Whakaoti i te {l}{11a_{1}+8a_{2}=-8}{8a_{1}+11a_{2}=-26/5}

$$\left\{ \begin{array} { l } { 11 a _ { 1 } + 8 a _ { 2 } = - 8 } \\ { 8 a _ { 1 } + 11 a _ { 2 } = - \frac { 26 } { 5 } } \end{array} \right.$$

Whakaoti mō a_1, a_2

$$a_{1}=-\frac{232}{285}\approx -0.814035088$$

$$a_{2}=\frac{34}{285}\approx 0.119298246$$

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.

$$11a_{1}+8a_{2}=-8,8a_{1}+11a_{2}=-\frac{26}{5}$$

Kōwhiria tētahi o ngā whārite ka whakaotia mō te a_{1} mā te wehe i te a_{1} i te taha mauī o te tohu ōrite.

$$11a_{1}+8a_{2}=-8$$

Me tango 8a_{2} mai i ngā taha e rua o te whārite.

$$11a_{1}=-8a_{2}-8$$

Whakawehea ngā taha e rua ki te 11.

$$a_{1}=\frac{1}{11}\left(-8a_{2}-8\right)$$

Whakareatia \frac{1}{11} ki te -8a_{2}-8.

$$a_{1}=-\frac{8}{11}a_{2}-\frac{8}{11}$$

Whakakapia te \frac{-8a_{2}-8}{11} mō te a_{1} ki tērā atu whārite, 8a_{1}+11a_{2}=-\frac{26}{5}.

$$8\left(-\frac{8}{11}a_{2}-\frac{8}{11}\right)+11a_{2}=-\frac{26}{5}$$

Whakareatia 8 ki te \frac{-8a_{2}-8}{11}.

$$-\frac{64}{11}a_{2}-\frac{64}{11}+11a_{2}=-\frac{26}{5}$$

Tāpiri -\frac{64a_{2}}{11} ki te 11a_{2}.

$$\frac{57}{11}a_{2}-\frac{64}{11}=-\frac{26}{5}$$

Me tāpiri \frac{64}{11} ki ngā taha e rua o te whārite.

$$\frac{57}{11}a_{2}=\frac{34}{55}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{57}{11}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$a_{2}=\frac{34}{285}$$

Whakaurua te \frac{34}{285} mō a_{2} ki a_{1}=-\frac{8}{11}a_{2}-\frac{8}{11}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō a_{1} hāngai tonu.

$$a_{1}=-\frac{8}{11}\times \frac{34}{285}-\frac{8}{11}$$

Whakareatia -\frac{8}{11} ki te \frac{34}{285} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$a_{1}=-\frac{272}{3135}-\frac{8}{11}$$

Tāpiri -\frac{8}{11} ki te -\frac{272}{3135} 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.

$$a_{1}=-\frac{232}{285}$$

Kua oti te pūnaha te whakatau.

$$a_{1}=-\frac{232}{285},a_{2}=\frac{34}{285}$$

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.

$$11a_{1}+8a_{2}=-8,8a_{1}+11a_{2}=-\frac{26}{5}$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}11&8\\8&11\end{matrix}\right)\left(\begin{matrix}a_{1}\\a_{2}\end{matrix}\right)=\left(\begin{matrix}-8\\-\frac{26}{5}\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}11&8\\8&11\end{matrix}\right).

$$inverse(\left(\begin{matrix}11&8\\8&11\end{matrix}\right))\left(\begin{matrix}11&8\\8&11\end{matrix}\right)\left(\begin{matrix}a_{1}\\a_{2}\end{matrix}\right)=inverse(\left(\begin{matrix}11&8\\8&11\end{matrix}\right))\left(\begin{matrix}-8\\-\frac{26}{5}\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}a_{1}\\a_{2}\end{matrix}\right)=inverse(\left(\begin{matrix}11&8\\8&11\end{matrix}\right))\left(\begin{matrix}-8\\-\frac{26}{5}\end{matrix}\right)$$

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

$$\left(\begin{matrix}a_{1}\\a_{2}\end{matrix}\right)=inverse(\left(\begin{matrix}11&8\\8&11\end{matrix}\right))\left(\begin{matrix}-8\\-\frac{26}{5}\end{matrix}\right)$$

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.

$$\left(\begin{matrix}a_{1}\\a_{2}\end{matrix}\right)=\left(\begin{matrix}\frac{11}{11\times 11-8\times 8}&-\frac{8}{11\times 11-8\times 8}\\-\frac{8}{11\times 11-8\times 8}&\frac{11}{11\times 11-8\times 8}\end{matrix}\right)\left(\begin{matrix}-8\\-\frac{26}{5}\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}a_{1}\\a_{2}\end{matrix}\right)=\left(\begin{matrix}\frac{11}{57}&-\frac{8}{57}\\-\frac{8}{57}&\frac{11}{57}\end{matrix}\right)\left(\begin{matrix}-8\\-\frac{26}{5}\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}a_{1}\\a_{2}\end{matrix}\right)=\left(\begin{matrix}\frac{11}{57}\left(-8\right)-\frac{8}{57}\left(-\frac{26}{5}\right)\\-\frac{8}{57}\left(-8\right)+\frac{11}{57}\left(-\frac{26}{5}\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}a_{1}\\a_{2}\end{matrix}\right)=\left(\begin{matrix}-\frac{232}{285}\\\frac{34}{285}\end{matrix}\right)$$

Tangohia ngā huānga poukapa a_{1} me a_{2}.

$$a_{1}=-\frac{232}{285},a_{2}=\frac{34}{285}$$

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.

$$11a_{1}+8a_{2}=-8,8a_{1}+11a_{2}=-\frac{26}{5}$$

Kia ōrite ai a 11a_{1} me 8a_{1}, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 8 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 11.

$$8\times 11a_{1}+8\times 8a_{2}=8\left(-8\right),11\times 8a_{1}+11\times 11a_{2}=11\left(-\frac{26}{5}\right)$$

Whakarūnātia.

$$88a_{1}+64a_{2}=-64,88a_{1}+121a_{2}=-\frac{286}{5}$$

Me tango 88a_{1}+121a_{2}=-\frac{286}{5} mai i 88a_{1}+64a_{2}=-64 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$88a_{1}-88a_{1}+64a_{2}-121a_{2}=-64+\frac{286}{5}$$

Tāpiri 88a_{1} ki te -88a_{1}. Ka whakakore atu ngā kupu 88a_{1} me -88a_{1}, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$64a_{2}-121a_{2}=-64+\frac{286}{5}$$

Tāpiri 64a_{2} ki te -121a_{2}.

$$-57a_{2}=-64+\frac{286}{5}$$

Tāpiri -64 ki te \frac{286}{5}.

$$-57a_{2}=-\frac{34}{5}$$

Whakawehea ngā taha e rua ki te -57.

$$a_{2}=\frac{34}{285}$$

Whakaurua te \frac{34}{285} mō a_{2} ki 8a_{1}+11a_{2}=-\frac{26}{5}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō a_{1} hāngai tonu.

$$8a_{1}+11\times \frac{34}{285}=-\frac{26}{5}$$

Whakareatia 11 ki te \frac{34}{285}.

$$8a_{1}+\frac{374}{285}=-\frac{26}{5}$$

Me tango \frac{374}{285} mai i ngā taha e rua o te whārite.

$$8a_{1}=-\frac{1856}{285}$$

Whakawehea ngā taha e rua ki te 8.

$$a_{1}=-\frac{232}{285}$$

Kua oti te pūnaha te whakatau.

$$a_{1}=-\frac{232}{285},a_{2}=\frac{34}{285}$$