Whakaoti i te {l}{361x+463y=-102}{463x+361y=102}

$$\left\{ \begin{array} { l } { 361 x + 463 y = - 102 } \\ { 463 x + 361 y = 102 } \end{array} \right.$$

Whakaoti mō x, y

$$x=1$$

$$y=-1$$

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.

$$361x+463y=-102,463x+361y=102$$

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

$$361x+463y=-102$$

Me tango 463y mai i ngā taha e rua o te whārite.

$$361x=-463y-102$$

Whakawehea ngā taha e rua ki te 361.

$$x=\frac{1}{361}\left(-463y-102\right)$$

Whakareatia \frac{1}{361} ki te -463y-102.

$$x=-\frac{463}{361}y-\frac{102}{361}$$

Whakakapia te \frac{-463y-102}{361} mō te x ki tērā atu whārite, 463x+361y=102.

$$463\left(-\frac{463}{361}y-\frac{102}{361}\right)+361y=102$$

Whakareatia 463 ki te \frac{-463y-102}{361}.

$$-\frac{214369}{361}y-\frac{47226}{361}+361y=102$$

Tāpiri -\frac{214369y}{361} ki te 361y.

$$-\frac{84048}{361}y-\frac{47226}{361}=102$$

Me tāpiri \frac{47226}{361} ki ngā taha e rua o te whārite.

$$-\frac{84048}{361}y=\frac{84048}{361}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{84048}{361}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-1$$

Whakaurua te -1 mō y ki x=-\frac{463}{361}y-\frac{102}{361}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{463}{361}\left(-1\right)-\frac{102}{361}$$

Whakareatia -\frac{463}{361} ki te -1.

$$x=\frac{463-102}{361}$$

Tāpiri -\frac{102}{361} ki te \frac{463}{361} 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.

$$x=1$$

Kua oti te pūnaha te whakatau.

$$x=1,y=-1$$

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.

$$361x+463y=-102,463x+361y=102$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}361&463\\463&361\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-102\\102\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}361&463\\463&361\end{matrix}\right).

$$inverse(\left(\begin{matrix}361&463\\463&361\end{matrix}\right))\left(\begin{matrix}361&463\\463&361\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}361&463\\463&361\end{matrix}\right))\left(\begin{matrix}-102\\102\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}361&463\\463&361\end{matrix}\right))\left(\begin{matrix}-102\\102\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}361&463\\463&361\end{matrix}\right))\left(\begin{matrix}-102\\102\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{361}{361\times 361-463\times 463}&-\frac{463}{361\times 361-463\times 463}\\-\frac{463}{361\times 361-463\times 463}&\frac{361}{361\times 361-463\times 463}\end{matrix}\right)\left(\begin{matrix}-102\\102\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{361}{84048}&\frac{463}{84048}\\\frac{463}{84048}&-\frac{361}{84048}\end{matrix}\right)\left(\begin{matrix}-102\\102\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{361}{84048}\left(-102\right)+\frac{463}{84048}\times 102\\\frac{463}{84048}\left(-102\right)-\frac{361}{84048}\times 102\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}1\\-1\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=1,y=-1$$

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.

$$361x+463y=-102,463x+361y=102$$

Kia ōrite ai a 361x me 463x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 463 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 361.

$$463\times 361x+463\times 463y=463\left(-102\right),361\times 463x+361\times 361y=361\times 102$$

Whakarūnātia.

$$167143x+214369y=-47226,167143x+130321y=36822$$

Me tango 167143x+130321y=36822 mai i 167143x+214369y=-47226 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$167143x-167143x+214369y-130321y=-47226-36822$$

Tāpiri 167143x ki te -167143x. Ka whakakore atu ngā kupu 167143x me -167143x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$214369y-130321y=-47226-36822$$

Tāpiri 214369y ki te -130321y.

$$84048y=-47226-36822$$

Tāpiri -47226 ki te -36822.

$$84048y=-84048$$

Whakawehea ngā taha e rua ki te 84048.

$$y=-1$$

Whakaurua te -1 mō y ki 463x+361y=102. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$463x+361\left(-1\right)=102$$

Whakareatia 361 ki te -1.

$$463x-361=102$$

Me tāpiri 361 ki ngā taha e rua o te whārite.

$$463x=463$$

Whakawehea ngā taha e rua ki te 463.

$$x=1$$

Kua oti te pūnaha te whakatau.

$$x=1,y=-1$$