s4/100-s1/60-s1/300=0 s1/60-s2/70=0 s1/300+s4/30-s3/100=0 s2/70-s4/100-s4/30+s3/100=0

$$- frac{s_{1}}{300} + – frac{s_{1}}{60} + frac{s_{4}}{100} = 0$$

s1 s2
— – — = 0
60 70

$$frac{s_{1}}{60} – frac{s_{2}}{70} = 0$$

s1 s4 s3
— + — – — = 0
300 30 100

$$- frac{s_{3}}{100} + frac{s_{1}}{300} + frac{s_{4}}{30} = 0$$

s2 s4 s4 s3
— – — – — + — = 0
70 100 30 100

$$frac{s_{3}}{100} + – frac{s_{4}}{30} + frac{s_{2}}{70} – frac{s_{4}}{100} = 0$$

$$s_{11} = frac{s_{4}}{2}$$
=
$$frac{s_{4}}{2}$$
=

0.5*s4

$$s_{21} = frac{7 s_{4}}{12}$$
=
$$frac{7 s_{4}}{12}$$
=

0.583333333333333*s4

$$s_{31} = frac{7 s_{4}}{2}$$
=
$$frac{7 s_{4}}{2}$$
=

3.5*s4

Дана система ур-ний
$$- frac{s_{1}}{300} + – frac{s_{1}}{60} + frac{s_{4}}{100} = 0$$
$$frac{s_{1}}{60} – frac{s_{2}}{70} = 0$$
$$- frac{s_{3}}{100} + frac{s_{1}}{300} + frac{s_{4}}{30} = 0$$
$$frac{s_{3}}{100} + – frac{s_{4}}{30} + frac{s_{2}}{70} – frac{s_{4}}{100} = 0$$

Приведём систему ур-ний к каноническому виду
$$- frac{s_{1}}{50} + frac{s_{4}}{100} = 0$$
$$frac{s_{1}}{60} – frac{s_{2}}{70} = 0$$
$$frac{s_{1}}{300} – frac{s_{3}}{100} + frac{s_{4}}{30} = 0$$
$$frac{s_{2}}{70} + frac{s_{3}}{100} – frac{13 s_{4}}{300} = 0$$
Запишем систему линейных ур-ний в матричном виде
$$left[begin{matrix}- frac{1}{50} & 0 & 0 & frac{1}{100} & 0\frac{1}{60} & – frac{1}{70} & 0 & 0 & 0\frac{1}{300} & 0 & – frac{1}{100} & frac{1}{30} & 0 & frac{1}{70} & frac{1}{100} & – frac{13}{300} & 0end{matrix}right]$$
В 1 ом столбце
$$left[begin{matrix}- frac{1}{50}\frac{1}{60}\frac{1}{300}end{matrix}right]$$
делаем так, чтобы все элементы, кроме
1 го элемента равнялись нулю.
– Для этого берём 1 ую строку
$$left[begin{matrix}- frac{1}{50} & 0 & 0 & frac{1}{100} & 0end{matrix}right]$$
,
и будем вычитать ее из других строк:
Из 2 ой строки вычитаем:
$$left[begin{matrix}- frac{1}{60} + frac{1}{60} & – frac{1}{70} & 0 & – frac{-1}{120} & 0end{matrix}right] = left[begin{matrix}0 & – frac{1}{70} & 0 & frac{1}{120} & 0end{matrix}right]$$
получаем
$$left[begin{matrix}- frac{1}{50} & 0 & 0 & frac{1}{100} & 0 & – frac{1}{70} & 0 & frac{1}{120} & 0\frac{1}{300} & 0 & – frac{1}{100} & frac{1}{30} & 0 & frac{1}{70} & frac{1}{100} & – frac{13}{300} & 0end{matrix}right]$$
Из 3 ой строки вычитаем:
$$left[begin{matrix}- frac{1}{300} + frac{1}{300} & 0 & – frac{1}{100} & – frac{-1}{600} + frac{1}{30} & 0end{matrix}right] = left[begin{matrix}0 & 0 & – frac{1}{100} & frac{7}{200} & 0end{matrix}right]$$
получаем
$$left[begin{matrix}- frac{1}{50} & 0 & 0 & frac{1}{100} & 0 & – frac{1}{70} & 0 & frac{1}{120} & 0 & 0 & – frac{1}{100} & frac{7}{200} & 0 & frac{1}{70} & frac{1}{100} & – frac{13}{300} & 0end{matrix}right]$$
Во 2 ом столбце
$$left[begin{matrix}0 – frac{1}{70}\frac{1}{70}end{matrix}right]$$
делаем так, чтобы все элементы, кроме
2 го элемента равнялись нулю.
– Для этого берём 2 ую строку
$$left[begin{matrix}0 & – frac{1}{70} & 0 & frac{1}{120} & 0end{matrix}right]$$
,
и будем вычитать ее из других строк:
Из 4 ой строки вычитаем:
$$left[begin{matrix}0 & – frac{1}{70} + frac{1}{70} & frac{1}{100} & – frac{13}{300} – – frac{1}{120} & 0end{matrix}right] = left[begin{matrix}0 & 0 & frac{1}{100} & – frac{7}{200} & 0end{matrix}right]$$
получаем
$$left[begin{matrix}- frac{1}{50} & 0 & 0 & frac{1}{100} & 0 & – frac{1}{70} & 0 & frac{1}{120} & 0 & 0 & – frac{1}{100} & frac{7}{200} & 0 & 0 & frac{1}{100} & – frac{7}{200} & 0end{matrix}right]$$
В 3 ом столбце
$$left[begin{matrix}0 – frac{1}{100}\frac{1}{100}end{matrix}right]$$
делаем так, чтобы все элементы, кроме
3 го элемента равнялись нулю.
– Для этого берём 3 ую строку
$$left[begin{matrix}0 & 0 & – frac{1}{100} & frac{7}{200} & 0end{matrix}right]$$
,
и будем вычитать ее из других строк:
Из 4 ой строки вычитаем:
$$left[begin{matrix}0 & 0 & – frac{1}{100} + frac{1}{100} & – frac{7}{200} – – frac{7}{200} & 0end{matrix}right] = left[begin{matrix}0 & 0 & 0 & 0 & 0end{matrix}right]$$
получаем
$$left[begin{matrix}- frac{1}{50} & 0 & 0 & frac{1}{100} & 0 & – frac{1}{70} & 0 & frac{1}{120} & 0 & 0 & – frac{1}{100} & frac{7}{200} & 0 & 0 & 0 & 0 & 0end{matrix}right]$$

Все почти готово – осталось только найти неизвестные, решая элементарные ур-ния:
$$- frac{x_{1}}{50} + frac{x_{4}}{100} = 0$$
$$- frac{x_{2}}{70} + frac{x_{4}}{120} = 0$$
$$- frac{x_{3}}{100} + frac{7 x_{4}}{200} = 0$$
$$0 – 0 = 0$$
Получаем ответ:
$$x_{1} = frac{x_{4}}{2}$$
$$x_{2} = frac{7 x_{4}}{12}$$
$$x_{3} = frac{7 x_{4}}{2}$$
где x4 – свободные переменные

s11 = 0.2222219838036431
s21 = 0.2592589811042503
s31 = 1.555553886625502
s41 = 0.4444439676072862

s12 = -3.777779049343533
s22 = -4.407408890900789
s32 = -26.44445334540473
s42 = -7.555558098687066

s13 = 2.000001059638129
s23 = 2.333334569577818
s33 = 14.00000741746691
s43 = 4.000002119276259

s14 = 3.777778627143966
s24 = 4.407408398334627
s34 = 26.44445039000776
s44 = 7.555557254287932

s15 = -2.000000635368956
s25 = -2.333334074597116
s35 = -14.0000044475827
s45 = -4.000001270737913

s16 = 0.6666666666666667
s26 = 0.7777777777777778
s36 = 4.666666666666667
s46 = 1.333333333333333

s17 = 5.111108991834852
s27 = 5.962960490473995
s37 = 35.77776294284397
s47 = 10.2222179836697

s18 = 5.555557674003972
s28 = 6.481483953004634
s38 = 38.8889037180278
s48 = 11.11111534800794

s19 = 0.666666719234652
s29 = 0.777777839107094
s39 = 4.666667034642564
s49 = 1.333333438469304

s110 = 3.333335878120528
s210 = 3.888891857807283
s310 = 23.3333511468437
s410 = 6.666671756241057

s111 = -0.2222226990593805
s211 = -0.2592598155692772
s311 = -1.555558893415663
s411 = -0.4444453981187609

s112 = 4.666668362087674
s212 = 5.444446422435619
s312 = 32.66667853461372
s412 = 9.333336724175347

s113 = 4.222223917643229
s213 = 4.925927903917101
s313 = 29.5555674235026
s413 = 8.444447835286458

s114 = 3.777776507867707
s214 = 4.407405925845658
s314 = 26.44443555507395
s414 = 7.555553015735414

s115 = 4.666665501064724
s215 = 5.444443084575512
s315 = 32.66665850745307
s415 = 9.333331002129449

s116 = 3.333333546916644
s216 = 3.888889138069418
s316 = 23.33333482841651
s416 = 6.666667093833287

s117 = 3.333333334989018
s217 = 3.888888890820521
s317 = 23.33333334492313
s417 = 6.666666669978036

s118 = 1.555556191338433
s218 = 1.814815556561505
s318 = 10.88889333936903
s418 = 3.111112382676866

s119 = -1.555555767483181
s219 = -1.814815062063712
s319 = -10.88889037238227
s419 = -3.111111534966363

s120 = 3.333334394627147
s220 = 3.888890127065005
s320 = 23.33334076239003
s420 = 6.666668789254294

s121 = 3.333334182699521
s221 = 3.888889879816108
s321 = 23.33333927889665
s421 = 6.666668365399043

s122 = 0.2222218116124471
s222 = 0.2592587802145216
s322 = 1.55555268128713
s422 = 0.4444436232248942

s123 = 0.2222226055132018
s223 = 0.2592597064320688
s323 = 1.555558238592413
s423 = 0.4444452110264036

s124 = 0.666666931576199
s224 = 0.7777780868388988
s324 = 4.666668521033393
s424 = 1.333333863152398

s125 = 0.2222226990593804
s225 = 0.2592598155692771
s325 = 1.555558893415663
s425 = 0.4444453981187608

s126 = 4.222222222222222
s226 = 4.925925925925926
s326 = 29.55555555555556
s426 = 8.444444444444444

s127 = 3.333332275350889
s227 = 3.888887654576037
s327 = 23.33332592745622
s427 = 6.666664550701777

s128 = -0.2222215202119616
s228 = -0.2592584402472885
s328 = -1.555550641483731
s428 = -0.4444430404239231

s129 = 4.666667302449544
s229 = 5.444445186191135
s329 = 32.66667111714681
s429 = 9.333334604899089

s130 = 1.555551742513974
s230 = 1.814810366266304
s330 = 10.88886219759782
s430 = 3.111103485027949

s131 = 0.2222225268681844
s231 = 0.2592596146795485
s331 = 1.555557688077291
s431 = 0.4444450537363688

s132 = -3.777778626316124
s232 = -4.407408397368811
s332 = -26.44445038421286
s432 = -7.555557252632247

s133 = -0.2222222220152617
s233 = -0.2592592590178053
s333 = -1.555555554106832
s433 = -0.4444444440305234

s134 = 2.444444763163726
s234 = 2.851852223691013
s334 = 17.11111334214608
s434 = 4.888889526327451

s135 = 3.777776931722959
s235 = 4.407406420343452
s335 = 26.44443852206071
s435 = 7.555553863445918

s136 = 1.111111110904151
s236 = 1.296296296054842
s336 = 7.777777776329054
s436 = 2.222222221808301

s137 = 2.444442961778906
s237 = 2.85185012207539
s337 = 17.11110073245234
s437 = 4.888885923557811

s138 = 1.555555873446994
s238 = 1.81481518568816
s338 = 10.88889111412896
s438 = 3.111111746893989

s139 = -3.777778202460872
s239 = -4.407407902871017
s339 = -26.4444474172261
s439 = -7.555556404921744

s140 = 1.555556404921744
s240 = 1.814815805742034
s340 = 10.88889483445221
s440 = 3.111112809843487

s141 = 3.33333375884427
s241 = 3.888889385318315
s341 = 23.33333631190989
s441 = 6.666667517688539

s142 = 1.555555449591743
s242 = 1.814814691190366
s342 = 10.8888881471422
s442 = 3.111110899183485

s143 = 4.666666666666667
s243 = 5.444444444444444
s343 = 32.66666666666667
s443 = 9.333333333333333

s144 = 1.55555714501275
s244 = 1.814816669181541
s344 = 10.88890001508925
s444 = 3.111114290025499

s145 = 2.000000635782878
s245 = 2.333334075080024
s345 = 14.00000445048014
s445 = 4.000001271565755

s146 = -0.2222217586305407
s246 = -0.2592587184022974
s346 = -1.555552310413785
s446 = -0.4444435172610813

s147 = 3.333333123061392
s247 = 3.888888643571624
s347 = 23.33333186142974
s447 = 6.666666246122784

s148 = -0.2222218776328696
s248 = -0.2592588572383479
s348 = -1.555553143430087
s448 = -0.4444437552657393

s149 = 5.111114501953125
s249 = 5.962966918945313
s349 = 35.77780151367188
s449 = 10.22222900390625

s150 = 0.2222222089767456
s250 = 0.2592592438062032
s350 = 1.555555462837219
s450 = 0.4444444179534912

s151 = -3.333333333333333
s251 = -3.888888888888889
s351 = -23.33333333333333
s451 = -6.666666666666667

s152 = 4.222222646077474
s252 = 4.92592642042372
s352 = 29.55555852254232
s452 = 8.444445292154948

s153 = 4.222223175896539
s253 = 4.925927038545962
s353 = 29.55556223127577
s453 = 8.444446351793077

s154 = 2.444442325996028
s254 = 2.851849380328699
s354 = 17.1110962819722
s454 = 4.888884651992056

s155 = 3.333336090048154
s255 = 3.88889210505618
s355 = 23.33335263033708
s455 = 6.666672180096308

s156 = -1.555555131596824
s256 = -1.814814320196294
s356 = -10.88888592117776
s456 = -3.111110263193647

s157 = -0.222223957379659
s257 = -0.2592612836096022
s357 = -1.555567701657613
s457 = -0.4444479147593181

s158 = -1.111111110904151
s258 = -1.296296296054842
s358 = -7.777777776329054
s458 = -2.222222221808301

s159 = 1.555555767483181
s259 = 1.814815062063712
s359 = 10.88889037238227
s459 = 3.111111534966363

s160 = -1.555556297302246
s260 = -1.814815680185954
s360 = -10.88889408111572
s460 = -3.111112594604492

s161 = 2.000000211927626
s261 = 2.33333358058223
s361 = 14.00000148349338
s461 = 4.000000423855252

s162 = -5.555557674003972
s262 = -6.481483953004634
s362 = -38.8889037180278
s462 = -11.11111534800794

s163 = 1.555556510885557
s263 = 1.814815929366483
s363 = 10.8888955761989
s463 = 3.111113021771113

s164 = 3.333335242337651
s264 = 3.888891116060592
s364 = 23.33334669636355
s464 = 6.666670484675301

s165 = 3.777779898709721
s265 = 4.407409881828008
s365 = 26.44445929096805
s465 = 7.555559797419442

s166 = 2.888888994852702
s266 = 3.370370493994819
s366 = 20.22222296396891
s466 = 5.777777989705404

s167 = -2.000000847296582
s267 = -2.333334321846013
s367 = -14.00000593107608
s467 = -4.000001694593165

s168 = 3.333333333333333
s268 = 3.888888888888889
s368 = 23.33333333333333
s468 = 6.666666666666667

s169 = 3.333334606554773
s269 = 3.888890374313902
s369 = 23.33334224588341
s469 = 6.666669213109546

s170 = -0.2222222487131755
s270 = -0.2592592901653714
s370 = -1.555555740992228
s470 = -0.444444497426351

s171 = 2.000000425510936
s271 = 2.333333829762759
s371 = 14.00000297857655
s471 = 4.000000851021873

s172 = 2.000000423855252
s272 = 2.333333827831127
s372 = 14.00000296698676
s472 = 4.000000847710503

s173 = -2.000001271255314
s273 = -2.333334816464533
s373 = -14.0000088987872
s473 = -4.000002542510629

s174 = 1.555554919772678
s274 = 1.814814073068124
s374 = 10.88888443840875
s474 = 3.111109839545356

s175 = -1.999999999586079
s275 = -2.333333332850425
s375 = -13.99999999710255
s475 = -3.999999999172158

s176 = 3.333331003785133
s276 = 3.888886171082656
s376 = 23.33331702649593
s476 = 6.666662007570267

s177 = 3.777778203288714
s277 = 4.407407903836833
s377 = 26.4444474230210
s477 = 7.555556406577428

s178 = 4.666667620340983
s278 = 5.44444555706448
s378 = 32.66667334238688
s478 = 9.333335240681966

s179 = -0.2222219041238229
s279 = -0.259258888144460
s379 = -1.55555332886676
s479 = -0.4444438082476457

s180 = 2.888888041178385
s280 = 3.370369381374783
s380 = 20.2222162882487
s480 = 5.777776082356771

s181 = 2.00000000000000
s281 = 2.333333333333333
s381 = 14.0000000000000
s481 = 4.00000000000000

s182 = 2.444443915453222
s282 = 2.851851234695426
s382 = 17.11110740817255
s482 = 4.888887830906444

s183 = 5.111112806532118
s283 = 5.962964940954138
s383 = 35.77778964572483
s483 = 10.22222561306424

s184 = 2.888888782925076
s284 = 3.370370246745922
s384 = 20.22222148047553
s484 = 5.777777565850152

s185 = 2.444444021417035
s285 = 2.851851358319874
s385 = 17.11110814991924
s485 = 4.88888804283407

s186 = 4.222222328186035
s286 = 4.925926049550374
s386 = 29.55555629730225
s486 = 8.44444465637207