Whakaoti i te {l}{12a+4b=-4}{3a-9b=-21}

$$\left. \begin{array} { l } { 12 a + 4 b = - 4 } \\ { 3 a - 9 b = - 21 } \end{array} \right.$$

Whakaoti mō a, b

$$a=-1$$

$$b=2$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Hei whakaoti i ētahi whārite takirua mā te whakakapinga, me whakaoti tētahi whārite i te tuatahi mō tētahi o ngā taurangi. Ka whakakapi i te otinga mō taua taurangi ki tērā o ngā whārite.

$$12a+4b=-4,3a-9b=-21$$

Kōwhiria tētahi o ngā whārite ka whakaotia mō te a mā te wehe i te a i te taha mauī o te tohu ōrite.

$$12a+4b=-4$$

Me tango 4b mai i ngā taha e rua o te whārite.

$$12a=-4b-4$$

Whakawehea ngā taha e rua ki te 12.

$$a=\frac{1}{12}\left(-4b-4\right)$$

Whakareatia \frac{1}{12} ki te -4b-4.

$$a=-\frac{1}{3}b-\frac{1}{3}$$

Whakakapia te \frac{-b-1}{3} mō te a ki tērā atu whārite, 3a-9b=-21.

$$3\left(-\frac{1}{3}b-\frac{1}{3}\right)-9b=-21$$

Whakareatia 3 ki te \frac{-b-1}{3}.

$$-b-1-9b=-21$$

Tāpiri -b ki te -9b.

$$-10b-1=-21$$

Me tāpiri 1 ki ngā taha e rua o te whārite.

$$-10b=-20$$

Whakawehea ngā taha e rua ki te -10.

$$b=2$$

Whakaurua te 2 mō b ki a=-\frac{1}{3}b-\frac{1}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō a hāngai tonu.

$$a=-\frac{1}{3}\times 2-\frac{1}{3}$$

Whakareatia -\frac{1}{3} ki te 2.

$$a=\frac{-2-1}{3}$$

Tāpiri -\frac{1}{3} ki te -\frac{2}{3} mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$a=-1$$

Kua oti te pūnaha te whakatau.

$$a=-1,b=2$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Tuhia ngā whārite ki te tānga ngahuru ka whakamahi i ngā poukapa hei whakaoti i te pūnaha o ngā whārite.

$$12a+4b=-4,3a-9b=-21$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}12&4\\3&-9\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}-4\\-21\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}12&4\\3&-9\end{matrix}\right).

$$inverse(\left(\begin{matrix}12&4\\3&-9\end{matrix}\right))\left(\begin{matrix}12&4\\3&-9\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}12&4\\3&-9\end{matrix}\right))\left(\begin{matrix}-4\\-21\end{matrix}\right)$$

Ko te hua o tētahi poukapa me te kōaro ko te poukapa tuakiri.

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}12&4\\3&-9\end{matrix}\right))\left(\begin{matrix}-4\\-21\end{matrix}\right)$$

Whakareatia ngā poukapa kei te taha mauī o te tohu ōrite.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}12&4\\3&-9\end{matrix}\right))\left(\begin{matrix}-4\\-21\end{matrix}\right)$$

Mō te poukapa 2\times 2 \left(\begin{matrix}a&b\\c&d\end{matrix}\right), ko te \left(\begin{matrix}\frac{d}{ad-bc}&\frac{-b}{ad-bc}\\\frac{-c}{ad-bc}&\frac{a}{ad-bc}\end{matrix}\right) te poukapa kōaro, nō reira ka taea te tuhi anō te whārite poukapa hei rapanga whakarea poukapa.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}-\frac{9}{12\left(-9\right)-4\times 3}&-\frac{4}{12\left(-9\right)-4\times 3}\\-\frac{3}{12\left(-9\right)-4\times 3}&\frac{12}{12\left(-9\right)-4\times 3}\end{matrix}\right)\left(\begin{matrix}-4\\-21\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}\frac{3}{40}&\frac{1}{30}\\\frac{1}{40}&-\frac{1}{10}\end{matrix}\right)\left(\begin{matrix}-4\\-21\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}\frac{3}{40}\left(-4\right)+\frac{1}{30}\left(-21\right)\\\frac{1}{40}\left(-4\right)-\frac{1}{10}\left(-21\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}-1\\2\end{matrix}\right)$$

Tangohia ngā huānga poukapa a me b.

$$a=-1,b=2$$

Ngā Upane e Whakamahi i te Tangohanga

Hei whakaoti mā te tangohanga, ko ngā tau whakarea o tētahi o ngā taurangi me mātua ōrite i ngā whārite e rua kia whakakorehia ai te taurangi ina tangohia tētahi whārite mai i tētahi atu.

$$12a+4b=-4,3a-9b=-21$$

Kia ōrite ai a 12a me 3a, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 3 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 12.

$$3\times 12a+3\times 4b=3\left(-4\right),12\times 3a+12\left(-9\right)b=12\left(-21\right)$$

Whakarūnātia.

$$36a+12b=-12,36a-108b=-252$$

Me tango 36a-108b=-252 mai i 36a+12b=-12 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$36a-36a+12b+108b=-12+252$$

Tāpiri 36a ki te -36a. Ka whakakore atu ngā kupu 36a me -36a, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$12b+108b=-12+252$$

Tāpiri 12b ki te 108b.

$$120b=-12+252$$

Tāpiri -12 ki te 252.

$$120b=240$$

Whakawehea ngā taha e rua ki te 120.

$$b=2$$

Whakaurua te 2 mō b ki 3a-9b=-21. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō a hāngai tonu.

$$3a-9\times 2=-21$$

Whakareatia -9 ki te 2.

$$3a-18=-21$$

Me tāpiri 18 ki ngā taha e rua o te whārite.

$$3a=-3$$

Whakawehea ngā taha e rua ki te 3.

$$a=-1$$

Kua oti te pūnaha te whakatau.

$$a=-1,b=2$$