6f 6e 12 30 0d 0a 00 |en equat|ion.0...|
|00002610| 20 20 20 20 11 31 32 11 | 33 78 20 11 31 2b 20 34 |    .12.|3x .1+ 4|
|00002620| 20 3d 20 34 11 33 78 20 | 11 31 2d 20 34 20 2d 20 | = 4.3x |.1- 4 - |
|00002630| 32 11 33 78 20 20 20 20 | 20 20 20 20 20 20 20 11 |2.3x    |       .|
|00002640| 32 12 31 52 65 6d 6f 76 | 65 20 70 61 72 65 6e 74 |2.1Remov|e parent|
|00002650| 68 65 73 65 73 12 30 0d | 0a 00 20 20 20 20 11 31 |heses.0.|..    .1|
|00002660| 32 11 33 78 20 11 31 2b | 20 34 20 3d 20 32 11 33 |2.3x .1+| 4 = 2.3|
|00002670| 78 20 11 31 2d 20 34 20 | 20 20 20 20 20 20 20 20 |x .1- 4 |        |
|00002680| 20 20 20 20 20 20 20 11 | 32 12 31 43 6f 6d 62 69 |       .|2.1Combi|
|00002690| 6e 65 20 6c 69 6b 65 20 | 74 65 72 6d 73 12 30 0d |ne like |terms.0.|
|000026a0| 0a 00 20 20 20 20 20 20 | 20 20 20 11 31 34 20 3d |..      |   .14 =|
|000026b0| 20 2d 34 20 20 20 20 20 | 20 20 20 20 20 20 20 20 | -4     |        |
|000026c0| 20 20 20 20 20 20 20 11 | 32 12 31 53 75 62 74 72 |       .|2.1Subtr|
|000026d0| 61 63 74 20 32 78 12 30 | 0d 0a 00 11 31 53 69 6e |act 2x.0|....1Sin|
|000026e0| 63 65 20 74 68 69 73 20 | 6c 61 73 74 20 73 74 65 |ce this |last ste|
|000026f0| 70 20 70 72 6f 64 75 63 | 65 73 20 61 20 63 6f 6e |p produc|es a con|
|00002700| 74 72 61 64 69 63 74 69 | 6f 6e 2c 20 77 65 20 73 |tradicti|on, we s|
|00002710| 61 79 20 74 68 61 74 20 | 74 68 65 20 67 69 76 65 |ay that |the give|
|00002720| 6e 20 0d 0a 00 65 71 75 | 61 74 69 6f 6e 20 68 61 |n ...equ|ation ha|
|00002730| 73 20 6e 6f 20 73 6f 6c | 75 74 69 6f 6e 2e 0d 0a |s no sol|ution...|
|00002740| 00 00 17 4f 04 0d 0b 00 | 56 65 72 79 20 67 6f 6f |...O....|Very goo|
|00002750| 64 2e 20 20 54 68 69 73 | 20 65 71 75 61 74 69 6f |d.  This| equatio|
|00002760| 6e 20 73 69 6d 70 6c 69 | 66 69 65 73 20 74 6f 20 |n simpli|fies to |
|00002770| 34 20 3d 20 2d 34 2c 20 | 77 68 69 63 68 20 69 73 |4 = -4, |which is|
|00002780| 20 6e 6f 74 20 70 6f 73 | 73 69 62 6c 65 20 66 6f | not pos|sible fo|
|00002790| 72 0d 0a 00 61 6e 79 20 | 76 61 6c 75 65 20 6f 66 |r...any |value of|
|000027a0| 20 11 33 78 11 31 2e 0d | 0a 00 00 17 4f 09 0d 0b | .3x.1..|....O...|
|000027b0| 00 57 72 6f 6e 67 2e 20 | 20 57 65 20 75 73 65 20 |.Wrong. | We use |
|000027c0| 63 72 6f 73 73 20 6d 75 | 6c 74 69 70 6c 69 63 61 |cross mu|ltiplica|
|000027d0| 74 69 6f 6e 20 74 6f 20 | 73 6f 6c 76 65 20 74 68 |tion to |solve th|
|000027e0| 65 20 67 69 76 65 6e 20 | 65 71 75 61 74 69 6f 6e |e given |equation|
|000027f0| 20 61 73 20 66 6f 6c 6c | 6f 77 73 2e 12 30 0d 0a | as foll|ows..0..|
|00002800| 00 0d 0b 00 20 20 20 20 | 20 33 11 33 78 20 11 31 |....    | 3.3x .1|
|00002810| 2d 20 35 20 20 20 32 0d | 0b 00 20 20 20 20 20 11 |- 5   2.|..     .|
|00002820| 34 32 32 32 32 32 32 20 | 11 31 3d 20 11 34 32 20 |4222222 |.1= .42 |
|00002830| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00002840| 20 20 20 20 20 20 11 32 | 12 31 47 69 76 65 6e 20 |      .2|.1Given |
|00002850| 65 71 75 61 74 69 6f 6e | 12 30 0d 0b 00 20 20 20 |equation|.0...   |
|00002860| 20 20 11 31 35 11 33 78 | 20 11 31 2b 20 32 20 20 |  .15.3x| .1+ 2  |
|00002870| 20 33 0d 0a 00 20 20 33 | 28 33 11 33 78 20 11 31 | 3...  3|(3.3x .1|
|00002880| 2d 20 35 29 20 3d 20 32 | 28 35 11 33 78 20 11 31 |- 5) = 2|(5.3x .1|
|00002890| 2b 20 32 29 11 32 2c 20 | 11 33 78 20 11 34 3d 20 |+ 2).2, |.3x .4= |
|000028a0| 11 31 2d 32 2f 35 20 20 | 20 20 20 11 32 12 31 43 |.1-2/5  |   .2.1C|
|000028b0| 72 6f 73 73 20 6d 75 6c | 74 69 70 6c 79 12 30 0d |ross mul|tiply.0.|
|000028c0| 0a 00 20 20 20 20 11 31 | 39 11 33 78 20 11 31 2d |..    .1|9.3x .1-|
|000028d0| 20 31 35 20 3d 20 31 30 | 11 33 78 20 11 31 2b 20 | 15 = 10|.3x .1+ |
|000028e0| 34 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |4       |        |
|000028f0| 20 20 11 32 12 31 52 65 | 6d 6f 76 65 20 70 61 72 |  .2.1Re|move par|
|00002900| 65 6e 74 68 65 73 65 73 | 12 30 0d 0a 00 20 20 20 |entheses|.0...   |
|00002910| 20 20 20 20 20 20 11 31 | 2d 11 33 78 20 11 31 3d |      .1|-.3x .1=|
|00002920| 20 31 39 20 20 20 20 20 | 20 20 20 20 20 20 20 20 | 19     |        |
|00002930| 20 20 20 20 20 20 20 20 | 20 11 32 12 31 41 64 64 |        | .2.1Add|
|00002940| 20 31 35 20 61 6e 64 20 | 73 75 62 74 72 61 63 74 | 15 and |subtract|
|00002950| 20 31 30 78 12 30 0d 0a | 00 20 20 20 20 20 20 20 | 10x.0..|.       |
|00002960| 20 20 20 11 33 78 20 11 | 31 3d 20 2d 31 39 20 20 |   .3x .|1= -19  |
|00002970| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00002980| 20 20 20 11 32 12 31 44 | 69 76 69 64 65 20 62 79 |   .2.1D|ivide by|
|00002990| 20 2d 31 12 30 0d 0a 00 | 00 17 4f 09 0d 0b 00 57 | -1.0...|..O....W|
|000029a0| 72 6f 6e 67 2e 20 20 57 | 65 20 75 73 65 20 63 72 |rong.  W|e use cr|
|000029b0| 6f 73 73 20 6d 75 6c 74 | 69 70 6c 69 63 61 74 69 |oss mult|iplicati|
|000029c0| 6f 6e 20 74 6f 20 73 6f | 6c 76 65 20 74 68 65 20 |on to so|lve the |
|000029d0| 67 69 76 65 6e 20 65 71 | 75 61 74 69 6f 6e 20 61 |given eq|uation a|
|000029e0| 73 20 66 6f 6c 6c 6f 77 | 73 2e 12 30 0d 0a 00 0d |s follow|s..0....|
|000029f0| 0b 00 20 20 20 20 20 33 | 11 33 78 20 11 31 2d 20 |..     3|.3x .1- |
|00002a00| 35 20 20 20 32 0d 0b 00 | 20 20 20 20 20 11 34 32 |5   2...|     .42|
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|00002a20| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
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|00002a40| 75 61 74 69 6f 6e 12 30 | 0d 0b 00 20 20 20 20 20 |uation.0|...     |
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|00002a80| 32 29 11 32 2c 20 11 33 | 78 20 11 34 3d 20 11 31 |2).2, .3|x .4= .1|
|00002a90| 2d 32 2f 35 20 20 20 20 | 20 11 32 12 31 43 72 6f |-2/5    | .2.1Cro|
|00002aa0| 73 73 20 6d 75 6c 74 69 | 70 6c 79 12 30 0d 0a 00 |ss multi|ply.0...|
|00002ab0| 20 20 20 20 11 31 39 11 | 33 78 20 11 31 2d 20 31 |    .19.|3x .1- 1|
|00002ac0| 35 20 3d 20 31 30 11 33 | 78 20 11 31 2b 20 34 20 |5 = 10.3|x .1+ 4 |
|00002ad0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
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|00002b10| 39 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |9       |        |
|00002b20| 20 20 20 20 20 20 20 11 | 32 12 31 41 64 64 20 31 |       .|2.1Add 1|
|00002b30| 35 20 61 6e 64 20 73 75 | 62 74 72 61 63 74 20 31 |5 and su|btract 1|
|00002b40| 30 78 12 30 0d 0a 00 20 | 20 20 20 20 20 20 20 20 |0x.0... |        |
|00002b50| 20 11 33 78 20 11 31 3d | 20 2d 31 39 20 20 20 20 | .3x .1=| -19    |
|00002b60| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00002b70| 20 11 32 12 31 44 69 76 | 69 64 65 20 62 79 20 2d | .2.1Div|ide by -|
|00002b80| 31 12 30 0d 0a 00 00 17 | 4f 03 0d 0b 00 56 65 72 |1.0.....|O....Ver|
|00002b90| 79 20 67 6f 6f 64 2e 20 | 20 43 68 65 63 6b 20 74 |y good. | Check t|
|00002ba0| 68 69 73 20 73 6f 6c 75 | 74 69 6f 6e 20 69 6e 20 |his solu|tion in |
|00002bb0| 74 68 65 20 6f 72 69 67 | 69 6e 61 6c 20 65 71 75 |the orig|inal equ|
|00002bc0| 61 74 69 6f 6e 2e 0d 0a | 00 00 17 4f 03 0d 0b 00 |ation...|...O....|
|00002bd0| 49 6e 63 6f 72 72 65 63 | 74 2e 20 20 54 68 65 72 |Incorrec|t.  Ther|
|00002be0| 65 20 69 73 20 61 20 73 | 6f 6c 75 74 69 6f 6e 20 |e is a s|olution |
|00002bf0| 74 6f 20 74 68 69 73 20 | 65 71 75 61 74 69 6f 6e |to this |equation|
|00002c00| 2e 20 20 54 72 79 20 61 | 67 61 69 6e 2e 0d 0a 00 |.  Try a|gain....|
|00002c10| 00 17 4f 04 0d 0b 00 56 | 65 72 79 20 67 6f 6f 64 |..O....V|ery good|
|00002c20| 2e 20 20 54 72 79 20 63 | 68 65 63 6b 69 6e 67 20 |.  Try c|hecking |
|00002c30| 74 68 69 73 20 61 6e 73 | 77 65 72 20 62 79 20 70 |this ans|wer by p|
|00002c40| 6c 75 67 67 69 6e 67 20 | 11 33 78 20 11 31 3d 20 |lugging |.3x .1= |
|00002c50| 33 20 69 6e 20 74 68 65 | 20 6f 72 69 67 69 6e 61 |3 in the| origina|
|00002c60| 6c 20 0d 0a 00 65 71 75 | 61 74 69 6f 6e 2e 0d 0a |l ...equ|ation...|
|00002c70| 00 00 17 4f 09 20 20 20 | 20 20 20 20 20 20 20 20 |...O.   |        |
|00002c80| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00002c90| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 11 32 |        |      .2|
|00002ca0| 32 0d 0b 00 11 31 57 72 | 6f 6e 67 2e 20 20 54 68 |2....1Wr|ong.  Th|
|00002cb0| 65 20 6c 65 61 73 74 20 | 63 6f 6d 6d 6f 6e 20 64 |e least |common d|
|00002cc0| 65 6e 6f 6d 69 6e 61 74 | 6f 72 20 69 73 20 11 33 |enominat|or is .3|
|00002cd0| 78 20 20 11 31 2d 20 34 | 20 3d 20 28 11 33 78 20 |x  .1- 4| = (.3x |
|00002ce0| 11 31 2b 20 32 29 28 11 | 33 78 20 11 31 2d 20 32 |.1+ 2)(.|3x .1- 2|
|00002cf0| 29 2e 20 20 0d 0a 00 4d | 75 6c 74 69 70 6c 79 69 |).  ...M|ultiplyi|
|00002d00| 6e 67 20 65 61 63 68 20 | 74 65 72 6d 20 62 79 20 |ng each |term by |
|00002d10| 74 68 69 73 20 4c 43 44 | 20 70 72 6f 64 75 63 65 |this LCD| produce|
|00002d20| 73 20 74 68 65 20 66 6f | 6c 6c 6f 77 69 6e 67 2e |s the fo|llowing.|
|00002d30| 12 30 0d 0a 00 20 20 20 | 31 20 20 20 20 20 20 20 |.0...   |1       |
|00002d40| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 31 20 |        |      1 |
|00002d50| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00002d60| 20 20 20 20 20 36 0d 0b | 00 20 11 34 32 32 32 32 |     6..|. .42222|
|00002d70| 32 11 31 28 11 33 78 20 | 11 31 2b 20 32 29 28 11 |2.1(.3x |.1+ 2)(.|
|00002d80| 33 78 20 11 31 2d 20 32 | 29 20 2b 20 11 34 32 32 |3x .1- 2|) + .422|
|00002d90| 32 32 32 11 31 28 11 33 | 78 20 11 31 2b 20 32 29 |222.1(.3|x .1+ 2)|
|00002da0| 28 11 33 78 20 11 31 2d | 20 32 29 20 3d 20 11 34 |(.3x .1-| 2) = .4|
|00002db0| 32 32 32 32 32 32 11 31 | 28 11 33 78 20 11 31 2b |222222.1|(.3x .1+|
|00002dc0| 20 32 29 28 11 33 78 20 | 11 31 2d 20 32 29 0d 0b | 2)(.3x |.1- 2)..|
|00002dd0| 00 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |.       |        |
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|00002e10| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00002e20| 11 33 78 20 11 31 2b 20 | 32 20 20 20 20 20 20 20 |.3x .1+ |2       |
|00002e30| 20 20 20 20 20 20 20 20 | 20 20 11 33 78 20 20 11 |        |  .3x  .|
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|00002e60| 20 20 20 20 11 33 78 20 | 11 31 2b 20 32 20 2b 20 |    .3x |.1+ 2 + |
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|00002e90| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00002ea0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00002eb0| 20 20 20 20 20 20 32 11 | 33 78 20 11 31 3d 20 36 |      2.|3x .1= 6|
|00002ec0| 0d 0a 00 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |...     |        |
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|00002f20| 20 20 20 20 20 11 32 32 | 0d 0b 00 11 31 57 72 6f |     .22|....1Wro|
|00002f30| 6e 67 2e 20 20 54 68 65 | 20 6c 65 61 73 74 20 63 |ng.  The| least c|
|00002f40| 6f 6d 6d 6f 6e 20 64 65 | 6e 6f 6d 69 6e 61 74 6f |ommon de|nominato|
|00002f50| 72 20 69 73 20 11 33 78 | 20 20 11 31 2d 20 34 20 |r is .3x|  .1- 4 |
|00002f60| 3d 20 28 11 33 78 20 11 | 31 2b 20 32 29 28 11 33 |= (.3x .|1+ 2)(.3|
|00002f70| 78 20 11 31 2d 20 32 29 | 2e 20 20 0d 0a 00 4d 75 |x .1- 2)|.  ...Mu|
|00002f80| 6c 74 69 70 6c 79 69 6e | 67 20 65 61 63 68 20 74 |ltiplyin|g each t|
|00002f90| 65 72 6d 20 62 79 20 74 | 68 69 73 20 4c 43 44 20 |erm by t|his LCD |
|00002fa0| 70 72 6f 64 75 63 65 73 | 20 74 68 65 20 66 6f 6c |produces| the fol|
|00002fb0| 6c 6f 77 69 6e 67 2e 12 | 30 0d 0a 00 20 20 20 31 |lowing..|0...   1|
|00002fc0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
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|00002fe0| 20 20 20 20 20 20 20 20 | 20 20 20 20 36 0d 0b 00 |        |    6...|
|00002ff0| 20 11 34 32 32 32 32 32 | 11 31 28 11 33 78 20 11 | .422222|.1(.3x .|
|00003000| 31 2b 20 32 29 28 11 33 | 78 20 11 31 2d 20 32 29 |1+ 2)(.3|x .1- 2)|
|00003010| 20 2b 20 11 34 32 32 32 | 32 32 11 31 28 11 33 78 | + .4222|22.1(.3x|
|00003020| 20 11 31 2b 20 32 29 28 | 11 33 78 20 11 31 2d 20 | .1+ 2)(|.3x .1- |
|00003030| 32 29 20 3d 20 11 34 32 | 32 32 32 32 32 11 31 28 |2) = .42|22222.1(|
|00003040| 11 33 78 20 11 31 2b 20 | 32 29 28 11 33 78 20 11 |.3x .1+ |2)(.3x .|
|00003050| 31 2d 20 32 29 0d 0b 00 | 20 20 20 20 20 20 20 20 |1- 2)...|        |
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|00003070| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00003080| 20 20 20 20 20 20 11 32 | 32 0d 0b 00 20 11 33 78 |      .2|2... .3x|
|00003090| 20 11 31 2d 20 32 20 20 | 20 20 20 20 20 20 20 20 | .1- 2  |        |
|000030a0| 20 20 20 20 20 20 20 11 | 33 78 20 11 31 2b 20 32 |       .|3x .1+ 2|
|000030b0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
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|000030d0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|000030e0| 20 20 20 20 20 20 20 20 | 20 20 20 11 33 78 20 11 |        |   .3x .|
|000030f0| 31 2b 20 32 20 2b 20 11 | 33 78 20 11 31 2d 20 32 |1+ 2 + .|3x .1- 2|
|00003100| 20 3d 20 36 2c 20 11 33 | 78 20 11 34 3d 20 2b 11 | = 6, .3|x .4= +.|
|00003110| 31 32 0d 0a 00 20 20 20 | 20 20 20 20 20 20 20 20 |12...   |        |
|00003120| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00003130| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 32 11 33 |        |     2.3|
|00003140| 78 20 11 31 3d 20 36 0d | 0a 00 20 20 20 20 20 20 |x .1= 6.|..      |
|00003150| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
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|00003170| 20 20 20 11 33 78 20 11 | 31 3d 20 33 0d 0a 00 00 |   .3x .|1= 3....|
|00003180| 17 4f 03 0d 0b 00 53 6f | 72 72 79 2c 20 74 68 69 |.O....So|rry, thi|
|00003190| 73 20 65 71 75 61 74 69 | 6f 6e 20 64 6f 65 73 20 |s equati|on does |
|000031a0| 68 61 76 65 20 61 20 73 | 6f 6c 75 74 69 6f 6e 2e |have a s|olution.|
|000031b0| 20 20 54 72 79 20 61 67 | 61 69 6e 2e 0d 0a 00 00 |  Try ag|ain.....|
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|000031d0| 65 20 6c 65 61 73 74 20 | 63 6f 6d 6d 6f 6e 20 64 |e least |common d|
|000031e0| 65 6e 6f 6d 69 6e 61 74 | 6f 72 20 69 73 20 28 33 |enominat|or is (3|
|000031f0| 11 33 78 20 11 31 2b 20 | 31 29 28 33 11 33 78 20 |.3x .1+ |1)(3.3x |
|00003200| 11 31 2d 20 31 29 2e 20 | 20 4d 75 6c 74 69 70 6c |.1- 1). | Multipl|
|00003210| 79 69 6e 67 20 65 61 63 | 68 0d 0a 00 74 65 72 6d |ying eac|h...term|
|00003220| 20 62 79 20 74 68 69 73 | 20 4c 43 44 20 70 72 6f | by this| LCD pro|
|00003230| 64 75 63 65 73 20 74 68 | 65 20 66 6f 6c 6c 6f 77 |duces th|e follow|
|00003240| 69 6e 67 2e 20 12 30 0d | 0a 00 20 20 20 20 20 20 |ing. .0.|..      |
|00003250| 33 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |3       |        |
|00003260| 20 20 20 20 20 20 20 20 | 36 11 33 78 0d 0b 00 20 |        |6.3x... |
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|00003280| 78 20 11 31 2b 20 31 29 | 28 33 11 33 78 20 11 31 |x .1+ 1)|(3.3x .1|
|00003290| 2d 20 31 29 20 2d 20 11 | 34 32 32 32 32 32 32 11 |- 1) - .|4222222.|
|000032a0| 31 28 33 11 33 78 20 11 | 31 2b 20 31 29 28 33 11 |1(3.3x .|1+ 1)(3.|
|000032b0| 33 78 20 11 31 2d 20 31 | 29 20 3d 20 2d 32 28 33 |3x .1- 1|) = -2(3|
|000032c0| 11 33 78 20 11 31 2b 20 | 31 29 28 33 11 33 78 20 |.3x .1+ |1)(3.3x |
|000032d0| 11 31 2d 20 31 29 0d 0b | 00 20 20 20 33 11 33 78 |.1- 1)..|.   3.3x|
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|00003300| 2d 20 31 0d 0a 00 20 20 | 20 20 20 20 20 20 20 20 |- 1...  |        |
|00003310| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00003320| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
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|00003380| 11 31 2b 20 31 29 20 3d | 20 2d 32 28 39 11 33 78 |.1+ 1) =| -2(9.3x|
|00003390| 20 20 11 31 2d 20 31 29 | 0d 0a 00 20 20 20 20 20 |  .1- 1)|...     |
|000033a0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|000033b0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|000033c0| 20 20 20 20 20 20 20 11 | 32 32 20 20 20 20 20 20 |       .|22      |
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|000033e0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|000033f0| 20 20 20 20 20 20 20 20 | 20 20 11 31 39 11 33 78 |        |  .19.3x|
|00003400| 20 11 31 2d 20 33 20 2d | 20 31 38 11 33 78 20 20 | .1- 3 -| 18.3x  |
|00003410| 11 31 2d 20 36 11 33 78 | 20 11 31 3d 20 2d 31 38 |.1- 6.3x| .1= -18|
|00003420| 11 33 78 20 20 11 31 2b | 20 32 0d 0a 00 20 20 20 |.3x  .1+| 2...   |
|00003430| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
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|00003450| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 33 11 33 |        |     3.3|
|00003460| 78 20 11 31 3d 20 35 0d | 0a 00 20 20 20 20 20 20 |x .1= 5.|..      |
|00003470| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
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|00003490| 20 20 20 20 20 20 20 20 | 20 20 20 11 33 78 20 11 |        |   .3x .|
|000034a0| 31 3d 20 35 2f 33 0d 0a | 00 00 17 4f 0b 0d 0b 00 |1= 5/3..|...O....|
|000034b0| 57 72 6f 6e 67 2e 20 20 | 54 68 65 20 6c 65 61 73 |Wrong.  |The leas|
|000034c0| 74 20 63 6f 6d 6d 6f 6e | 20 64 65 6e 6f 6d 69 6e |t common| denomin|
|000034d0| 61 74 6f 72 20 69 73 20 | 28 33 11 33 78 20 11 31 |ator is |(3.3x .1|
|000034e0| 2b 20 31 29 28 33 11 33 | 78 20 11 31 2d 20 31 29 |+ 1)(3.3|x .1- 1)|
|000034f0| 2e 20 20 4d 75 6c 74 69 | 70 6c 79 69 6e 67 20 65 |.  Multi|plying e|
|00003500| 61 63 68 0d 0a 00 74 65 | 72 6d 20 62 79 20 74 68 |ach...te|rm by th|
|00003510| 69 73 20 4c 43 44 20 70 | 72 6f 64 75 63 65 73 20 |is LCD p|roduces |
|00003520| 74 68 65 20 66 6f 6c 6c | 6f 77 69 6e 67 2e 20 12 |the foll|owing. .|
|00003530| 30 0d 0a 00 20 20 20 20 | 20 20 33 20 20 20 20 20 |0...    |  3     |
|00003540| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00003550| 20 20 36 11 33 78 0d 0b | 00 20 20 20 11 34 32 32 |  6.3x..|.   .422|
|00003560| 32 32 32 32 11 31 28 33 | 11 33 78 20 11 31 2b 20 |2222.1(3|.3x .1+ |
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|00003580| 20 11 34 32 32 32 32 32 | 32 11 31 28 33 11 33 78 | .422222|2.1(3.3x|
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|000035a0| 20 31 29 20 3d 20 2d 32 | 28 33 11 33 78 20 11 31 | 1) = -2|(3.3x .1|
|000035b0| 2b 20 31 29 28 33 11 33 | 78 20 11 31 2d 20 31 29 |+ 1)(3.3|x .1- 1)|
|000035c0| 0d 0b 00 20 20 20 33 11 | 33 78 20 11 31 2b 20 31 |...   3.|3x .1+ 1|
|000035d0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
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|000035f0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00003600| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00003610| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00003620| 20 20 20 20 20 20 20 20 | 20 20 11 32 32 0d 0b 00 |        |  .22...|
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|00003640| 20 20 20 20 20 20 20 20 | 20 20 20 20 11 31 33 28 |        |    .13(|
|00003650| 33 11 33 78 20 11 31 2d | 20 31 29 20 2d 20 36 11 |3.3x .1-| 1) - 6.|
|00003660| 33 78 11 31 28 33 11 33 | 78 20 11 31 2b 20 31 29 |3x.1(3.3|x .1+ 1)|
|00003670| 20 3d 20 2d 32 28 39 11 | 33 78 20 20 11 31 2d 20 | = -2(9.|3x  .1- |
|00003680| 31 29 0d 0a 00 20 20 20 | 20 20 20 20 20 20 20 20 |1)...   |        |
|00003690| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
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|000036b0| 20 11 32 32 20 20 20 20 | 20 20 20 20 20 20 20 20 | .22    |        |
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|000036d0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|000036e0| 20 20 20 20 11 31 39 11 | 33 78 20 11 31 2d 20 33 |    .19.|3x .1- 3|
|000036f0| 20 2d 20 31 38 11 33 78 | 20 20 11 31 2d 20 36 11 | - 18.3x|  .1- 6.|
|00003700| 33 78 20 11 31 3d 20 2d | 31 38 11 33 78 20 20 11 |3x .1= -|18.3x  .|
|00003710| 31 2b 20 32 0d 0a 00 20 | 20 20 20 20 20 20 20 20 |1+ 2... |        |
|00003720| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00003730| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00003740| 20 20 20 20 20 20 20 33 | 11 33 78 20 11 31 3d 20 |       3|.3x .1= |
|00003750| 35 0d 0a 00 20 20 20 20 | 20 20 20 20 20 20 20 20 |5...    |        |
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|00003770| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00003780| 20 20 20 20 20 11 33 78 | 20 11 31 3d 20 35 2f 33 |     .3x| .1= 5/3|
|00003790| 0d 0a 00 00 17 4f 03 0d | 0b 00 47 6f 6f 64 2e 20 |.....O..|..Good. |
|000037a0| 20 54 72 79 20 63 68 65 | 63 6b 69 6e 67 20 79 6f | Try che|cking yo|
|000037b0| 75 72 20 61 6e 73 77 65 | 72 20 69 6e 20 74 68 65 |ur answe|r in the|
|000037c0| 20 6f 72 69 67 69 6e 61 | 6c 20 65 71 75 61 74 69 | origina|l equati|
|000037d0| 6f 6e 2e 0d 0a 00 00 17 | 4f 03 0d 0b 00 53 6f 72 |on......|O....Sor|
|000037e0| 72 79 2c 20 74 68 69 73 | 20 65 71 75 61 74 69 6f |ry, this| equatio|
|000037f0| 6e 20 64 6f 65 73 20 68 | 61 76 65 20 61 20 73 6f |n does h|ave a so|
|00003800| 6c 75 74 69 6f 6e 2e 20 | 20 54 72 79 20 61 67 61 |lution. | Try aga|
|00003810| 69 6e 2e 0d 0a 00 00 17 | 4f 03 0d 0b 00 52 69 67 |in......|O....Rig|
|00003820| 68 74 2e 20 20 43 68 65 | 63 6b 20 74 68 69 73 20 |ht.  Che|ck this |
|00003830| 73 6f 6c 75 74 69 6f 6e | 20 69 6e 20 74 68 65 20 |solution| in the |
|00003840| 6f 72 69 67 69 6e 61 6c | 20 65 71 75 61 74 69 6f |original| equatio|
|00003850| 6e 2e 0d 0a 00 00 17 4f | 0a 0d 0b 00 57 72 6f 6e |n......O|....Wron|
|00003860| 67 2e 20 20 57 65 20 73 | 6f 6c 76 65 20 66 6f 72 |g.  We s|olve for|
|00003870| 20 11 33 78 20 11 31 61 | 73 20 66 6f 6c 6c 6f 77 | .3x .1a|s follow|
|00003880| 73 2e 12 30 0d 0a 00 20 | 20 20 20 20 20 20 20 20 |s..0... |        |
|00003890| 20 20 20 11 32 32 20 20 | 20 20 20 20 20 20 20 20 |   .22  |        |
|000038a0| 32 0d 0b 00 20 20 20 20 | 20 11 31 28 11 33 78 20 |2...    | .1(.3x |
|000038b0| 11 31 2b 20 32 29 20 20 | 3d 20 28 11 33 78 20 11 |.1+ 2)  |= (.3x .|
|000038c0| 31 2b 20 34 29 20 20 2d | 20 31 32 20 20 20 20 20 |1+ 4)  -| 12     |
|000038d0| 20 20 20 20 20 20 20 20 | 20 20 11 32 12 31 47 69 |        |  .2.1Gi|
|000038e0| 76 65 6e 20 65 71 75 61 | 74 69 6f 6e 12 30 0d 0a |ven equa|tion.0..|
|000038f0| 00 20 20 20 20 32 20 20 | 20 20 20 20 20 20 20 20 |.    2  |        |
|00003900| 20 20 32 0d 0b 00 20 20 | 20 11 33 78 20 11 31 2b |  2...  | .3x .1+|
|00003910| 20 34 11 33 78 20 11 31 | 2b 20 34 20 3d 20 11 33 | 4.3x .1|+ 4 = .3|
|00003920| 78 20 20 11 31 2b 20 38 | 11 33 78 20 11 31 2b 20 |x  .1+ 8|.3x .1+ |
|00003930| 31 36 20 2d 20 31 32 20 | 20 20 20 20 20 20 20 20 |16 - 12 |        |
|00003940| 20 20 11 32 12 31 42 69 | 6e 6f 6d 69 61 6c 20 73 |  .2.1Bi|nomial s|
|00003950| 71 75 61 72 65 64 12 30 | 0d 0a 00 20 20 20 20 20 |quared.0|...     |
|00003960| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00003970| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00003980| 20 20 20 20 20 20 20 12 | 31 20 20 20 20 20 20 20 |       .|1       |
|00003990| 20 20 20 32 12 30 0d 0b | 00 20 20 20 20 20 20 20 |   2.0..|.       |
|000039a0| 11 31 34 11 33 78 20 11 | 31 2b 20 34 20 3d 20 38 |.14.3x .|1+ 4 = 8|
|000039b0| 11 33 78 20 11 31 2b 20 | 34 20 20 20 20 20 20 20 |.3x .1+ |4       |
|000039c0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 11 |        |       .|
|000039d0| 32 12 31 53 75 62 74 72 | 61 63 74 20 78 20 20 20 |2.1Subtr|act x   |
|000039e0| 20 12 30 0d 0a 00 0d 0b | 00 20 20 20 20 20 20 20 | .0.....|.       |
|000039f0| 20 20 20 11 31 2d 34 11 | 33 78 20 11 31 3d 20 30 |   .1-4.|3x .1= 0|
|00003a00| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00003a10| 20 20 20 20 20 20 20 20 | 20 20 20 11 32 12 31 53 |        |   .2.1S|
|00003a20| 75 62 74 72 61 63 74 20 | 34 20 61 6e 64 20 38 78 |ubtract |4 and 8x|
|00003a30| 12 30 0d 0a 00 0d 0b 00 | 20 20 20 20 20 20 20 20 |.0......|        |
|00003a40| 20 20 20 20 11 33 78 20 | 11 31 3d 20 30 20 20 20 |    .3x |.1= 0   |
|00003a50| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00003a60| 20 20 20 20 20 20 20 20 | 11 32 12 31 53 6f 6c 75 |        |.2.1Solu|
|00003a70| 74 69 6f 6e 12 30 0d 0a | 00 00 17 4f 0a 0d 0b 00 |tion.0..|...O....|
|00003a80| 57 72 6f 6e 67 2e 20 20 | 57 65 20 73 6f 6c 76 65 |Wrong.  |We solve|
|00003a90| 20 66 6f 72 20 11 33 78 | 20 11 31 61 73 20 66 6f | for .3x| .1as fo|
|00003aa0| 6c 6c 6f 77 73 2e 12 30 | 0d 0a 00 20 20 20 20 20 |llows..0|...     |
|00003ab0| 20 20 20 20 20 20 20 11 | 32 32 20 20 20 20 20 20 |       .|22      |
|00003ac0| 20 20 20 20 32 0d 0b 00 | 20 20 20 20 20 11 31 28 |    2...|     .1(|
|00003ad0| 11 33 78 20 11 31 2b 20 | 32 29 20 20 3d 20 28 11 |.3x .1+ |2)  = (.|
|00003ae0| 33 78 20 11 31 2b 20 34 | 29 20 20 2d 20 31 32 20 |3x .1+ 4|)  - 12 |
|00003af0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 11 32 |        |      .2|
|00003b00| 12 31 47 69 76 65 6e 20 | 65 71 75 61 74 69 6f 6e |.1Given |equation|
|00003b10| 12 30 0d 0a 00 20 20 20 | 20 32 20 20 20 20 20 20 |.0...   | 2      |
|00003b20| 20 20 20 20 20 20 32 0d | 0b 00 20 20 20 11 33 78 |      2.|..   .3x|
|00003b30| 20 11 31 2b 20 34 11 33 | 78 20 11 31 2b 20 34 20 | .1+ 4.3|x .1+ 4 |
|00003b40| 3d 20 11 33 78 20 20 11 | 31 2b 20 38 11 33 78 20 |= .3x  .|1+ 8.3x |
|00003b50| 11 31 2b 20 31 36 20 2d | 20 31 32 20 20 20 20 20 |.1+ 16 -| 12     |
|00003b60| 20 20 20 20 20 20 11 32 | 12 31 42 69 6e 6f 6d 69 |      .2|.1Binomi|
|00003b70| 61 6c 20 73 71 75 61 72 | 65 64 12 30 0d 0a 00 20 |al squar|ed.0... |
|00003b80| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00003b90| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00003ba0| 20 20 20 20 20 20 20 20 | 20 20 20 12 31 20 20 20 |        |   .1   |
|00003bb0| 20 20 20 20 20 20 20 32 | 12 30 0d 0b 00 20 20 20 |       2|.0...   |
|00003bc0| 20 20 20 20 11 31 34 11 | 33 78 20 11 31 2b 20 34 |    .14.|3x .1+ 4|
|00003bd0| 20 3d 20 38 11 33 78 20 | 11 31 2b 20 34 20 20 20 | = 8.3x |.1+ 4   |
|00003be0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00003bf0| 20 20 20 11 32 12 31 53 | 75 62 74 72 61 63 74 20 |   .2.1S|ubtract |
|00003c00| 78 20 20 20 20 12 30 0d | 0a 00 0d 0b 00 20 20 20 |x    .0.|.....   |
|00003c10| 20 20 20 20 20 20 20 11 | 31 2d 34 11 33 78 20 11 |       .|1-4.3x .|
|00003c20| 31 3d 20 30 20 20 20 20 | 20 20 20 20 20 20 20 20 |1= 0    |        |
|00003c30| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 11 |        |       .|
|00003c40| 32 12 31 53 75 62 74 72 | 61 63 74 20 34 20 61 6e |2.1Subtr|act 4 an|
|00003c50| 64 20 38 78 12 30 0d 0a | 00 0d 0b 00 20 20 20 20 |d 8x.0..|....    |
|00003c60| 20 20 20 20 20 20 20 20 | 11 33 78 20 11 31 3d 20 |        |.3x .1= |
|00003c70| 30 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |0       |        |
|00003c80| 20 20 20 20 20 20 20 20 | 20 20 20 20 11 32 12 31 |        |    .2.1|
|00003c90| 53 6f 6c 75 74 69 6f 6e | 12 30 0d 0a 00 00 17 4f |Solution|.0.....O|
|00003ca0| 03 0d 0b 00 53 6f 72 72 | 79 2c 20 74 68 69 73 20 |....Sorr|y, this |
|00003cb0| 65 71 75 61 74 69 6f 6e | 20 64 6f 65 73 20 68 61 |equation| does ha|
|00003cc0| 76 65 20 61 20 73 6f 6c | 75 74 69 6f 6e 2e 20 20 |ve a sol|ution.  |
|00003cd0| 54 72 79 20 61 67 61 69 | 6e 2e 0d 0a 00 00 17 4f |Try agai|n......O|
|00003ce0| 07 0d 0b 00 57 72 6f 6e | 67 2e 20 20 57 69 74 68 |....Wron|g.  With|
|00003cf0| 20 74 68 65 20 61 69 64 | 20 6f 66 20 61 20 63 61 | the aid| of a ca|
|00003d00| 6c 63 75 6c 61 74 6f 72 | 2c 20 77 65 20 73 6f 6c |lculator|, we sol|
|00003d10| 76 65 20 61 73 20 66 6f | 6c 6c 6f 77 73 2e 12 30 |ve as fo|llows..0|
|00003d20| 0d 0a 00 0d 0b 00 20 20 | 20 20 20 20 30 2e 33 37 |......  |    0.37|
|00003d30| 35 11 33 78 20 11 31 2b | 20 30 2e 38 32 35 28 33 |5.3x .1+| 0.825(3|
|00003d40| 30 30 20 2d 20 11 33 78 | 11 31 29 20 3d 20 32 30 |00 - .3x|.1) = 20|
|00003d50| 30 0d 0a 00 20 20 20 20 | 20 20 30 2e 33 37 35 11 |0...    |  0.375.|
|00003d60| 33 78 20 11 31 2b 20 32 | 34 37 2e 35 20 2d 20 30 |3x .1+ 2|47.5 - 0|
|00003d70| 2e 38 32 35 11 33 78 20 | 11 31 3d 20 32 30 30 0d |.825.3x |.1= 200.|
|00003d80| 0a 00 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |..      |        |
|00003d90| 20 20 20 20 20 20 20 20 | 20 2d 30 2e 34 35 11 33 |        | -0.45.3|
|00003da0| 78 20 11 31 3d 20 2d 34 | 37 2e 35 0d 0a 00 20 20 |x .1= -4|7.5...  |
|00003db0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00003dc0| 20 20 20 20 20 20 20 20 | 20 20 11 33 78 20 11 34 |        |  .3x .4|
|00003dd0| 7e 20 11 31 31 30 35 2e | 35 35 36 20 20 20 20 20 |~ .1105.|556     |
|00003de0| 20 20 20 20 20 11 32 12 | 31 52 6f 75 6e 64 20 74 |     .2.|1Round t|
|00003df0| 6f 20 74 68 72 65 65 20 | 70 6c 61 63 65 73 12 30 |o three |places.0|
|00003e00| 0d 0a 00 00 17 4f 03 0d | 0b 00 52 69 67 68 74 2e |.....O..|..Right.|
|00003e10| 20 20 54 68 65 20 73 6f | 6c 75 74 69 6f 6e 20 74 |  The so|lution t|
|00003e20| 6f 20 74 68 72 65 65 20 | 64 65 63 69 6d 61 6c 20 |o three |decimal |
|00003e30| 70 6c 61 63 65 73 20 69 | 73 20 31 30 35 2e 35 35 |places i|s 105.55|
|00003e40| 36 2e 20 20 4e 6f 74 65 | 20 74 68 61 74 20 62 65 |6.  Note| that be|
|00003e50| 63 61 75 73 65 0d 0a 00 | 6f 66 20 72 6f 75 6e 64 |cause...|of round|
|00003e60| 2d 6f 66 66 20 65 72 72 | 6f 72 2c 20 61 20 63 68 |-off err|or, a ch|
|00003e70| 65 63 6b 20 6f 66 20 61 | 20 64 65 63 69 6d 61 6c |eck of a| decimal|
|00003e80| 20 73 6f 6c 75 74 69 6f | 6e 20 6d 61 79 20 6e 6f | solutio|n may no|
|00003e90| 74 20 79 69 65 6c 64 20 | 65 78 61 63 74 6c 79 20 |t yield |exactly |
|00003ea0| 74 68 65 0d 0a 00 73 61 | 6d 65 20 76 61 6c 75 65 |the...sa|me value|
|00003eb0| 73 20 66 6f 72 20 62 6f | 74 68 20 73 69 64 65 73 |s for bo|th sides|
|00003ec0| 20 6f 66 20 74 68 65 20 | 6f 72 69 67 69 6e 61 6c | of the |original|
|00003ed0| 20 65 71 75 61 74 69 6f | 6e 2e 20 20 54 68 65 20 | equatio|n.  The |
|00003ee0| 64 69 66 66 65 72 65 6e | 63 65 2c 20 0d 0a 00 68 |differen|ce, ...h|
|00003ef0| 6f 77 65 76 65 72 2c 20 | 77 69 6c 6c 20 75 73 75 |owever, |will usu|
|00003f00| 61 6c 6c 79 20 62 65 20 | 71 75 69 74 65 20 73 6d |ally be |quite sm|
|00003f10| 61 6c 6c 2e 0d 0a 00 00 | 17 4f 07 0d 0b 00 57 72 |all.....|.O....Wr|
|00003f20| 6f 6e 67 2e 20 20 57 69 | 74 68 20 74 68 65 20 61 |ong.  Wi|th the a|
|00003f30| 69 64 20 6f 66 20 61 20 | 63 61 6c 63 75 6c 61 74 |id of a |calculat|
|00003f40| 6f 72 2c 20 77 65 20 73 | 6f 6c 76 65 20 61 73 20 |or, we s|olve as |
|00003f50| 66 6f 6c 6c 6f 77 73 2e | 12 30 0d 0a 00 0d 0b 00 |follows.|.0......|
|00003f60| 20 20 20 20 20 20 30 2e | 33 37 35 11 33 78 20 11 |      0.|375.3x .|
|00003f70| 31 2b 20 30 2e 38 32 35 | 28 33 30 30 20 2d 20 11 |1+ 0.825|(300 - .|
|00003f80| 33 78 11 31 29 20 3d 20 | 32 30 30 0d 0a 00 20 20 |3x.1) = |200...  |
|00003f90| 20 20 20 20 30 2e 33 37 | 35 11 33 78 20 11 31 2b |    0.37|5.3x .1+|
|00003fa0| 20 32 34 37 2e 35 20 2d | 20 30 2e 38 32 35 11 33 | 247.5 -| 0.825.3|
|00003fb0| 78 20 11 31 3d 20 32 30 | 30 0d 0a 00 20 20 20 20 |x .1= 20|0...    |
|00003fc0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00003fd0| 20 20 20 2d 30 2e 34 35 | 11 33 78 20 11 31 3d 20 |   -0.45|.3x .1= |
|00003fe0| 2d 34 37 2e 35 0d 0a 00 | 20 20 20 20 20 20 20 20 |-47.5...|        |
|00003ff0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00004000| 20 20 20 20 11 33 78 20 | 11 34 7e 20 11 31 31 30 |    .3x |.4~ .110|
|00004010| 35 2e 35 35 36 20 20 20 | 20 20 20 20 20 20 20 11 |5.556   |       .|
|00004020| 32 12 31 52 6f 75 6e 64 | 20 74 6f 20 74 68 72 65 |2.1Round| to thre|
|00004030| 65 20 70 6c 61 63 65 73 | 12 30 0d 0a 00 00 17 4f |e places|.0.....O|
|00004040| 07 0d 0b 00 57 72 6f 6e | 67 2e 20 20 57 69 74 68 |....Wron|g.  With|
|00004050| 20 74 68 65 20 61 69 64 | 20 6f 66 20 61 20 63 61 | the aid| of a ca|
|00004060| 6c 63 75 6c 61 74 6f 72 | 2c 20 77 65 20 73 6f 6c |lculator|, we sol|
|00004070| 76 65 20 61 73 20 66 6f | 6c 6c 6f 77 73 2e 12 30 |ve as fo|llows..0|
|00004080| 0d 0a 00 0d 0b 00 20 20 | 20 20 20 20 30 2e 33 37 |......  |    0.37|
|00004090| 35 11 33 78 20 11 31 2b | 20 30 2e 38 32 35 28 33 |5.3x .1+| 0.825(3|
|000040a0| 30 30 20 2d 20 11 33 78 | 11 31 29 20 3d 20 32 30 |00 - .3x|.1) = 20|
|000040b0| 30 0d 0a 00 20 20 20 20 | 20 20 30 2e 33 37 35 11 |0...    |  0.375.|
|000040c0| 33 78 20 11 31 2b 20 32 | 34 37 2e 35 20 2d 20 30 |3x .1+ 2|47.5 - 0|
|000040d0| 2e 38 32 35 11 33 78 20 | 11 31 3d 20 32 30 30 0d |.825.3x |.1= 200.|
|000040e0| 0a 00 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |..      |        |
|000040f0| 20 20 20 20 20 20 20 20 | 20 2d 30 2e 34 35 11 33 |        | -0.45.3|
|00004100| 78 20 11 31 3d 20 2d 34 | 37 2e 35 0d 0a 00 20 20 |x .1= -4|7.5...  |
|00004110| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00004120| 20 20 20 20 20 20 20 20 | 20 20 11 33 78 20 11 34 |        |  .3x .4|
|00004130| 7e 20 11 31 31 30 35 2e | 35 35 36 20 20 20 20 20 |~ .1105.|556     |
|00004140| 20 20 20 20 20 11 32 12 | 31 52 6f 75 6e 64 20 74 |     .2.|1Round t|
|00004150| 6f 20 74 68 72 65 65 20 | 70 6c 61 63 65 73 12 30 |o three |places.0|
|00004160| 0d 0a 00 53 65 63 74 69 | 6f 6e 20 31 2e 32 20 20 |...Secti|on 1.2  |
|00004170| 4c 69 6e 65 61 72 20 45 | 71 75 61 74 69 6f 6e 73 |Linear E|quations|
|00004180| 0d 0a 00 0d 0a 00 0d 0b | 00 20 20 20 20 0e 78 31 |........|.    .x1|
|00004190| 2d 32 2d 31 0e 54 75 74 | 6f 72 69 61 6c 20 45 78 |-2-1.Tut|orial Ex|
|000041a0| 65 72 63 69 73 65 20 20 | 31 0f 0d 0a 00 0d 0b 00 |ercise  |1.......|
|000041b0| 20 20 20 20 0e 78 31 2d | 32 2d 32 0e 54 75 74 6f |    .x1-|2-2.Tuto|
|000041c0| 72 69 61 6c 20 45 78 65 | 72 63 69 73 65 20 20 32 |rial Exe|rcise  2|
|000041d0| 0f 20 20 20 20 0e 78 31 | 2d 32 2d 33 0e 54 75 74 |.    .x1|-2-3.Tut|
|000041e0| 6f 72 69 61 6c 20 45 78 | 65 72 63 69 73 65 20 20 |orial Ex|ercise  |
|000041f0| 33 0f 20 20 20 20 0e 78 | 31 2d 32 2d 34 0e 54 75 |3.    .x|1-2-4.Tu|
|00004200| 74 6f 72 69 61 6c 20 45 | 78 65 72 63 69 73 65 20 |torial E|xercise |
|00004210| 20 34 0f 20 20 20 20 0e | 78 31 2d 32 2d 35 0e 54 | 4.    .|x1-2-5.T|
|00004220| 75 74 6f 72 69 61 6c 20 | 45 78 65 72 63 69 73 65 |utorial |Exercise|
|00004230| 20 20 35 0f 0d 0a 00 0d | 0b 00 20 20 20 20 0e 78 |  5.....|..    .x|
|00004240| 31 2d 32 2d 36 0e 54 75 | 74 6f 72 69 61 6c 20 45 |1-2-6.Tu|torial E|
|00004250| 78 65 72 63 69 73 65 20 | 20 36 0f 20 20 20 20 0e |xercise | 6.    .|
|00004260| 78 31 2d 32 2d 37 0e 54 | 75 74 6f 72 69 61 6c 20 |x1-2-7.T|utorial |
|00004270| 45 78 65 72 63 69 73 65 | 20 20 37 0f 20 20 20 20 |Exercise|  7.    |
|00004280| 0e 78 31 2d 32 2d 38 0e | 54 75 74 6f 72 69 61 6c |.x1-2-8.|Tutorial|
|00004290| 20 45 78 65 72 63 69 73 | 65 20 20 38 0f 20 20 20 | Exercis|e  8.   |
|000042a0| 20 0e 78 31 2d 32 2d 39 | 0e 54 75 74 6f 72 69 61 | .x1-2-9|.Tutoria|
|000042b0| 6c 20 45 78 65 72 63 69 | 73 65 20 20 39 0f 0d 0a |l Exerci|se  9...|
|000042c0| 00 0d 0b 00 20 20 20 20 | 0e 78 31 2d 32 2d 31 30 |....    |.x1-2-10|
|000042d0| 0e 54 75 74 6f 72 69 61 | 6c 20 45 78 65 72 63 69 |.Tutoria|l Exerci|
|000042e0| 73 65 20 31 30 0f 0d 0a | 00 0d 0b 00 20 20 20 20 |se 10...|....    |
|000042f0| 0e 78 31 2d 32 2d 31 31 | 0e 54 75 74 6f 72 69 61 |.x1-2-11|.Tutoria|
|00004300| 6c 20 45 78 65 72 63 69 | 73 65 20 31 31 0f 0d 0a |l Exerci|se 11...|
|00004310| 00 0d 0b 00 20 20 20 20 | 0e 78 31 2d 32 2d 31 32 |....    |.x1-2-12|
|00004320| 0e 54 75 74 6f 72 69 61 | 6c 20 45 78 65 72 63 69 |.Tutoria|l Exerci|
|00004330| 73 65 20 31 32 0f 0d 0a | 00 0d 0b 00 20 20 20 20 |se 12...|....    |
|00004340| 0e 78 31 2d 32 2d 31 33 | 0e 54 75 74 6f 72 69 61 |.x1-2-13|.Tutoria|
|00004350| 6c 20 45 78 65 72 63 69 | 73 65 20 31 33 0f 0d 0a |l Exerci|se 13...|
|00004360| 00 53 65 63 74 69 6f 6e | 20 31 2e 32 20 20 4c 69 |.Section| 1.2  Li|
|00004370| 6e 65 61 72 20 45 71 75 | 61 74 69 6f 6e 73 0d 0b |near Equ|ations..|
|00004380| 00 20 31 2e 20 20 44 65 | 74 65 72 6d 69 6e 65 20 |. 1.  De|termine |
|00004390| 77 68 65 74 68 65 72 20 | 65 61 63 68 20 6f 66 20 |whether |each of |
|000043a0| 74 68 65 20 66 6f 6c 6c | 6f 77 69 6e 67 20 65 71 |the foll|owing eq|
|000043b0| 75 61 74 69 6f 6e 73 20 | 69 73 20 61 6e 20 69 64 |uations |is an id|
|000043c0| 65 6e 74 69 74 79 20 6f | 72 20 61 20 0d 0a 00 20 |entity o|r a ... |
|000043d0| 20 20 20 20 63 6f 6e 64 | 69 74 69 6f 6e 61 6c 20 |    cond|itional |
|000043e0| 65 71 75 61 74 69 6f 6e | 2e 0d 0a 00 0d 0b 00 20 |equation|....... |
|000043f0| 20 20 20 20 49 2e 20 32 | 28 11 33 78 20 11 31 2d |    I. 2|(.3x .1-|
|00004400| 20 31 29 20 3d 20 32 11 | 33 78 20 11 31 2d 20 32 | 1) = 2.|3x .1- 2|
|00004410| 20 20 20 49 49 2e 20 32 | 28 11 33 78 20 11 31 2d |   II. 2|(.3x .1-|
|00004420| 20 31 29 20 3d 20 33 11 | 33 78 20 11 31 2b 20 34 | 1) = 3.|3x .1+ 4|
|00004430| 20 20 20 49 49 49 2e 20 | 32 28 11 33 78 20 11 31 |   III. |2(.3x .1|
|00004440| 2b 20 31 29 20 3d 20 32 | 11 33 78 20 11 31 2b 20 |+ 1) = 2|.3x .1+ |
|00004450| 31 0d 0a 00 0d 0a 00 20 | 20 20 20 20 0e 32 2d 31 |1...... |    .2-1|
|00004460| 2d 31 61 0e 28 61 29 0f | 20 41 6c 6c 20 74 68 72 |-1a.(a).| All thr|
|00004470| 65 65 20 61 72 65 20 69 | 64 65 6e 74 69 74 69 65 |ee are i|dentitie|
|00004480| 73 2e 0d 0a 00 0d 0b 00 | 20 20 20 20 20 0e 32 2d |s.......|     .2-|
|00004490| 31 2d 31 62 0e 28 62 29 | 0f 20 41 6c 6c 20 74 68 |1-1b.(b)|. All th|
|000044a0| 72 65 65 20 61 72 65 20 | 63 6f 6e 64 69 74 69 6f |ree are |conditio|
|000044b0| 6e 61 6c 20 65 71 75 61 | 74 69 6f 6e 73 2e 0d 0a |nal equa|tions...|
|000044c0| 00 0d 0b 00 20 20 20 20 | 20 0e 32 2d 31 2d 31 63 |....    | .2-1-1c|
|000044d0| 0e 28 63 29 0f 20 49 20 | 61 6e 64 20 49 49 49 20 |.(c). I |and III |
|000044e0| 61 72 65 20 69 64 65 6e | 74 69 74 69 65 73 2c 20 |are iden|tities, |
|000044f0| 49 49 20 69 73 20 61 20 | 63 6f 6e 64 69 74 69 6f |II is a |conditio|
|00004500| 6e 61 6c 20 65 71 75 61 | 74 69 6f 6e 2e 0d 0a 00 |nal equa|tion....|
|00004510| 0d 0b 00 20 20 20 20 20 | 0e 32 2d 31 2d 31 64 0e |...     |.2-1-1d.|
|00004520| 28 64 29 0f 20 49 20 69 | 73 20 61 6e 20 69 64 65 |(d). I i|s an ide|
|00004530| 6e 74 69 74 79 2c 20 49 | 49 20 61 6e 64 20 49 49 |ntity, I|I and II|
|00004540| 49 20 61 72 65 20 63 6f | 6e 64 69 74 69 6f 6e 61 |I are co|nditiona|
|00004550| 6c 20 65 71 75 61 74 69 | 6f 6e 73 2e 0d 0a 00 0d |l equati|ons.....|
|00004560| 0a 00 59 6f 75 20 6d 61 | 79 20 77 61 6e 74 20 74 |..You ma|y want t|
|00004570| 6f 20 72 65 76 69 65 77 | 20 10 31 2d 32 2d 32 0e |o review| .1-2-2.|
|00004580| 65 31 2d 32 2d 31 0e 47 | 75 69 64 65 64 20 45 78 |e1-2-1.G|uided Ex|
|00004590| 61 6d 70 6c 65 20 31 2e | 0f 0d 0a 00 53 65 63 74 |ample 1.|....Sect|
|000045a0| 69 6f 6e 20 31 2e 32 20 | 20 4c 69 6e 65 61 72 20 |ion 1.2 | Linear |
|000045b0| 45 71 75 61 74 69 6f 6e | 73 0d 0b 00 31 30 2e 20 |Equation|s...10. |
|000045c0| 20 44 65 74 65 72 6d 69 | 6e 65 20 77 68 69 63 68 | Determi|ne which|
|000045d0| 20 6f 6e 65 20 6f 66 20 | 74 68 65 20 66 6f 6c 6c | one of |the foll|
|000045e0| 6f 77 69 6e 67 20 76 61 | 6c 75 65 73 20 6f 66 20 |owing va|lues of |
|000045f0| 11 33 78 20 11 31 69 73 | 20 61 20 73 6f 6c 75 74 |.3x .1is| a solut|
|00004600| 69 6f 6e 20 74 6f 20 74 | 68 65 0d 0a 00 20 20 20 |ion to t|he...   |
|00004610| 20 20 20 20 20 20 20 20 | 20 20 20 11 33 78 20 20 |        |   .3x  |
|00004620| 20 20 11 31 33 0d 0b 00 | 20 20 20 20 20 65 71 75 |  .13...|     equ|
|00004630| 61 74 69 6f 6e 20 11 34 | 32 20 11 31 2b 20 11 34 |ation .4|2 .1+ .4|
|00004640| 32 32 20 11 31 3d 20 31 | 2e 0d 0b 00 20 20 20 20 |22 .1= 1|....    |
|00004650| 20 20 20 20 20 20 20 20 | 20 20 34 20 20 20 34 11 |        |  4   4.|
|00004660| 33 78 0d 0a 00 0d 0a 00 | 20 20 20 20 20 11 31 0e |3x......|     .1.|
|00004670| 32 2d 31 2d 31 35 61 0e | 28 61 29 0f 20 11 33 78 |2-1-15a.|(a). .3x|
|00004680| 20 11 31 3d 20 2d 31 20 | 20 20 20 20 20 20 20 20 | .1= -1 |        |
|00004690| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 0e 32 |        |      .2|
|000046a0| 2d 31 2d 31 35 62 0e 28 | 62 29 0f 20 11 33 78 20 |-1-15b.(|b). .3x |
|000046b0| 11 31 3d 20 32 0d 0a 00 | 0d 0b 00 20 20 20 20 20 |.1= 2...|...     |
|000046c0| 0e 32 2d 31 2d 31 35 63 | 0e 28 63 29 0f 20 11 33 |.2-1-15c|.(c). .3|
|000046d0| 78 20 11 31 3d 20 33 20 | 20 20 20 20 20 20 20 20 |x .1= 3 |        |
|000046e0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 0e |        |       .|
|000046f0| 32 2d 31 2d 31 35 64 0e | 28 64 29 0f 20 11 33 78 |2-1-15d.|(d). .3x|
|00004700| 20 11 31 3d 20 34 0d 0a | 00 0d 0a 00 59 6f 75 20 | .1= 4..|....You |
|00004710| 6d 61 79 20 77 61 6e 74 | 20 74 6f 20 72 65 76 69 |may want| to revi|
|00004720| 65 77 20 10 31 2d 32 2d | 32 0e 69 31 2d 32 2d 31 |ew .1-2-|2.i1-2-1|
|00004730| 0e 49 6e 74 65 67 72 61 | 74 65 64 20 45 78 61 6d |.Integra|ted Exam|
|00004740| 70 6c 65 20 31 2e 0f 0d | 0a 00 53 65 63 74 69 6f |ple 1...|..Sectio|
|00004750| 6e 20 31 2e 32 20 20 4c | 69 6e 65 61 72 20 45 71 |n 1.2  L|inear Eq|
|00004760| 75 61 74 69 6f 6e 73 0d | 0b 00 31 31 2e 20 20 53 |uations.|..11.  S|
|00004770| 6f 6c 76 65 3a 20 30 2e | 33 11 33 78 20 11 31 2b |olve: 0.|3.3x .1+|
|00004780| 20 31 2e 35 20 3d 20 38 | 2e 34 20 2e 0d 0a 00 0d | 1.5 = 8|.4 .....|
|00004790| 0a 00 20 20 20 20 20 0e | 32 2d 31 2d 33 33 61 0e |..     .|2-1-33a.|
|000047a0| 28 61 29 0f 20 32 33 2f | 31 30 20 20 20 20 20 20 |(a). 23/|10      |
|000047b0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|000047c0| 20 20 0e 32 2d 31 2d 33 | 33 62 0e 28 62 29 0f 20 |  .2-1-3|3b.(b). |
|000047d0| 31 33 0d 0a 00 0d 0b 00 | 20 20 20 20 20 0e 32 2d |13......|     .2-|
|000047e0| 31 2d 33 33 63 0e 28 63 | 29 0f 20 32 33 20 20 20 |1-33c.(c|). 23   |
|000047f0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00004800| 20 20 20 20 20 20 20 20 | 0e 32 2d 31 2d 33 33 64 |        |.2-1-33d|
|00004810| 0e 28 64 29 0f 20 4e 6f | 74 20 70 6f 73 73 69 62 |.(d). No|t possib|
|00004820| 6c 65 0d 0a 00 0d 0a 00 | 59 6f 75 20 6d 61 79 20 |le......|You may |
|00004830| 77 61 6e 74 20 74 6f 20 | 72 65 76 69 65 77 20 10 |want to |review .|
|00004840| 31 2d 32 2d 32 0e 69 31 | 2d 32 2d 32 0e 49 6e 74 |1-2-2.i1|-2-2.Int|
|00004850| 65 67 72 61 74 65 64 20 | 45 78 61 6d 70 6c 65 20 |egrated |Example |
|00004860| 32 2e 0f 0d 0a 00 53 65 | 63 74 69 6f 6e 20 31 2e |2.....Se|ction 1.|
|00004870| 32 20 20 4c 69 6e 65 61 | 72 20 45 71 75 61 74 69 |2  Linea|r Equati|
|00004880| 6f 6e 73 0d 0b 00 20 20 | 20 20 20 20 20 20 20 20 |ons...  |        |
|00004890| 20 20 20 20 20 20 20 20 | 20 11 32 32 20 20 20 20 |        | .22    |
|000048a0| 20 20 20 20 20 20 32 0d | 0b 00 11 31 31 32 2e 20 |      2.|...112. |
|000048b0| 20 53 6f 6c 76 65 3a 20 | 28 11 33 78 20 11 31 2b | Solve: |(.3x .1+|
|000048c0| 20 32 29 20 20 3d 20 28 | 11 33 78 20 11 31 2b 20 | 2)  = (|.3x .1+ |
|000048d0| 34 29 20 20 2d 20 31 32 | 0d 0a 00 0d 0a 00 20 20 |4)  - 12|......  |
|000048e0| 20 20 20 0e 61 32 2d 31 | 2d 35 33 61 0e 28 61 29 |   .a2-1|-53a.(a)|
|000048f0| 0f 20 30 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |. 0     |        |
|00004900| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 0e |        |       .|
|00004910| 61 32 2d 31 2d 35 33 62 | 0e 28 62 29 0f 20 2d 32 |a2-1-53b|.(b). -2|
|00004920| 0d 0a 00 0d 0b 00 20 20 | 20 20 20 0e 61 32 2d 31 |......  |   .a2-1|
|00004930| 2d 35 33 63 0e 28 63 29 | 0f 20 2d 33 2f 34 20 20 |-53c.(c)|. -3/4  |
|00004940| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00004950| 20 20 20 20 20 20 20 0e | 61 32 2d 31 2d 35 33 64 |       .|a2-1-53d|
|00004960| 0e 28 64 29 0f 20 4e 6f | 20 73 6f 6c 75 74 69 6f |.(d). No| solutio|
|00004970| 6e 0d 0a 00 0d 0a 00 59 | 6f 75 20 6d 61 79 20 77 |n......Y|ou may w|
|00004980| 61 6e 74 20 74 6f 20 72 | 65 76 69 65 77 20 10 31 |ant to r|eview .1|
|00004990| 2d 32 2d 32 0e 69 31 2d | 32 2d 33 0e 49 6e 74 65 |-2-2.i1-|2-3.Inte|
|000049a0| 67 72 61 74 65 64 20 45 | 78 61 6d 70 6c 65 20 33 |grated E|xample 3|
|000049b0| 2e 0f 0d 0a 00 53 65 63 | 74 69 6f 6e 20 31 2e 32 |.....Sec|tion 1.2|
|000049c0| 20 20 4c 69 6e 65 61 72 | 20 45 71 75 61 74 69 6f |  Linear| Equatio|
|000049d0| 6e 73 0d 0b 00 31 33 2e | 20 20 55 73 65 20 79 6f |ns...13.|  Use yo|
|000049e0| 75 72 20 63 61 6c 63 75 | 6c 61 74 6f 72 20 74 6f |ur calcu|lator to|
|000049f0| 20 73 6f 6c 76 65 20 30 | 2e 33 37 35 11 33 78 20 | solve 0|.375.3x |
|00004a00| 11 31 2b 20 30 2e 38 32 | 35 28 33 30 30 20 2d 20 |.1+ 0.82|5(300 - |
|00004a10| 11 33 78 11 31 29 20 3d | 20 32 30 30 20 66 6f 72 |.3x.1) =| 200 for|
|00004a20| 20 11 33 78 20 11 31 61 | 6e 64 20 0d 0a 00 20 20 | .3x .1a|nd ...  |
|00004a30| 20 20 20 72 6f 75 6e 64 | 20 79 6f 75 72 20 61 6e |   round| your an|
|00004a40| 73 77 65 72 20 74 6f 20 | 74 68 72 65 65 20 64 65 |swer to |three de|
|00004a50| 63 69 6d 61 6c 20 70 6c | 61 63 65 73 2e 0d 0a 00 |cimal pl|aces....|
|00004a60| 0d 0a 00 20 20 20 20 20 | 0e 61 32 2d 31 2d 36 31 |...     |.a2-1-61|
|00004a70| 61 0e 28 61 29 0f 20 37 | 36 2e 30 30 30 20 20 20 |a.(a). 7|6.000   |
|00004a80| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00004a90| 20 20 20 20 0e 61 32 2d | 31 2d 36 31 62 0e 28 62 |    .a2-|1-61b.(b|
|00004aa0| 29 0f 20 31 30 35 2e 35 | 35 36 0d 0a 00 0d 0b 00 |). 105.5|56......|
|00004ab0| 20 20 20 20 20 0e 61 32 | 2d 31 2d 36 31 63 0e 28 |     .a2|-1-61c.(|
|00004ac0| 63 29 0f 20 2d 39 39 34 | 2e 34 34 34 20 20 20 20 |c). -994|.444    |
|00004ad0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00004ae0| 20 0e 61 32 2d 31 2d 36 | 31 64 0e 28 64 29 0f 20 | .a2-1-6|1d.(d). |
|00004af0| 4e 6f 20 73 6f 6c 75 74 | 69 6f 6e 0d 0a 00 0d 0a |No solut|ion.....|
|00004b00| 00 59 6f 75 20 6d 61 79 | 20 77 61 6e 74 20 74 6f |.You may| want to|
|00004b10| 20 72 65 76 69 65 77 20 | 10 31 2d 32 2d 32 0e 69 | review |.1-2-2.i|
|00004b20| 31 2d 32 2d 34 0e 49 6e | 74 65 67 72 61 74 65 64 |1-2-4.In|tegrated|
|00004b30| 20 45 78 61 6d 70 6c 65 | 20 34 2e 0f 0d 0a 00 53 | Example| 4.....S|
|00004b40| 65 63 74 69 6f 6e 20 31 | 2e 32 20 20 4c 69 6e 65 |ection 1|.2  Line|
|00004b50| 61 72 20 45 71 75 61 74 | 69 6f 6e 73 0d 0b 00 20 |ar Equat|ions... |
|00004b60| 32 2e 20 20 53 6f 6c 76 | 65 3a 20 32 33 11 33 78 |2.  Solv|e: 23.3x|
|00004b70| 20 11 31 2d 20 34 20 3d | 20 34 32 2e 0d 0a 00 0d | .1- 4 =| 42.....|
|00004b80| 0a 00 20 20 20 20 20 0e | 32 2d 31 2d 32 31 61 0e |..     .|2-1-21a.|
|00004b90| 28 61 29 0f 20 32 20 20 | 20 20 20 20 20 20 20 20 |(a). 2  |        |
|00004ba0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00004bb0| 20 20 0e 32 2d 31 2d 32 | 31 62 0e 28 62 29 0f 20 |  .2-1-2|1b.(b). |
|00004bc0| 34 38 0d 0a 00 0d 0b 00 | 20 20 20 20 20 20 20 20 |48......|        |
|00004bd0| 20 33 38 20 20 20 20 20 | 20 20 20 20 20 20 20 20 | 38     |        |
|00004be0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00004bf0| 20 20 34 32 0d 0b 00 20 | 20 20 20 20 0e 32 2d 31 |  42... |    .2-1|
|00004c00| 2d 32 31 63 0e 28 63 29 | 0f 20 11 34 32 32 20 20 |-21c.(c)|. .422  |
|00004c10| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00004c20| 20 20 20 20 20 20 20 20 | 20 11 31 0e 32 2d 31 2d |        | .1.2-1-|
|00004c30| 32 31 64 0e 28 64 29 0f | 20 11 34 32 32 0d 0b 00 |21d.(d).| .422...|
|00004c40| 20 20 20 20 20 20 20 20 | 20 11 31 32 33 20 20 20 |        | .123   |
|00004c50| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00004c60| 20 20 20 20 20 20 20 20 | 20 20 20 20 32 33 0d 0a |        |    23..|
|00004c70| 00 0d 0a 00 59 6f 75 20 | 6d 61 79 20 77 61 6e 74 |....You |may want|
|00004c80| 20 74 6f 20 72 65 76 69 | 65 77 20 10 31 2d 32 2d | to revi|ew .1-2-|
|00004c90| 32 0e 65 31 2d 32 2d 32 | 0e 47 75 69 64 65 64 20 |2.e1-2-2|.Guided |
|00004ca0| 45 78 61 6d 70 6c 65 20 | 32 2e 0f 0d 0a 00 53 65 |Example |2.....Se|
|00004cb0| 63 74 69 6f 6e 20 31 2e | 32 20 20 4c 69 6e 65 61 |ction 1.|2  Linea|
|00004cc0| 72 20 45 71 75 61 74 69 | 6f 6e 73 0d 0b 00 20 33 |r Equati|ons... 3|
|00004cd0| 2e 20 20 53 6f 6c 76 65 | 3a 20 33 28 11 33 78 20 |.  Solve|: 3(.3x |
|00004ce0| 11 31 2b 20 32 29 20 2d | 20 39 20 3d 20 34 28 11 |.1+ 2) -| 9 = 4(.|
|00004cf0| 33 78 20 11 31 2d 20 33 | 29 2e 0d 0a 00 0d 0a 00 |3x .1- 3|).......|
|00004d00| 20 20 20 20 20 0e 61 32 | 2d 31 2d 32 35 61 0e 28 |     .a2|-1-25a.(|
|00004d10| 61 29 0f 20 34 20 20 20 | 20 20 20 20 20 20 20 20 |a). 4   |        |
|00004d20| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00004d30| 20 0e 61 32 2d 31 2d 32 | 35 62 0e 28 62 29 0f 20 | .a2-1-2|5b.(b). |
|00004d40| 2d 34 0d 0a 00 0d 0b 00 | 20 20 20 20 20 0e 61 32 |-4......|     .a2|
|00004d50| 2d 31 2d 32 35 63 0e 28 | 63 29 0f 20 39 20 20 20 |-1-25c.(|c). 9   |
|00004d60| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00004d70| 20 20 20 20 20 20 20 20 | 20 0e 61 32 2d 31 2d 32 |        | .a2-1-2|
|00004d80| 35 64 0e 28 64 29 0f 20 | 4e 6f 20 73 6f 6c 75 74 |5d.(d). |No solut|
|00004d90| 69 6f 6e 0d 0a 00 0d 0a | 00 59 6f 75 20 6d 61 79 |ion.....|.You may|
|00004da0| 20 77 61 6e 74 20 74 6f | 20 72 65 76 69 65 77 20 | want to| review |
|00004db0| 10 31 2d 32 2d 32 0e 65 | 31 2d 32 2d 32 0e 47 75 |.1-2-2.e|1-2-2.Gu|
|00004dc0| 69 64 65 64 20 45 78 61 | 6d 70 6c 65 20 32 2e 0f |ided Exa|mple 2..|
|00004dd0| 0d 0a 00 53 65 63 74 69 | 6f 6e 20 31 2e 32 20 20 |...Secti|on 1.2  |
|00004de0| 4c 69 6e 65 61 72 20 45 | 71 75 61 74 69 6f 6e 73 |Linear E|quations|
|00004df0| 0d 0b 00 20 20 20 20 20 | 20 20 20 20 20 20 20 37 |...     |       7|
|00004e00| 11 33 78 20 20 20 11 31 | 31 20 20 20 20 20 20 20 |.3x   .1|1       |
|00004e10| 32 0d 0b 00 20 34 2e 20 | 20 53 6f 6c 76 65 3a 20 |2... 4. | Solve: |
|00004e20| 11 34 32 32 20 11 31 2b | 20 11 34 32 20 11 31 3d |.422 .1+| .42 .1=|
|00004e30| 20 11 33 78 20 11 31 2d | 20 11 34 32 20 11 31 2e | .3x .1-| .42 .1.|
|00004e40| 0d 0b 00 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |...     |        |
|00004e50| 36 20 20 20 33 20 20 20 | 20 20 20 20 33 0d 0a 00 |6   3   |    3...|
|00004e60| 0d 0a 00 20 20 20 20 20 | 0e 61 32 2d 31 2d 32 39 |...     |.a2-1-29|
|00004e70| 61 0e 28 61 29 0f 20 2d | 31 2f 32 20 20 20 20 20 |a.(a). -|1/2     |
|00004e80| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00004e90| 20 20 20 20 0e 61 32 2d | 31 2d 32 39 62 0e 28 62 |    .a2-|1-29b.(b|
|00004ea0| 29 0f 20 2d 36 0d 0a 00 | 0d 0b 00 20 20 20 20 20 |). -6...|...     |
|00004eb0| 0e 61 32 2d 31 2d 32 39 | 63 0e 28 63 29 0f 20 2d |.a2-1-29|c.(c). -|
|00004ec0| 32 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |2       |        |
|00004ed0| 20 20 20 20 20 20 20 20 | 20 20 20 20 0e 61 32 2d |        |    .a2-|
|00004ee0| 31 2d 32 39 64 0e 28 64 | 29 0f 20 2d 34 0d 0a 00 |1-29d.(d|). -4...|
|00004ef0| 0d 0a 00 59 6f 75 20 6d | 61 79 20 77 61 6e 74 20 |...You m|ay want |
|00004f00| 74 6f 20 72 65 76 69 65 | 77 20 10 31 2d 32 2d 32 |to revie|w .1-2-2|
|00004f10| 0e 65 31 2d 32 2d 33 0e | 47 75 69 64 65 64 20 45 |.e1-2-3.|Guided E|
|00004f20| 78 61 6d 70 6c 65 20 33 | 2e 0f 0d 0a 00 53 65 63 |xample 3|.....Sec|
|00004f30| 74 69 6f 6e 20 31 2e 32 | 20 20 4c 69 6e 65 61 72 |tion 1.2|  Linear|
|00004f40| 20 45 71 75 61 74 69 6f | 6e 73 0d 0b 00 20 35 2e | Equatio|ns... 5.|
|00004f50| 20 20 53 6f 6c 76 65 3a | 20 32 11 33 78 20 11 31 |  Solve:| 2.3x .1|
|00004f60| 2b 20 34 20 3d 20 34 28 | 11 33 78 20 11 31 2d 20 |+ 4 = 4(|.3x .1- |
|00004f70| 31 29 20 2d 20 32 11 33 | 78 11 31 2e 0d 0a 00 0d |1) - 2.3|x.1.....|
|00004f80| 0a 00 20 20 20 20 20 0e | 61 32 2d 31 2d 33 35 61 |..     .|a2-1-35a|
|00004f90| 0e 28 61 29 0f 20 30 20 | 20 20 20 20 20 20 20 20 |.(a). 0 |        |
|00004fa0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00004fb0| 20 20 20 0e 61 32 2d 31 | 2d 33 35 62 0e 28 62 29 |   .a2-1|-35b.(b)|
|00004fc0| 0f 20 2d 32 0d 0a 00 0d | 0b 00 20 20 20 20 20 0e |. -2....|..     .|
|00004fd0| 61 32 2d 31 2d 33 35 63 | 0e 28 63 29 0f 20 2d 38 |a2-1-35c|.(c). -8|
|00004fe0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00004ff0| 20 20 20 20 20 20 20 20 | 20 20 20 0e 61 32 2d 31 |        |   .a2-1|
|00005000| 2d 33 35 64 0e 28 64 29 | 0f 20 4e 6f 20 73 6f 6c |-35d.(d)|. No sol|
|00005010| 75 74 69 6f 6e 0d 0a 00 | 0d 0a 00 59 6f 75 20 6d |ution...|...You m|
|00005020| 61 79 20 77 61 6e 74 20 | 74 6f 20 72 65 76 69 65 |ay want |to revie|
|00005030| 77 20 10 31 2d 32 2d 32 | 0e 65 31 2d 32 2d 34 0e |w .1-2-2|.e1-2-4.|
|00005040| 47 75 69 64 65 64 20 45 | 78 61 6d 70 6c 65 20 34 |Guided E|xample 4|
|00005050| 2e 0f 0d 0a 00 53 65 63 | 74 69 6f 6e 20 31 2e 32 |.....Sec|tion 1.2|
|00005060| 20 20 4c 69 6e 65 61 72 | 20 45 71 75 61 74 69 6f |  Linear| Equatio|
|00005070| 6e 73 0d 0b 00 20 20 20 | 20 20 20 20 20 20 20 20 |ns...   |        |
|00005080| 20 33 11 33 78 20 11 31 | 2d 20 35 20 20 20 32 0d | 3.3x .1|- 5   2.|
|00005090| 0b 00 20 36 2e 20 20 53 | 6f 6c 76 65 3a 20 11 34 |.. 6.  S|olve: .4|
|000050a0| 32 32 32 32 32 32 20 11 | 31 3d 20 11 34 32 20 11 |222222 .|1= .42 .|
|000050b0| 31 2e 0d 0b 00 20 20 20 | 20 20 20 20 20 20 20 20 |1....   |        |
|000050c0| 20 35 11 33 78 20 11 31 | 2b 20 32 20 20 20 33 0d | 5.3x .1|+ 2   3.|
|000050d0| 0a 00 0d 0a 00 20 20 20 | 20 20 20 20 20 20 20 20 |.....   |        |
|000050e0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|000050f0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00005100| 31 34 0d 0b 00 20 20 20 | 20 20 0e 61 32 2d 31 2d |14...   |  .a2-1-|
|00005110| 33 39 61 0e 28 61 29 0f | 20 2d 37 20 20 20 20 20 |39a.(a).| -7     |
|00005120| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00005130| 20 20 20 20 20 20 0e 61 | 32 2d 31 2d 33 39 62 0e |      .a|2-1-39b.|
|00005140| 28 62 29 0f 20 2d 11 34 | 32 32 0d 0b 00 20 20 20 |(b). -.4|22...   |
|00005150| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00005160| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00005170| 20 20 20 20 20 20 20 20 | 20 11 31 39 0d 0a 00 0d |        | .19....|
|00005180| 0b 00 20 20 20 20 20 0e | 61 32 2d 31 2d 33 39 63 |..     .|a2-1-39c|
|00005190| 0e 28 63 29 0f 20 2d 31 | 39 20 20 20 20 20 20 20 |.(c). -1|9       |
|000051a0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|000051b0| 20 20 20 0e 61 32 2d 31 | 2d 33 39 64 0e 28 64 29 |   .a2-1|-39d.(d)|
|000051c0| 0f 20 4e 6f 20 73 6f 6c | 75 74 69 6f 6e 0d 0a 00 |. No sol|ution...|
|000051d0| 0d 0a 00 59 6f 75 20 6d | 61 79 20 77 61 6e 74 20 |...You m|ay want |
|000051e0| 74 6f 20 72 65 76 69 65 | 77 20 10 31 2d 32 2d 32 |to revie|w .1-2-2|
|000051f0| 0e 65 31 2d 32 2d 35 0e | 47 75 69 64 65 64 20 45 |.e1-2-5.|Guided E|
|00005200| 78 61 6d 70 6c 65 20 35 | 2e 0f 0d 0a 00 53 65 63 |xample 5|.....Sec|
|00005210| 74 69 6f 6e 20 31 2e 32 | 20 20 4c 69 6e 65 61 72 |tion 1.2|  Linear|
|00005220| 20 45 71 75 61 74 69 6f | 6e 73 0d 0b 00 20 20 20 | Equatio|ns...   |
|00005230| 20 20 20 20 20 20 20 20 | 20 20 20 31 20 20 20 20 |        |   1    |
|00005240| 20 20 20 31 20 20 20 20 | 20 20 20 20 36 0d 0b 00 |   1    |    6...|
|00005250| 20 37 2e 20 20 53 6f 6c | 76 65 3a 20 11 34 32 32 | 7.  Sol|ve: .422|
|00005260| 32 32 32 20 11 31 2b 20 | 11 34 32 32 32 32 32 20 |222 .1+ |.422222 |
|00005270| 11 31 3d 20 11 34 32 32 | 32 32 32 32 20 11 31 2e |.1= .422|2222 .1.|
|00005280| 0d 0b 00 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |...     |        |
|00005290| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|000052a0| 11 32 32 0d 0b 00 20 20 | 20 20 20 20 20 20 20 20 |.22...  |        |
|000052b0| 20 20 11 33 78 20 11 31 | 2d 20 32 20 20 20 11 33 |  .3x .1|- 2   .3|
|000052c0| 78 20 11 31 2b 20 32 20 | 20 20 11 33 78 20 20 11 |x .1+ 2 |  .3x  .|
|000052d0| 31 2d 20 34 0d 0a 00 0d | 0a 00 20 20 20 20 20 0e |1- 4....|..     .|
|000052e0| 61 32 2d 31 2d 34 33 61 | 0e 28 61 29 0f 20 33 20 |a2-1-43a|.(a). 3 |
|000052f0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00005300| 20 20 20 20 20 20 20 20 | 20 20 20 0e 61 32 2d 31 |        |   .a2-1|
|00005310| 2d 34 33 62 0e 28 62 29 | 0f 20 31 0d 0a 00 0d 0b |-43b.(b)|. 1.....|
|00005320| 00 20 20 20 20 20 0e 61 | 32 2d 31 2d 34 33 63 0e |.     .a|2-1-43c.|
|00005330| 28 63 29 0f 20 37 20 20 | 20 20 20 20 20 20 20 20 |(c). 7  |        |
|00005340| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00005350| 20 20 0e 61 32 2d 31 2d | 34 33 64 0e 28 64 29 0f |  .a2-1-|43d.(d).|
|00005360| 20 4e 6f 20 73 6f 6c 75 | 74 69 6f 6e 0d 0a 00 0d | No solu|tion....|
|00005370| 0a 00 59 6f 75 20 6d 61 | 79 20 77 61 6e 74 20 74 |..You ma|y want t|
|00005380| 6f 20 72 65 76 69 65 77 | 20 10 31 2d 32 2d 32 0e |o review| .1-2-2.|
|00005390| 65 31 2d 32 2d 36 0e 47 | 75 69 64 65 64 20 45 78 |e1-2-6.G|uided Ex|
|000053a0| 61 6d 70 6c 65 20 36 2e | 0f 0d 0a 00 53 65 63 74 |ample 6.|....Sect|
|000053b0| 69 6f 6e 20 31 2e 32 20 | 20 4c 69 6e 65 61 72 20 |ion 1.2 | Linear |
|000053c0| 45 71 75 61 74 69 6f 6e | 73 0d 0b 00 20 20 20 20 |Equation|s...    |
|000053d0| 20 20 20 20 20 20 20 20 | 20 20 20 33 20 20 20 20 |        |   3    |
|000053e0| 20 20 20 36 11 33 78 0d | 0b 00 20 11 31 38 2e 20 |   6.3x.|.. .18. |
|000053f0| 20 53 6f 6c 76 65 3a 20 | 11 34 32 32 32 32 32 32 | Solve: |.4222222|
|00005400| 20 11 31 2d 20 11 34 32 | 32 32 32 32 32 20 11 31 | .1- .42|22222 .1|
|00005410| 3d 20 2d 32 2e 0d 0b 00 | 20 20 20 20 20 20 20 20 |= -2....|        |
|00005420| 20 20 20 20 33 11 33 78 | 20 11 31 2b 20 31 20 20 |    3.3x| .1+ 1  |
|00005430| 20 33 11 33 78 20 11 31 | 2d 20 31 0d 0a 00 0d 0a | 3.3x .1|- 1.....|
|00005440| 00 20 20 20 20 20 20 20 | 20 20 20 31 20 20 20 20 |.       |   1    |
|00005450| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00005460| 20 20 20 20 20 20 20 20 | 20 20 20 20 31 0d 0b 00 |        |    1...|
|00005470| 20 20 20 20 20 0e 61 32 | 2d 31 2d 34 37 61 0e 28 |     .a2|-1-47a.(|
|00005480| 61 29 0f 20 2d 11 34 32 | 20 20 20 20 20 20 20 20 |a). -.42|        |
|00005490| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|000054a0| 20 20 20 11 31 0e 61 32 | 2d 31 2d 34 37 62 0e 28 |   .1.a2|-1-47b.(|
|000054b0| 62 29 0f 20 2d 11 34 32 | 0d 0b 00 20 20 20 20 20 |b). -.42|...     |
|000054c0| 20 20 20 20 20 11 31 33 | 20 20 20 20 20 20 20 20 |     .13|        |
|000054d0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|000054e0| 20 20 20 20 20 20 20 20 | 39 0d 0a 00 0d 0b 00 20 |        |9...... |
|000054f0| 20 20 20 20 20 20 20 20 | 35 0d 0b 00 20 20 20 20 |        |5...    |
|00005500| 20 0e 61 32 2d 31 2d 34 | 37 63 0e 28 63 29 0f 20 | .a2-1-4|7c.(c). |
|00005510| 11 34 32 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |.42     |        |
|00005520| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 11 |        |       .|
|00005530| 31 0e 61 32 2d 31 2d 34 | 37 64 0e 28 64 29 0f 20 |1.a2-1-4|7d.(d). |
|00005540| 4e 6f 20 73 6f 6c 75 74 | 69 6f 6e 0d 0b 00 20 20 |No solut|ion...  |
|00005550| 20 20 20 20 20 20 20 33 | 0d 0a 00 0d 0a 00 59 6f |       3|......Yo|
|00005560| 75 20 6d 61 79 20 77 61 | 6e 74 20 74 6f 20 72 65 |u may wa|nt to re|
|00005570| 76 69 65 77 20 10 31 2d | 32 2d 32 0e 65 31 2d 32 |view .1-|2-2.e1-2|
|00005580| 2d 36 0e 47 75 69 64 65 | 64 20 45 78 61 6d 70 6c |-6.Guide|d Exampl|
|00005590| 65 20 36 2e 0f 0d 0a 00 | 53 65 63 74 69 6f 6e 20 |e 6.....|Section |
|000055a0| 31 2e 32 20 20 4c 69 6e | 65 61 72 20 45 71 75 61 |1.2  Lin|ear Equa|
|000055b0| 74 69 6f 6e 73 20 20 20 | 20 20 20 20 20 20 20 20 |tions   |        |
|000055c0| 20 20 20 20 20 31 30 20 | 20 20 20 20 34 20 20 20 |     10 |    4   |
|000055d0| 20 20 35 0d 0b 00 20 39 | 2e 20 20 53 6f 6c 76 65 |  5... 9|.  Solve|
|000055e0| 3a 20 11 34 32 32 32 32 | 32 32 32 32 20 11 31 2b |: .42222|2222 .1+|
|000055f0| 20 11 34 32 20 11 31 3d | 20 11 34 32 32 32 32 32 | .42 .1=| .422222|
|00005600| 20 11 31 2e 0d 0b 00 20 | 20 20 20 20 20 20 20 20 | .1.... |        |
|00005610| 20 20 20 11 33 78 11 31 | 28 11 33 78 20 11 31 2d |   .3x.1|(.3x .1-|
|00005620| 20 32 29 20 20 20 11 33 | 78 20 20 20 78 20 11 31 | 2)   .3|x   x .1|
|00005630| 2d 20 32 0d 0a 00 0d 0a | 00 20 20 20 20 20 0e 32 |- 2.....|.     .2|
|00005640| 2d 31 2d 35 31 61 0e 28 | 61 29 0f 20 32 20 20 20 |-1-51a.(|a). 2   |
|00005650| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00005660| 20 20 20 20 20 20 20 20 | 20 0e 32 2d 31 2d 35 31 |        | .2-1-51|
|00005670| 62 0e 28 62 29 0f 20 38 | 0d 0a 00 0d 0b 00 20 20 |b.(b). 8|......  |
|00005680| 20 20 20 0e 32 2d 31 2d | 35 31 63 0e 28 63 29 0f |   .2-1-|51c.(c).|
|00005690| 20 32 30 20 20 20 20 20 | 20 20 20 20 20 20 20 20 | 20     |        |
|000056a0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 0e 32 |        |      .2|
|000056b0| 2d 31 2d 35 31 64 0e 28 | 64 29 0f 20 4e 6f 20 73 |-1-51d.(|d). No s|
|000056c0| 6f 6c 75 74 69 6f 6e 0d | 0a 00 0d 0a 00 59 6f 75 |olution.|.....You|
|000056d0| 20 6d 61 79 20 77 61 6e | 74 20 74 6f 20 72 65 76 | may wan|t to rev|
|000056e0| 69 65 77 20 10 31 2d 32 | 2d 32 0e 65 31 2d 32 2d |iew .1-2|-2.e1-2-|
|000056f0| 37 0e 47 75 69 64 65 64 | 20 45 78 61 6d 70 6c 65 |7.Guided| Example|
|00005700| 20 37 2e 0f 0d 0a 00 14 | 00 00 00 14 02 00 00 50 | 7......|.......P|
|00005710| 00 04 00 14 00 00 00 10 | 00 00 00 32 2d 31 2d 31 |........|...2-1-1|
|00005720| 35 61 00 2c 02 00 00 3a | 02 00 00 50 00 04 00 2c |5a.,...:|...P...,|
|00005730| 02 00 00 28 02 00 00 32 | 2d 31 2d 31 35 62 00 6a |...(...2|-1-15b.j|
|00005740| 04 00 00 92 00 00 00 50 | 00 04 00 6a 04 00 00 66 |.......P|...j...f|
|00005750| 04 00 00 32 2d 31 2d 31 | 35 63 00 00 05 00 00 36 |...2-1-1|5c.....6|
|00005760| 02 00 00 50 00 04 00 00 | 05 00 00 fc 04 00 00 32 |...P....|.......2|
|00005770| 2d 31 2d 31 35 64 00 3a | 07 00 00 40 01 00 00 50 |-1-15d.:|...@...P|
|00005780| 00 04 00 3a 07 00 00 36 | 07 00 00 32 2d 31 2d 31 |...:...6|...2-1-1|
|00005790| 61 00 7e 08 00 00 d1 00 | 00 00 50 00 04 00 7e 08 |a.~.....|..P...~.|
|000057a0| 00 00 7a 08 00 00 32 2d | 31 2d 31 62 00 53 09 00 |..z...2-|1-1b.S..|
|000057b0| 00 fa 00 00 00 50 00 04 | 00 53 09 00 00 4f 09 00 |.....P..|.S...O..|
|000057c0| 00 32 2d 31 2d 31 63 00 | 51 0a 00 00 5e 00 00 00 |.2-1-1c.|Q...^...|
|000057d0| 50 00 04 00 51 0a 00 00 | 4d 0a 00 00 32 2d 31 2d |P...Q...|M...2-1-|
|000057e0| 31 64 00 b3 0a 00 00 3c | 00 00 00 50 00 04 00 b3 |1d.....<|...P....|
|000057f0| 0a 00 00 af 0a 00 00 32 | 2d 31 2d 32 31 61 00 f3 |.......2|-1-21a..|
|00005800| 0a 00 00 ff 00 00 00 50 | 00 04 00 f3 0a 00 00 ef |.......P|........|
|00005810| 0a 00 00 32 2d 31 2d 32 | 31 62 00 f6 0b 00 00 ff |...2-1-2|1b......|
|00005820| 00 00 00 50 00 04 00 f6 | 0b 00 00 f2 0b 00 00 32 |...P....|.......2|
|00005830| 2d 31 2d 32 31 63 00 f9 | 0c 00 00 ff 00 00 00 50 |-1-21c..|.......P|
|00005840| 00 04 00 f9 0c 00 00 f5 | 0c 00 00 32 2d 31 2d 32 |........|...2-1-2|
|00005850| 31 64 00 fc 0d 00 00 34 | 01 00 00 50 00 04 00 fc |1d.....4|...P....|
|00005860| 0d 00 00 f8 0d 00 00 32 | 2d 31 2d 33 33 61 00 34 |.......2|-1-33a.4|
|00005870| 0f 00 00 36 01 00 00 50 | 00 04 00 34 0f 00 00 30 |...6...P|...4...0|
|00005880| 0f 00 00 32 2d 31 2d 33 | 33 62 00 6e 10 00 00 3f |...2-1-3|3b.n...?|
|00005890| 00 00 00 50 00 04 00 6e | 10 00 00 6a 10 00 00 32 |...P...n|...j...2|
|000058a0| 2d 31 2d 33 33 63 00 b1 | 10 00 00 36 01 00 00 50 |-1-33c..|...6...P|
|000058b0| 00 04 00 b1 10 00 00 ad | 10 00 00 32 2d 31 2d 33 |........|...2-1-3|
|000058c0| 33 64 00 eb 11 00 00 36 | 02 00 00 50 00 04 00 eb |3d.....6|...P....|
|000058d0| 11 00 00 e7 11 00 00 32 | 2d 31 2d 35 31 61 00 25 |.......2|-1-51a.%|
|000058e0| 14 00 00 36 02 00 00 50 | 00 04 00 25 14 00 00 21 |...6...P|...%...!|
|000058f0| 14 00 00 32 2d 31 2d 35 | 31 62 00 5f 16 00 00 36 |...2-1-5|1b._...6|
|00005900| 02 00 00 50 00 04 00 5f | 16 00 00 5b 16 00 00 32 |...P..._|...[...2|
|00005910| 2d 31 2d 35 31 63 00 99 | 18 00 00 34 00 00 00 50 |-1-51c..|...4...P|
|00005920| 00 04 00 99 18 00 00 95 | 18 00 00 32 2d 31 2d 35 |........|...2-1-5|
|00005930| 31 64 00 d1 18 00 00 8e | 01 00 00 50 00 04 00 d1 |1d......|...P....|
|00005940| 18 00 00 cd 18 00 00 61 | 32 2d 31 2d 32 35 61 00 |.......a|2-1-25a.|
|00005950| 63 1a 00 00 8e 01 00 00 | 50 00 04 00 63 1a 00 00 |c.......|P...c...|
|00005960| 5f 1a 00 00 61 32 2d 31 | 2d 32 35 62 00 f5 1b 00 |_...a2-1|-25b....|
|00005970| 00 3f 00 00 00 50 00 04 | 00 f5 1b 00 00 f1 1b 00 |.?...P..|........|
|00005980| 00 61 32 2d 31 2d 32 35 | 63 00 38 1c 00 00 5f 00 |.a2-1-25|c.8..._.|
|00005990| 00 00 50 00 04 00 38 1c | 00 00 34 1c 00 00 61 32 |..P...8.|..4...a2|
|000059a0| 2d 31 2d 32 35 64 00 9b | 1c 00 00 ca 01 00 00 50 |-1-25d..|.......P|
|000059b0| 00 04 00 9b 1c 00 00 97 | 1c 00 00 61 32 2d 31 2d |........|...a2-1-|
|000059c0| 32 39 61 00 69 1e 00 00 | 3b 00 00 00 50 00 04 00 |29a.i...|;...P...|
|000059d0| 69 1e 00 00 65 1e 00 00 | 61 32 2d 31 2d 32 39 62 |i...e...|a2-1-29b|
|000059e0| 00 a8 1e 00 00 ca 01 00 | 00 50 00 04 00 a8 1e 00 |........|.P......|
|000059f0| 00 a4 1e 00 00 61 32 2d | 31 2d 32 39 63 00 76 20 |.....a2-|1-29c.v |
|00005a00| 00 00 ca 01 00 00 50 00 | 04 00 76 20 00 00 72 20 |......P.|..v ..r |
|00005a10| 00 00 61 32 2d 31 2d 32 | 39 64 00 44 22 00 00 a7 |..a2-1-2|9d.D"...|
|00005a20| 01 00 00 50 00 04 00 44 | 22 00 00 40 22 00 00 61 |...P...D|"..@"..a|
|00005a30| 32 2d 31 2d 33 35 61 00 | ef 23 00 00 a7 01 00 00 |2-1-35a.|.#......|
|00005a40| 50 00 04 00 ef 23 00 00 | eb 23 00 00 61 32 2d 31 |P....#..|.#..a2-1|
|00005a50| 2d 33 35 62 00 9a 25 00 | 00 a7 01 00 00 50 00 04 |-35b..%.|.....P..|
|00005a60| 00 9a 25 00 00 96 25 00 | 00 61 32 2d 31 2d 33 35 |..%...%.|.a2-1-35|
|00005a70| 63 00 45 27 00 00 65 00 | 00 00 50 00 04 00 45 27 |c.E'..e.|..P...E'|
|00005a80| 00 00 41 27 00 00 61 32 | 2d 31 2d 33 35 64 00 ae |..A'..a2|-1-35d..|
|00005a90| 27 00 00 ea 01 00 00 50 | 00 04 00 ae 27 00 00 aa |'......P|....'...|
|00005aa0| 27 00 00 61 32 2d 31 2d | 33 39 61 00 9c 29 00 00 |'..a2-1-|39a..)..|
|00005ab0| ea 01 00 00 50 00 04 00 | 9c 29 00 00 98 29 00 00 |....P...|.)...)..|
|00005ac0| 61 32 2d 31 2d 33 39 62 | 00 8a 2b 00 00 3f 00 00 |a2-1-39b|..+..?..|
|00005ad0| 00 50 00 04 00 8a 2b 00 | 00 86 2b 00 00 61 32 2d |.P....+.|..+..a2-|
|00005ae0| 31 2d 33 39 63 00 cd 2b | 00 00 43 00 00 00 50 00 |1-39c..+|..C...P.|
|00005af0| 04 00 cd 2b 00 00 c9 2b | 00 00 61 32 2d 31 2d 33 |...+...+|..a2-1-3|
|00005b00| 39 64 00 14 2c 00 00 5d | 00 00 00 50 00 04 00 14 |9d..,..]|...P....|
|00005b10| 2c 00 00 10 2c 00 00 61 | 32 2d 31 2d 34 33 61 00 |,...,..a|2-1-43a.|
|00005b20| 75 2c 00 00 83 02 00 00 | 50 00 04 00 75 2c 00 00 |u,......|P...u,..|
|00005b30| 71 2c 00 00 61 32 2d 31 | 2d 34 33 62 00 fc 2e 00 |q,..a2-1|-43b....|
|00005b40| 00 83 02 00 00 50 00 04 | 00 fc 2e 00 00 f8 2e 00 |.....P..|........|
|00005b50| 00 61 32 2d 31 2d 34 33 | 63 00 83 31 00 00 3c 00 |.a2-1-43|c..1..<.|
|00005b60| 00 00 50 00 04 00 83 31 | 00 00 7f 31 00 00 61 32 |..P....1|...1..a2|
|00005b70| 2d 31 2d 34 33 64 00 c3 | 31 00 00 e6 02 00 00 50 |-1-43d..|1......P|
|00005b80| 00 04 00 c3 31 00 00 bf | 31 00 00 61 32 2d 31 2d |....1...|1..a2-1-|
|00005b90| 34 37 61 00 ad 34 00 00 | e6 02 00 00 50 00 04 00 |47a..4..|....P...|
|00005ba0| ad 34 00 00 a9 34 00 00 | 61 32 2d 31 2d 34 37 62 |.4...4..|a2-1-47b|
|00005bb0| 00 97 37 00 00 3f 00 00 | 00 50 00 04 00 97 37 00 |..7..?..|.P....7.|
|00005bc0| 00 93 37 00 00 61 32 2d | 31 2d 34 37 63 00 da 37 |..7..a2-|1-47c..7|
|00005bd0| 00 00 3c 00 00 00 50 00 | 04 00 da 37 00 00 d6 37 |..<...P.|...7...7|
|00005be0| 00 00 61 32 2d 31 2d 34 | 37 64 00 1a 38 00 00 3b |..a2-1-4|7d..8..;|
|00005bf0| 00 00 00 50 00 04 00 1a | 38 00 00 16 38 00 00 61 |...P....|8...8..a|
|00005c00| 32 2d 31 2d 35 33 61 00 | 59 38 00 00 20 02 00 00 |2-1-53a.|Y8.. ...|
|00005c10| 50 00 04 00 59 38 00 00 | 55 38 00 00 61 32 2d 31 |P...Y8..|U8..a2-1|
|00005c20| 2d 35 33 62 00 7d 3a 00 | 00 20 02 00 00 50 00 04 |-53b.}:.|. ...P..|
|00005c30| 00 7d 3a 00 00 79 3a 00 | 00 61 32 2d 31 2d 35 33 |.}:..y:.|.a2-1-53|
|00005c40| 63 00 a1 3c 00 00 3c 00 | 00 00 50 00 04 00 a1 3c |c..<..<.|..P....<|
|00005c50| 00 00 9d 3c 00 00 61 32 | 2d 31 2d 35 33 64 00 e1 |...<..a2|-1-53d..|
|00005c60| 3c 00 00 22 01 00 00 50 | 00 04 00 e1 3c 00 00 dd |<.."...P|....<...|
|00005c70| 3c 00 00 61 32 2d 31 2d | 36 31 61 00 07 3e 00 00 |<..a2-1-|61a..>..|
|00005c80| 10 01 00 00 50 00 04 00 | 07 3e 00 00 03 3e 00 00 |....P...|.>...>..|
|00005c90| 61 32 2d 31 2d 36 31 62 | 00 1b 3f 00 00 22 01 00 |a2-1-61b|..?.."..|
|00005ca0| 00 50 00 04 00 1b 3f 00 | 00 17 3f 00 00 61 32 2d |.P....?.|..?..a2-|
|00005cb0| 31 2d 36 31 63 00 41 40 | 00 00 22 01 00 00 50 00 |1-61c.A@|.."...P.|
|00005cc0| 04 00 41 40 00 00 3d 40 | 00 00 61 32 2d 31 2d 36 |..A@..=@|..a2-1-6|
|00005cd0| 31 64 00 80 41 00 00 e1 | 01 00 00 4d 1d 00 00 63 |1d..A...|...M...c|
|00005ce0| 41 00 00 00 00 00 00 78 | 31 2d 32 00 7e 43 00 00 |A......x|1-2.~C..|
|00005cf0| 1e 02 00 00 4d 1d 00 00 | 61 43 00 00 00 00 00 00 |....M...|aC......|
|00005d00| 78 31 2d 32 2d 31 00 b9 | 45 00 00 91 01 00 00 4d |x1-2-1..|E......M|
|00005d10| 1d 00 00 9c 45 00 00 00 | 00 00 00 78 31 2d 32 2d |....E...|...x1-2-|
|00005d20| 31 30 00 67 47 00 00 ff | 00 00 00 4d 1d 00 00 4a |10.gG...|...M...J|
|00005d30| 47 00 00 00 00 00 00 78 | 31 2d 32 2d 31 31 00 83 |G......x|1-2-11..|
|00005d40| 48 00 00 32 01 00 00 4d | 1d 00 00 66 48 00 00 00 |H..2...M|...fH...|
|00005d50| 00 00 00 78 31 2d 32 2d | 31 32 00 d2 49 00 00 6d |...x1-2-|12..I..m|
|00005d60| 01 00 00 4d 1d 00 00 b5 | 49 00 00 00 00 00 00 78 |...M....|I......x|
|00005d70| 31 2d 32 2d 31 33 00 5c | 4b 00 00 52 01 00 00 4d |1-2-13.\|K..R...M|
|00005d80| 1d 00 00 3f 4b 00 00 00 | 00 00 00 78 31 2d 32 2d |...?K...|...x1-2-|
|00005d90| 32 00 cb 4c 00 00 08 01 | 00 00 4d 1d 00 00 ae 4c |2..L....|..M....L|
|00005da0| 00 00 00 00 00 00 78 31 | 2d 32 2d 33 00 f0 4d 00 |......x1|-2-3..M.|
|00005db0| 00 3d 01 00 00 4d 1d 00 | 00 d3 4d 00 00 00 00 00 |.=...M..|..M.....|
|00005dc0| 00 78 31 2d 32 2d 34 00 | 4a 4f 00 00 0b 01 00 00 |.x1-2-4.|JO......|
|00005dd0| 4d 1d 00 00 2d 4f 00 00 | 00 00 00 00 78 31 2d 32 |M...-O..|....x1-2|
|00005de0| 2d 35 00 72 50 00 00 9b | 01 00 00 4d 1d 00 00 55 |-5.rP...|...M...U|
|00005df0| 50 00 00 00 00 00 00 78 | 31 2d 32 2d 36 00 2a 52 |P......x|1-2-6.*R|
|00005e00| 00 00 82 01 00 00 4d 1d | 00 00 0d 52 00 00 00 00 |......M.|...R....|
|00005e10| 00 00 78 31 2d 32 2d 37 | 00 c9 53 00 00 cf 01 00 |..x1-2-7|..S.....|
|00005e20| 00 4d 1d 00 00 ac 53 00 | 00 00 00 00 00 78 31 2d |.M....S.|.....x1-|
|00005e30| 32 2d 38 00 b5 55 00 00 | 52 01 00 00 4d 1d 00 00 |2-8..U..|R...M...|
|00005e40| 98 55 00 00 00 00 00 00 | 78 31 2d 32 2d 39 00    |.U......|x1-2-9. |
+--------+-------------------------+-------------------------+--------+--------+