4.6E: Mazoezi

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Mazoezi hufanya kamili

Andika Matrix iliyoongezwa kwa Mfumo wa Ulinganisho

Katika mazoezi yafuatayo, weka kila mfumo wa equations linear kama tumbo la kuongezeka.

ⓐ$\left\{ \begin{array} {l} 3x−y=−1\\ 2y=2x+5\end{array} \right.$
ⓑ$\left\{ \begin{array} {l} 4x+3y=−2\\ x−2y−3z=7 \\ 2x−y+2z=−6 \end{array} \right.$

ⓐ$\left\{ \begin{array} {l} 2x+4y=−5\\ 3x−2y=2\end{array} \right.$
ⓑ$\left\{ \begin{array} {l} 3x−2y−z=−2\\ −2x+y=5 \\ 5x+4y+z=−1 \end{array} \right.$

Jibu: ⓐ$\left[ \begin{matrix} 2 &4 &−5 \\ 3 &−2 &2 \end{matrix} \right]$
ⓑ$\left[ \begin{matrix} 3 &−2 &−1 &−2 \\ −2 &1 &0 &5 \\ 5 &4 &1 &−1 \end{matrix} \right]$

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

ⓐ$\left\{ \begin{array} {l} 2x−5y=−3 \\ 4x=3y−1 \end{array} \right.$
ⓑ$\left\{ \begin{array} {l} 4x+3y−2z=−3 \\ −2x+y−3z=4 \\ −x−4y+5z=−2 \end{array} \right.$

Jibu: ⓐ$\left[ \begin{matrix} 2 &−5 &−3 \\ 4 &−3 &−1 \end{matrix} \right]$
ⓑ$\left[ \begin{matrix} 4 &3 &−2 &−3 \\ −2 &1 &−3 &4 \\ −1 &−4 &5 &−2 \end{matrix} \right]$

Andika mfumo wa equations unaofanana na tumbo la kuongezeka.

$\left[ \begin{array} {cc|c} 2 &−1 &4 \\ 1 &−3 &2 \end{array} \right]$

$\left[ \begin{array} {cc|c} 2 &−4 &-2 \\ 3 &−3 &-1 \end{array} \right]$

Jibu: $\left\{ \begin{array} {l} 2x−4y=−2 \\ 3x−3y=−1 \end{array} \right.$

$\left[ \begin{array} {ccc|c} 1 &0 &−3 &-1 \\ 1 &−2 &0 &-2 \\ 0 &−1 &2 &3 \end{array} \right]$

$\left[ \begin{array} {ccc|c} 2 &−2 &0 &-1 \\ 0 &2 &−1 &2 \\ 3 &0 &−1 &-2 \end{array} \right]$

Jibu: $\left\{ \begin{array} {l} 2x−2y=−1 \\ 2y−z=2 \\ 3x−z=−2 \end{array} \right.$

Matumizi Row Operations juu ya Matrix

Katika mazoezi yafuatayo, fanya shughuli zilizoonyeshwa kwenye matrices yaliyoongezeka.

$\left[ \begin{array} {cc|c} 6 &−4 &3 \\ 3 &−2 &1 \end{array} \right]$

ⓐ Kubadilishana safu 1 na 2

ⓑ Kuzidisha mstari 2 na 3

$\left[ \begin{array} {cc|c} 4 &−6 &-3 \\ 3 &2 &1 \end{array} \right]$

ⓐ Kubadilishana safu 1 na 2

ⓑ Kuzidisha mstari 1 na 4

Jibu: ⓐ$\left[ \begin{matrix} 3 &2 &1 \\ 4 &−6 &−3 \end{matrix} \right]$
ⓑ$\left[ \begin{matrix} 12 &8 &4 \\ 4 &−6 &−3 \end{matrix} \right]$
ⓒ$\left[ \begin{matrix} 12 &8 &4 \\ 24 &−10 &−5 \end{matrix} \right]$

$\left[ \begin{array} {ccc|c} 4 &−12 &−8 &16 \\ 4 &−2 &−3 &-1 \\ −6 &2 &−1 &-1 \end{array} \right]$

$\left[ \begin{array} {ccc|c} 6 &−5 &2 &3 \\ 2 &1 &−4 &5 \\ 3 &−3 &1 &-1 \end{array} \right]$

Jibu: ⓐ$\left[ \begin{matrix} 2 &1 &−4 &5 \\ 6 &−5 &2 &3 \\ 3 &−3 &1 &−1 \end{matrix} \right]$
ⓑ$\left[ \begin{matrix} 2 &1 &−4 &5 \\ 6 &−5 &2 &3 \\ 3 &−3 &1 &−1 \end{matrix} \right]$
ⓒ$\left[ \begin{matrix} 2 &1 &−4 &5 \\ 6 &−5 &2 &3 \\ −4 &7 &−6 &7 \end{matrix} \right]$

Fanya operesheni inayohitajika ya mstari ambayo itapata kuingia kwanza katika mstari wa 2 kuwa sifuri katika tumbo la kuongezeka:$\left[ \begin{array} {cc|c} 1 &2 &5 \\ −3 &−4 &-1 \end{array} \right]$

Fanya shughuli zinazohitajika za mstari ambazo zitapata kuingia kwanza katika mstari wa 2 na mstari wa 3 kuwa sifuri katika tumbo la kuongezeka:$\left[ \begin{array} {ccc|c} 1 &−2 &3 &-4 \\ 3 &−1 &−2 &5 \\ 2 &−3 &−4 &1 \end{array} \right]$

Jibu: $\left[ \begin{matrix} 1 &−2 &3 &−4 \\ 0 &5 &−11 &17 \\ 0 &1 &−10 &7 \end{matrix} \right]$

Kutatua Mifumo ya Equations Kutumia Matrices

Katika mazoezi yafuatayo, tatua kila mfumo wa equations kwa kutumia tumbo.

$\left\{ \begin{array} {l} 2x+y=2 \\ x−y=−2 \end{array} \right.$

$\left\{ \begin{array} {l} 3x+y=2 \\ x−y=2 \end{array} \right.$

Jibu: $(1,−1)$

$\left\{ \begin{array} {l} −x+2y=−2 \\ x+y=−4 \end{array} \right.$

$\left\{ \begin{array} {l} −2x+3y=3 \\ x+3y=12 \end{array} \right.$

Jibu: $(3,3)$

Katika mazoezi yafuatayo, tatua kila mfumo wa equations kwa kutumia tumbo.

$\left\{ \begin{array} {l} 2x−3y+z=19 \\ −3x+y−2z=−1 \\ 5x+y+z=0 \end{array} \right.$

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

Jibu: $(−2,5,2)$

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

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

Jibu: $(−3,−5,4)$

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

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

Jibu: $(−3,2,3)$

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

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

Jibu: $(−2,0,−3)$

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

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

Jibu: hakuna suluhisho

$\left\{ \begin{array} {l} x+2y−3z=−1 \\ x−3y+z=1 \\ 2x−y−2z=2 \end{array} \right.$

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

Jibu: hakuna suluhisho

$\left\{ \begin{array} {l} x−y+2z=−4 \\ 2x+y+3z=2 \\ −3x+3y−6z=12 \end{array} \right.$

$\left\{ \begin{array} {l} −x−3y+2z=14 \\ −x+2y−3z=−4 \\ 3x+y−2z=6 \end{array} \right.$

Jibu: infinitely wengi ufumbuzi$(x,y,z)$ ambapo$x=12z+4;\space y=12z−6;\space z$ ni idadi yoyote halisi

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

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

Jibu: infinitely wengi ufumbuzi$(x,y,z)$ ambapo$x=5z+2;\space y=−3z+1;\space z$ ni idadi yoyote halisi

Mazoezi ya kuandika

Tatua mfumo wa equations$\left\{ \begin{array} {l} x+y=10 \\ x−y=6\end{array} \right.$ ⓐ kwa kuchora na ⓑ kwa kubadilisha. ⓒ Ni njia ipi unayopendelea? Kwa nini?

Tatua mfumo wa equations$\left\{ \begin{array} {l} 3x+y=1 \\ 2x=y−8 \end{array} \right.$ kwa kubadilisha na kuelezea hatua zako zote kwa maneno.

Jibu: Majibu yatatofautiana.

Self Check

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

$Jedwali hili lina nguzo 4 safu 5 na mstari wa kichwa. Mstari wa kichwa huandika kila safu ninayoweza, kwa ujasiri, kwa msaada na hapana, siipati. safu ya kwanza ina kauli zifuatazo: Andika matrix augmented kwa mfumo wa milinganyo, Matumizi ya shughuli mfululizo juu ya tumbo, Kutatua mifumo ya milinganyo kutumia matrices, Andika Matrix augmented kwa mfumo wa milinganyo, Matumizi shughuli mstari juu ya tumbo. Nguzo zilizobaki ni tupu.$

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