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4.5E: Mazoezi

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    Mazoezi hufanya kamili

    Tambua Iwapo Triple Iliyoamriwa ni Suluhisho la Mfumo wa Ulinganisho wa Mstari wa Tatu na Vigezo

    Katika mazoezi yafuatayo, onyesha kama mara tatu zilizoamriwa ni suluhisho la mfumo.

    1. \(\left\{ \begin{array} {l} 2x−6y+z=3 \\ 3x−4y−3z=2 \\ 2x+3y−2z=3 \end{array} \right. \)

    ⓐ\((3,1,3)\)
    ⓑ\((4,3,7)\)

    2. \(\left\{ \begin{array} {l} -3x+y+z=-4 \\ -x+2y-2z=1 \\ 2x-y-z=-1 \end{array} \right. \)

    ⓐ\((−5,−7,4)\)
    ⓑ\((5,7,4)\)

    Jibu

    ⓐ hapana ⓑ ndiyo

    3. \(\left\{ \begin{array} {l} y−10z=−8 \\ 2x−y=2 \\ x−5z=3 \end{array} \right. \)

    ⓐ\((7,12,2)\)
    ⓑ\((2,2,1)\)

    4. \(\left\{ \begin{array} {l} x+3y−z=1 \\ 5y=\frac{2}{3}x \\ −2x−3y+z=−2 \end{array} \right. \)

    ⓐ\((−6,5,12)\)
    ⓑ\((5,\frac{4}{3},−3)\)

    Jibu

    ⓐ hapana ⓑ ndiyo

    Tatua Mfumo wa Ulinganisho wa Mstari na Vigezo vitatu

    Katika mazoezi yafuatayo, tatua mfumo wa equations.

    5. \(\left\{ \begin{array} {l} 5x+2y+z=5 \\ −3x−y+2z=6 \\ 2x+3y−3z=5 \end{array} \right. \)

    6. \(\left\{ \begin{array} {l} 6x−5y+2z=3 \\ 2x+y−4z=5 \\ 3x−3y+z=−1 \end{array} \right. \)

    Jibu

    \((4,5,2)\)

    7. \(\left\{ \begin{array} {l} 2x−5y+3z=8 \\ 3x−y+4z=7 \\ x+3y+2z=−3 \end{array} \right. \)

    8. \(\left\{ \begin{array} {l} 5x−3y+2z=−5 \\ 2x−y−z=4 \\ 3x−2y+2z=−7 \end{array} \right. \)

    Jibu

    \((7,12,−2)\)

    9. \(\left\{ \begin{array} {l} 3x−5y+4z=5 \\ 5x+2y+z=0 \\ 2x+3y−2z=3 \end{array} \right. \)

    10. \(\left\{ \begin{array} {l} 4x−3y+z=7 \\ 2x−5y−4z=3 \\ 3x−2y−2z=−7 \end{array} \right. \)

    Jibu

    \((−3,−5,4)\)

    11. \(\left\{ \begin{array} {l} 3x+8y+2z=−5 \\ 2x+5y−3z=0 \\ x+2y−2z=−1 \end{array} \right. \)

    12. \(\left\{ \begin{array} {l} 11x+9y+2z=−9 \\ 7x+5y+3z=−7 \\ 4x+3y+z=−3 \end{array} \right. \)

    Jibu

    \((2,−3,−2)\)

    13. \(\left\{ \begin{array} {l} \frac{1}{3}x−y−z=1 \\ x+\frac{5}{2}y+z=−2 \\ 2x+2y+\frac{1}{2}z=−4 \end{array} \right. \)

    14. \(\left\{ \begin{array} {l} x+\frac{1}{2}y+\frac{1}{2}z=0 \\ \frac{1}{5}x−\frac{1}{5}y+z=0 \\ \frac{1}{3}x−\frac{1}{3}y+2z=−1 \end{array} \right. \)

    Jibu

    \((6,−9,−3)\)

    15. \(\left\{ \begin{array} {l} x+\frac{1}{3}y−2z=−1 \\ \frac{1}{3}x+y+\frac{1}{2}z=0 \\ \frac{1}{2}x+\frac{1}{3}y−\frac{1}{2}z=−1 \end{array} \right. \)

    16. \(\left\{ \begin{array} {l} \frac{1}{3}x−y+\frac{1}{2}z=4 \\ \frac{2}{3}x+\frac{5}{2}y−4z=0 \\ x−\frac{1}{2}y+\frac{3}{2}z=2 \end{array} \right. \)

    Jibu

    \((3,−4,−2)\)

    17. \(\left\{ \begin{array} {l} x+2z=0 \\ 4y+3z=−2 \\ 2x−5y=3 \end{array} \right. \)

    18. \(\left\{ \begin{array} {l} 2x+5y=4 \\ 3y−z=\frac{3}{4} \\ x+3z=−3 \end{array} \right. \)

    Jibu

    \((−3,2,3)\)

    19. \(\left\{ \begin{array} {l} 2y+3z=−1 \\ 5x+3y=−6 \\ 7x+z=1 \end{array} \right. \)

    20. \(\left\{ \begin{array} {l} 3x−z=−3 \\ 5y+2z=−6 \\ 4x+3y=−8 \end{array} \right. \)

    Jibu

    \((−2,0,−3)\)

    21. \(\left\{ \begin{array} {l} 4x−3y+2z=0 \\ −2x+3y−7z=1 \\ 2x−2y+3z=6 \end{array} \right. \)

    22. \(\left\{ \begin{array} {l} x−2y+2z=1 \\ −2x+y−z=2 \\ x−y+z=5 \end{array} \right. \)

    Jibu

    hakuna suluhisho

    23. \(\left\{ \begin{array} {l} 2x+3y+z=1 \\ 2x+y+z=9 \\ 3x+4y+2z=20 \end{array} \right. \)

    24. \(\left\{ \begin{array} {l} x+4y+z=−8 \\ 4x−y+3z=9 \\ 2x+7y+z=0 \end{array} \right. \)

    Jibu

    \(x=\frac{203}{16};\space y=–\frac{25}{16};\space z=–\frac{231}{16};\)

    25. \(\left\{ \begin{array} {l} x+2y+z=4 \\ x+y−2z=3 \\ −2x−3y+z=−7 \end{array} \right. \)

    26. \(\left\{ \begin{array} {l} x+y−2z=3 \\ −2x−3y+z=−7 \\ x+2y+z=4 \end{array} \right. \)

    Jibu

    \((x,y,z)\)wapi\(x=5z+2;\space y=−3z+1;\space z\) nambari yoyote halisi

    27. \(\left\{ \begin{array} {l} x+y−3z=−1 \\ y−z=0 \\ −x+2y=1 \end{array} \right. \)

    28. \(\left\{ \begin{array} {l} x−2y+3z=1 \\ x+y−3z=7 \\ 3x−4y+5z=7 \end{array} \right. \)

    Jibu

    \((x,y,z)\)wapi\(x=5z−2;\space y=4z−3;\space z\) nambari yoyote halisi

    Tatua Maombi kwa kutumia Mifumo ya Ulinganisho wa mstari na Vigezo vitatu

    Katika mazoezi yafuatayo, tatua tatizo lililopewa.

    29. Jumla ya hatua za pembe za pembetatu ni 180. Jumla ya hatua za pembe ya pili na ya tatu ni mara mbili ya kipimo cha angle ya kwanza. Pembe ya tatu ni kumi na mbili zaidi ya pili. Pata hatua za pembe tatu.

    30. Jumla ya hatua za pembe za pembetatu ni 180. Jumla ya hatua za pembe ya pili na ya tatu ni mara tatu kipimo cha angle ya kwanza. Pembe ya tatu ni kumi na tano zaidi ya pili. Pata hatua za pembe tatu.

    Jibu

    42, 50, 58

    31. Baada ya kuangalia uzalishaji mkubwa wa muziki kwenye ukumbi wa michezo, watumishi wanaweza kununua zawadi. Ikiwa familia inununua mashati 4, video, na mnyama 1 ulioingizwa, jumla yao ni $135.

    Wanandoa hununua t-shirt 2, video, na wanyama 3 waliofunikwa kwa watoto wao na hutumia $115. Wanandoa wengine hununua t-shirt 2, video, na 1 stuffed mnyama na jumla yao ni $85. Ni gharama gani ya kila kitu?

    32. Kikundi cha vijana wa kanisa kinauza vitafunio ili kukusanya pesa ili kuhudhuria mkataba wao. Amy aliuza paundi 2 za pipi, masanduku 3 ya biskuti na 1 unaweza ya popcorn kwa mauzo ya jumla ya $65. Brian aliuza paundi 4 za pipi, masanduku 6 ya biskuti na makopo 3 ya popcorn kwa mauzo ya jumla ya $140. Paulina aliuza paundi 8 za pipi, masanduku 8 ya biskuti na makopo 5 ya popcorn kwa mauzo ya jumla ya $250. Ni gharama gani ya kila kitu?

    Jibu

    $20, $5, $10

    Mazoezi ya kuandika

    33. Kwa maneno yako mwenyewe kuelezea hatua za kutatua mfumo wa equations linear na vigezo vitatu kwa kuondoa.

    34. Unawezaje kujua wakati mfumo wa equations tatu linear na vigezo tatu haina ufumbuzi? Ufumbuzi mkubwa sana?

    Jibu

    Majibu yatatofautiana.

    Self Check

    ⓐ Baada ya kukamilisha mazoezi, tumia orodha hii ili kutathmini ujuzi wako wa malengo ya sehemu hii.

    Jedwali hili lina nguzo 4, safu 3 na mstari wa kichwa. Mstari wa kichwa huandika kila safu ninayoweza, kwa ujasiri, kwa msaada na hapana, siipati. Mstari wa kwanza una taarifa zifuatazo: kuamua kama amri mara tatu ni suluhisho la mfumo wa equations tatu linear na vigezo vitatu, kutatua mfumo wa equations linear na vigezo tatu, kutatua maombi kwa kutumia mifumo ya equations linear na vigezo tatu. Nguzo zilizobaki ni tupu.

    ⓑ Kwa kiwango cha 1-10, ungewezaje kupima ujuzi wako wa sehemu hii kwa kuzingatia majibu yako kwenye orodha? Unawezaje kuboresha hii?