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Ðîçâ’ÿçàííÿ. Øëÿõîì ïåðåòâîðåíü Ãàóññà îòðèìàºìî<br />
æ1 1 1 5ö æ1 1 1 5ö<br />
æ1 1 1 5ö<br />
2 1 1 4 0 3 3 6<br />
- - : - - - :<br />
3 0 0 9<br />
çè ÷ ø.<br />
çè2 2 2 10ø ÷ è ç0 0 0 0ø<br />
÷<br />
Íåõàé òåïåð õ 3 = ñ, äå ñ — äîâ³ëüíà ñòàëà. Òîä³ ðîçâ’ÿçîê<br />
äàíî¿ ñèñòåìè ìຠòàêèé âèãëÿä:<br />
(x 1 = 3; x 2 =2− c; x 3 = c).<br />
ßñíî, ùî çà ðàõóíîê äîâ³ëüíîñò³ ñòàëî¿ ñ ðîçãëÿíóòå ð³âíÿííÿ<br />
ìຠíåñê³í÷åííó ìíîæèíó ðîçâ’ÿçê³â.<br />
Òóò òåîðåìà Êðîíåêåðà–Êàïåëë³ òåæ ãàðíî ïðàöþº:<br />
rang A = rang A*=2.<br />
Ïðèêëàä 3.5.4. Íåõàé ôàáðèêà âèðîáëÿº äèòÿ÷å, æ³íî÷å<br />
³ ÷îëîâ³÷å âçóòòÿ. Ïðè öüîìó âèêîðèñòîâóºòüñÿ ñèðîâèíà<br />
òðüîõ âèä³â: Q 1 , Q 2 , Q 3 . Íàïðèêëàä, Q 1 ìîæå áóòè ñèðîâèíîþ<br />
ç íàòóðàëüíî¿ øê³ðè; Q 2 — ñèðîâèíîþ äëÿ âèãîòîâëåííÿ<br />
ï³äîøâ; Q 3 — ñèðîâèíîþ äëÿ âèãîòîâëåííÿ øíóðê³â. Íîðìè<br />
çàòðàò (â óìîâíèõ îäèíèöÿõ) êîæíî¿ ç íèõ íà îäíó ïàðó<br />
âçóòòÿ òà îáñÿã çàòðàò ñèðîâèíè íà 1 äåíü çàäàí³ ó âèãëÿä³<br />
ìàòðèö³ A òà âåêòîðà-ñòîâïöÿ b:<br />
æ1 2 3ö æ1300ö<br />
A= 1 3 4 , 1700<br />
b=<br />
.<br />
çè1 2 5 ÷ ø è ç1700ø<br />
÷<br />
Òóò ïåðøèé ðÿäîê ìàòðèö³ îçíà÷àº, ùî íà äèòÿ÷å, æ³íî÷å<br />
³ ÷îëîâ³÷å âçóòòÿ çà 1 äåíü âèòðà÷àºòüñÿ â³äïîâ³äíî îäíà<br />
îäèíèöÿ, äâ³ îäèíèö³ ³ òðè îäèíèö³ ñèðîâèíè Q 1 . Àíàëîã³÷íèé<br />
ñìèñë ìຠäðóãèé òà òðåò³é ðÿäîê. Ïðè öüîìó âåêòîðñòîâïåöü<br />
âèçíà÷ຠîáñÿã çàòðàò ñèðîâèíè çà 1 äåíü.<br />
Òðåáà çíàéòè ùîäåííèé îáñÿã âèïóñêó êîæíîãî âèäó<br />
âçóòòÿ.<br />
Ð î ç â ’ ÿ ç à í í ÿ. Íåõàé ùîäåííî ôàáðèêà âèïóñêຠõ 1<br />
ïàð äèòÿ÷îãî, õ 2 ïàð æ³íî÷îãî ³ õ 3 ïàð ÷îëîâ³÷îãî âçóòòÿ.<br />
Öåé ôàêò çàïèøåìî ó âèãëÿä³ âåêòîðà-ñòîâïöÿ:<br />
x<br />
æ ö x1<br />
= x<br />
2<br />
.<br />
ç çèx<br />
÷<br />
3 ø÷<br />
Òîä³, ó â³äïîâ³äíîñò³ äî çàòðàò ñèðîâèíè êîæíîãî âèäó<br />
áóäåìî ìàòè ìàòðè÷íå ð³âíÿííÿ Ax = b àáî ñèñòåìó<br />
ì x1+ 2x2 + 3x3<br />
=<br />
1300,<br />
ï<br />
íx1 + 3x2 + 4x3<br />
= 1700,<br />
ï ïî x1+ 2x2 + 5x3<br />
= 1700.<br />
Ðîçâ’ÿçîê ö³º¿ ñèñòåìè áóäåìî øóêàòè çà ïðàâèëîì<br />
Êðàìåðà.<br />
1 2 3 1 2 3 1300 2 3 13 2 3<br />
D= 1 3 4 = 0 1 1 = 2, D<br />
1<br />
= 1700 3 4 = 100× 17 3 4 =<br />
1 2 5 0 0 2 1700 2 5 17 2 5<br />
13 2 3 13 1 3<br />
5 1<br />
æ 1ö 5 1<br />
= ×- - = × 2 2 ç - × × × =- × - =<br />
çè 2÷<br />
ø<br />
2 1<br />
4 0 2<br />
201<br />
1+<br />
2<br />
100 0 100 2 2 5 0 1 200 ( 1) 600<br />
1 1300 3 1 1300 3 1 2 1300<br />
D = 1 1700 4 = 0 400 1 = 400, D = 1 3 1700 = 400.<br />
2 3<br />
1 1700 5 0 400 2 1 2 1700<br />
Çàñòîñóºìî òåïåð ôîðìóëè (3.3.2):<br />
D<br />
D<br />
D<br />
x = = x = = x = =<br />
D D D<br />
1 2<br />
3<br />
1<br />
300,<br />
2<br />
200,<br />
3<br />
200.<br />
Îòæå, çàäà÷ó ðîçâ’ÿçàíî ³ ìè ìîæåìî ñêàçàòè, ùî ôàáðèêà<br />
ùîäåííî âèïóñêຠ300 ïàð äèòÿ÷îãî ³ ïî 200 ïàð æ³íî-<br />
÷îãî òà ÷îëîâ³÷îãî âçóòòÿ.<br />
×èòà÷åâ³ ïðîïîíóºìî ðîç³áðàòèñÿ ç ïðèéîìàìè îá÷èñëåííÿ<br />
âèçíà÷íèê³â, à òàêîæ ðîçâ’ÿçàòè öþ çàäà÷ó ³íøèìè ìåòîäàìè.<br />
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