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11.6E: Mazoezi

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    Mazoezi hufanya kamili

    Zoezi\(\PageIndex{17}\) Solve a System of Nonlinear Equations Using Graphing

    Katika mazoezi yafuatayo, tatua mfumo wa equations kwa kutumia graphing.

    1. \(\left\{\begin{array}{l}{y=2 x+2} \\ {y=-x^{2}+2}\end{array}\right.\)
    2. \(\left\{\begin{array}{l}{y=6 x-4} \\ {y=2 x^{2}}\end{array}\right.\)
    3. \(\left\{\begin{array}{l}{x+y=2} \\ {x=y^{2}}\end{array}\right.\)
    4. \(\left\{\begin{array}{l}{x-y=-2} \\ {x=y^{2}}\end{array}\right.\)
    5. \(\left\{\begin{array}{l}{y=\frac{3}{2} x+3} \\ {y=-x^{2}+2}\end{array}\right.\)
    6. \(\left\{\begin{array}{l}{y=x-1} \\ {y=x^{2}+1}\end{array}\right.\)
    7. \(\left\{\begin{array}{l}{x=-2} \\ {x^{2}+y^{2}=4}\end{array}\right.\)
    8. \(\left\{\begin{array}{l}{y=-4} \\ {x^{2}+y^{2}=16}\end{array}\right.\)
    9. \(\left\{\begin{array}{l}{x=2} \\ {(x+2)^{2}+(y+3)^{2}=16}\end{array}\right.\)
    10. \(\left\{\begin{array}{l}{y=-1} \\ {(x-2)^{2}+(y-4)^{2}=25}\end{array}\right.\)
    11. \(\left\{\begin{array}{l}{y=-2 x+4} \\ {y=\sqrt{x}+1}\end{array}\right.\)
    12. \(\left\{\begin{array}{l}{y=-\frac{1}{2} x+2} \\ {y=\sqrt{x}-2}\end{array}\right.\)
    Jibu

    2.

    Grafu hii inaonyesha milinganyo ya mfumo, y ni sawa na 6 x minus 4 ambayo ni mstari na y ni sawa na 2 x squared ambayo ni parabola, kwenye ndege x y-kuratibu. Vertex ya parabola ni (0, 0) na parabola inafungua juu. Mstari una mteremko wa 6. Mstari na parabola huingiliana kwenye pointi (1, 2) na (2, 8), ambazo zimeandikwa. Ufumbuzi ni (1, 2) na (2, 8).
    Kielelezo 11.5.61

    4.

    Grafu hii inaonyesha milinganyo ya mfumo, x bala y ni sawa na hasi 2 ambayo ni mstari na x ni sawa na y squared ambayo ni parabola ya kufungua kulia, kwenye ndege ya kuratibu x y. Vertex ya parabola ni (0, 0) na inapita kupitia pointi (1, 1) na (1, hasi 1). Mstari una mteremko wa 1 na y-intercept saa 2. Mstari na parabola hazipatikani, hivyo mfumo hauna suluhisho.
    Kielelezo 11.5.62

    6.

    Grafu hii inaonyesha milinganyo ya mfumo, y ni x minus 1 ambayo ni mstari na y ni sawa na x squared plus 1 ambayo ni parabola ya kufungua juu, kwenye ndege ya kuratibu x y. Vertex ya parabola ni (0, 1) na inapita kupitia pointi (hasi 1, 2) na (1, 2). Mstari una mteremko wa 1 na y-intercept saa hasi 1. Mstari na parabola hazipatikani, hivyo mfumo hauna suluhisho.
    Kielelezo 11.5.63

    8.

    Grafu hii inaonyesha milinganyo ya mfumo, x ni sawa na hasi 2 ambayo ni mstari na x squared pamoja y squared ni sawa na 16 ambayo ni mduara, kwenye ndege x y-kuratibu. Mstari ni usawa. Katikati ya mduara ni (0, 0) na radius ya mduara ni 4. Mstari na mduara huingiliana (hasi 2, 0), hivyo suluhisho la mfumo ni (hasi 2, 0).
    Kielelezo 11.5.64

    10.

    Grafu hii inaonyesha equations ya mfumo, x ni sawa na 2 ambayo ni mstari na kiasi x bala 2 mwisho wingi squared pamoja wingi y bala 4 mwisho kiasi mraba ni sawa na 25 ambayo ni mduara, kwenye ndege x y kuratibu. Mstari ni usawa. Katikati ya mduara ni (2, 4) na radius ya mduara ni 5. Mstari na mduara huingiliana (2, hasi 1), hivyo suluhisho la mfumo ni (2, hasi 1).
    Kielelezo 11.5.65

    12.

    Grafu hii inaonyesha milinganyo ya mfumo, y ni sawa na nusu moja x pamoja na 2 ambayo ni mstari na y ni sawa na mizizi ya mraba ya x bala 2, kwenye ndege ya kuratibu x y. Curve kwa y ni sawa na mizizi ya mraba ya x minus 2 Curve kwa y ni sawa na mizizi ya mraba ya x plus 1 ambapo x ni kubwa kuliko au sawa na 0 na y ni kubwa kuliko au sawa na hasi 2. Mstari wa mstari na mraba wa mizizi huingiliana (4, 0), hivyo suluhisho ni (4, 0).
    Kielelezo 11.5.66
    Zoezi\(\PageIndex{18}\) Solve a System of Nonlinear Equations Using Substitution

    Katika mazoezi yafuatayo, tatua mfumo wa equations kwa kutumia badala.

    1. \(\left\{\begin{array}{l}{x^{2}+4 y^{2}=4} \\ {y=\frac{1}{2} x-1}\end{array}\right.\)
    2. \(\left\{\begin{array}{l}{9 x^{2}+y^{2}=9} \\ {y=3 x+3}\end{array}\right.\)
    3. \(\left\{\begin{array}{l}{9 x^{2}+y^{2}=9} \\ {y=x+3}\end{array}\right.\)
    4. \(\left\{\begin{array}{l}{9 x^{2}+4 y^{2}=36} \\ {x=2}\end{array}\right.\)
    5. \(\left\{\begin{array}{l}{4 x^{2}+y^{2}=4} \\ {y=4}\end{array}\right.\)
    6. \(\left\{\begin{array}{l}{x^{2}+y^{2}=169} \\ {x=12}\end{array}\right.\)
    7. \(\left\{\begin{array}{l}{3 x^{2}-y=0} \\ {y=2 x-1}\end{array}\right.\)
    8. \(\left\{\begin{array}{l}{2 y^{2}-x=0} \\ {y=x+1}\end{array}\right.\)
    9. \(\left\{\begin{array}{l}{y=x^{2}+3} \\ {y=x+3}\end{array}\right.\)
    10. \(\left\{\begin{array}{l}{y=x^{2}-4} \\ {y=x-4}\end{array}\right.\)
    11. \(\left\{\begin{array}{l}{x^{2}+y^{2}=25} \\ {x-y=1}\end{array}\right.\)
    12. \(\left\{\begin{array}{l}{x^{2}+y^{2}=25} \\ {2 x+y=10}\end{array}\right.\)
    Jibu

    2. \((-1,0),(0,3)\)

    4. \((2,0)\)

    6. \((12,-5),(12,5)\)

    8. Hakuna ufumbuzi

    10. \((0,-4),(1,-3)\)

    12. \((3,4),(5,0)\)

    Zoezi\(\PageIndex{19}\) Solve a System of Nonlinear Equations Using Elimination

    Katika mazoezi yafuatayo, tatua mfumo wa equations kwa kutumia kuondoa.

    1. \(\left\{\begin{array}{l}{x^{2}+y^{2}=16} \\ {x^{2}-2 y=8}\end{array}\right.\)
    2. \(\left\{\begin{array}{l}{x^{2}+y^{2}=16} \\ {x^{2}-y=4}\end{array}\right.\)
    3. \(\left\{\begin{array}{l}{x^{2}+y^{2}=4} \\ {x^{2}+2 y=1}\end{array}\right.\)
    4. \(\left\{\begin{array}{l}{x^{2}+y^{2}=4} \\ {x^{2}-y=2}\end{array}\right.\)
    5. \(\left\{\begin{array}{l}{x^{2}+y^{2}=9} \\ {x^{2}-y=3}\end{array}\right.\)
    6. \(\left\{\begin{array}{l}{x^{2}+y^{2}=4} \\ {y^{2}-x=2}\end{array}\right.\)
    7. \(\left\{\begin{array}{l}{x^{2}+y^{2}=25} \\ {2 x^{2}-3 y^{2}=5}\end{array}\right.\)
    8. \(\left\{\begin{array}{l}{x^{2}+y^{2}=20} \\ {x^{2}-y^{2}=-12}\end{array}\right.\)
    9. \(\left\{\begin{array}{l}{x^{2}+y^{2}=13} \\ {x^{2}-y^{2}=5}\end{array}\right.\)
    10. \(\left\{\begin{array}{l}{x^{2}+y^{2}=16} \\ {x^{2}-y^{2}=16}\end{array}\right.\)
    11. \(\left\{\begin{array}{l}{4 x^{2}+9 y^{2}=36} \\ {2 x^{2}-9 y^{2}=18}\end{array}\right.\)
    12. \(\left\{\begin{array}{l}{x^{2}-y^{2}=3} \\ {2 x^{2}+y^{2}=6}\end{array}\right.\)
    13. \(\left\{\begin{array}{l}{4 x^{2}-y^{2}=4} \\ {4 x^{2}+y^{2}=4}\end{array}\right.\)
    14. \(\left\{\begin{array}{l}{x^{2}-y^{2}=-5} \\ {3 x^{2}+2 y^{2}=30}\end{array}\right.\)
    15. \(\left\{\begin{array}{l}{x^{2}-y^{2}=1} \\ {x^{2}-2 y=4}\end{array}\right.\)
    16. \(\left\{\begin{array}{l}{2 x^{2}+y^{2}=11} \\ {x^{2}+3 y^{2}=28}\end{array}\right.\)
    Jibu

    2. \((0,-4),(-\sqrt{7}, 3),(\sqrt{7}, 3)\)

    4. \((0,-2),(-\sqrt{3}, 1),(\sqrt{3}, 1)\)

    6. \((-2,0),(1,-\sqrt{3}),(1, \sqrt{3})\)

    8. \((-2,-4),(-2,4),(2,-4),(2,4)\)

    10. \((-4,0),(4,0)\)

    12. \((-\sqrt{3}, 0),(\sqrt{3}, 0)\)

    14. \((-2,-3),(-2,3),(2,-3),(2,3)\)

    16. \((-1,-3),(-1,3),(1,-3),(1,3)\)

    Zoezi\(\PageIndex{20}\) Use a System of Nonlinear Equations to Solve Applications

    Katika mazoezi yafuatayo, tatua tatizo kwa kutumia mfumo wa equations.

    1. Jumla ya namba mbili ni\(−6\) na bidhaa ni\(8\). Kupata idadi.
    2. Jumla ya namba mbili ni\(11\) na bidhaa ni\(−42\). Kupata idadi.
    3. Jumla ya mraba ya namba mbili ni\(65\). Tofauti ya idadi ni\(3\). Kupata idadi.
    4. Jumla ya mraba ya namba mbili ni\(113\). Tofauti ya idadi ni\(1\). Kupata idadi.
    5. Tofauti ya mraba wa namba mbili ni\(15\). Tofauti ya mara mbili mraba wa namba ya kwanza na mraba wa namba ya pili ni\(30\). Kupata idadi.
    6. Tofauti ya mraba wa namba mbili ni\(20\). Tofauti ya mraba wa namba ya kwanza na mara mbili mraba wa namba ya pili ni\(4\). Kupata idadi.
    7. Mzunguko wa mstatili ni\(32\) inchi na eneo lake ni inchi\(63\) za mraba. Pata urefu na upana wa mstatili.
    8. Mzunguko wa mstatili ni\(52\) cm na eneo lake ni\(165\)\(\mathrm{cm}^{2}\). Pata urefu na upana wa mstatili.
    9. Dion kununuliwa microwave mpya. Ulalo wa mlango hupima\(17\) inchi. Mlango pia una eneo la inchi za\(120\) mraba. Urefu na upana wa mlango wa microwave ni nini?
    10. Jules alinunua microwave kwa jikoni yake. Ulalo wa mbele wa microwave hupima\(26\) inchi. Mbele pia ina eneo la inchi za\(240\) mraba. Urefu na upana wa microwave ni nini?
    11. Kirumi kupatikana widescreen TV kuuzwa, lakini si uhakika kama inafaa kituo chake burudani. TV ni\(60\)”. Ukubwa wa TV hupimwa kwenye diagonal ya skrini na widescreen ina urefu ambao ni mkubwa kuliko upana. Skrini pia ina eneo la inchi za\(1728\) mraba. Kituo chake cha burudani kina kuingizwa kwa TV yenye urefu wa\(50\) inchi na upana wa\(40\) inchi. Urefu na upana wa skrini ya TV ni nini na itafaa kituo cha burudani cha Kirumi?
    12. Donnette alipata TV ya widescreen katika mauzo ya karakana, lakini hajui kama itafaa kituo chake cha burudani. TV ni\(50\)”. Ukubwa wa TV hupimwa kwenye diagonal ya skrini na widescreen ina urefu ambao ni mkubwa kuliko upana. Skrini pia ina eneo la inchi za\(1200\) mraba. Kituo chake cha burudani kina kuingiza kwa TV yenye urefu wa\(38\) inchi na upana wa\(27\) inchi. Urefu na upana wa skrini ya TV ni nini na itafaa kituo cha burudani cha Donnette?
    Jibu

    2. \(-3\)na\(14\)

    4. \(-7\)na\(-8\) au\(8\)\(7\)

    6. \(-6\)na\(-4\) au\(-6\)\(6\) na\(4\) au\(-4\) au\(6\) na\(4\)

    8. Ikiwa urefu ni\(11\) cm, upana ni\(15\) cm. Ikiwa urefu ni\(15\) cm, upana ni\(11\) cm.

    10. Ikiwa urefu ni\(10\) inchi, upana ni\(24\) inchi. Ikiwa urefu ni\(24\) inchi, upana ni\(10\) inchi.

    12. Urefu ni\(40\) inchi na upana ni\(30\) inchi. TV haifai kituo cha burudani cha Donnette.

    Zoezi\(\PageIndex{21}\) Writing Exercises
    1. Kwa maneno yako mwenyewe, kuelezea faida na hasara za kutatua mfumo wa equations kwa kuchora.
    2. Eleza kwa maneno yako mwenyewe jinsi ya kutatua mfumo wa equations kwa kutumia mbadala.
    3. Eleza kwa maneno yako mwenyewe jinsi ya kutatua mfumo wa equations kwa kutumia kuondoa.
    4. Mduara na parabola vinaweza kuingiliana kwa njia ambazo zingeweza kusababisha\(0, 1, 2, 3,\) au\(4\) ufumbuzi. Chora mchoro wa kila uwezekano.
    Jibu

    2. Majibu inaweza kutofautiana

    4. Majibu inaweza kutofautiana

    Self Check

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

    Jedwali hili lina nguzo nne na safu tano. Mstari wa kwanza ni kichwa na inaandika kila safu, â € can â € €, â € kwa ujasiri, â € € â € kwa msaada fulani, â € na â € no-i donâ €™ t kupata! â € Katika mstari wa 2, naweza mara kutatua mfumo wa equations nonlinear kutumia graphing. Katika mstari wa 3, naweza kutatua mfumo wa equations nonlinear kutumia badala. Katika mstari wa 4, naweza mara kutatua mfumo wa equations nonlinear kutumia kuondoa. Katika mstari wa 5, naweza mara kutumia mfumo wa equations nonlinear kutatua maombi.
    Kielelezo 11.5.67

    b Baada ya kuangalia orodha, unafikiri umeandaliwa vizuri kwa sehemu inayofuata? Kwa nini au kwa nini?