Whakaoti i te {l}{217x+13ny=913}{131x+217y=827}

$$\left. \begin{array} { l } { 217 x + 13 n y = 913 } \\ { 131 x + 217 y = 827 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-\frac{198121-10751n}{1703n-47089}$$

$$y=-\frac{59856}{1703n-47089}$$

$$n\neq \frac{47089}{1703}$$

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.

$$217x+13ny=913,131x+217y=827$$

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

$$217x+13ny=913$$

Me tango 13ny mai i ngā taha e rua o te whārite.

$$217x=\left(-13n\right)y+913$$

Whakawehea ngā taha e rua ki te 217.

$$x=\frac{1}{217}\left(\left(-13n\right)y+913\right)$$

Whakareatia \frac{1}{217} ki te -13ny+913.

$$x=\left(-\frac{13n}{217}\right)y+\frac{913}{217}$$

Whakakapia te \frac{-13ny+913}{217} mō te x ki tērā atu whārite, 131x+217y=827.

$$131\left(\left(-\frac{13n}{217}\right)y+\frac{913}{217}\right)+217y=827$$

Whakareatia 131 ki te \frac{-13ny+913}{217}.

$$\left(-\frac{1703n}{217}\right)y+\frac{119603}{217}+217y=827$$

Tāpiri -\frac{1703ny}{217} ki te 217y.

$$\left(-\frac{1703n}{217}+217\right)y+\frac{119603}{217}=827$$

Me tango \frac{119603}{217} mai i ngā taha e rua o te whārite.

$$\left(-\frac{1703n}{217}+217\right)y=\frac{59856}{217}$$

Whakawehea ngā taha e rua ki te -\frac{1703n}{217}+217.

$$y=\frac{59856}{47089-1703n}$$

Whakaurua te \frac{59856}{47089-1703n} mō y ki x=\left(-\frac{13n}{217}\right)y+\frac{913}{217}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\left(-\frac{13n}{217}\right)\times \frac{59856}{47089-1703n}+\frac{913}{217}$$

Whakareatia -\frac{13n}{217} ki te \frac{59856}{47089-1703n}.

$$x=-\frac{778128n}{217\left(47089-1703n\right)}+\frac{913}{217}$$

Tāpiri \frac{913}{217} ki te -\frac{778128n}{217\left(47089-1703n\right)}.

$$x=\frac{198121-10751n}{47089-1703n}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{198121-10751n}{47089-1703n},y=\frac{59856}{47089-1703n}$$

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.

$$217x+13ny=913,131x+217y=827$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}217&13n\\131&217\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}913\\827\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}217&13n\\131&217\end{matrix}\right).

$$inverse(\left(\begin{matrix}217&13n\\131&217\end{matrix}\right))\left(\begin{matrix}217&13n\\131&217\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}217&13n\\131&217\end{matrix}\right))\left(\begin{matrix}913\\827\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}217&13n\\131&217\end{matrix}\right))\left(\begin{matrix}913\\827\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}217&13n\\131&217\end{matrix}\right))\left(\begin{matrix}913\\827\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{217}{217\times 217-13n\times 131}&-\frac{13n}{217\times 217-13n\times 131}\\-\frac{131}{217\times 217-13n\times 131}&\frac{217}{217\times 217-13n\times 131}\end{matrix}\right)\left(\begin{matrix}913\\827\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{217}{47089-1703n}&-\frac{13n}{47089-1703n}\\-\frac{131}{47089-1703n}&\frac{217}{47089-1703n}\end{matrix}\right)\left(\begin{matrix}913\\827\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{217}{47089-1703n}\times 913+\left(-\frac{13n}{47089-1703n}\right)\times 827\\\left(-\frac{131}{47089-1703n}\right)\times 913+\frac{217}{47089-1703n}\times 827\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{10751n-198121}{47089-1703n}\\\frac{59856}{47089-1703n}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-\frac{10751n-198121}{47089-1703n},y=\frac{59856}{47089-1703n}$$

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.

$$217x+13ny=913,131x+217y=827$$

Kia ōrite ai a 217x me 131x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 131 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 217.

$$131\times 217x+131\times 13ny=131\times 913,217\times 131x+217\times 217y=217\times 827$$

Whakarūnātia.

$$28427x+1703ny=119603,28427x+47089y=179459$$

Me tango 28427x+47089y=179459 mai i 28427x+1703ny=119603 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$28427x-28427x+1703ny-47089y=119603-179459$$

Tāpiri 28427x ki te -28427x. Ka whakakore atu ngā kupu 28427x me -28427x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$1703ny-47089y=119603-179459$$

Tāpiri 1703ny ki te -47089y.

$$\left(1703n-47089\right)y=119603-179459$$

Tāpiri 119603 ki te -179459.

$$\left(1703n-47089\right)y=-59856$$

Whakawehea ngā taha e rua ki te 1703n-47089.

$$y=-\frac{59856}{1703n-47089}$$

Whakaurua te -\frac{59856}{1703n-47089} mō y ki 131x+217y=827. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$131x+217\left(-\frac{59856}{1703n-47089}\right)=827$$

Whakareatia 217 ki te -\frac{59856}{1703n-47089}.

$$131x-\frac{12988752}{1703n-47089}=827$$

Me tāpiri \frac{12988752}{1703n-47089} ki ngā taha e rua o te whārite.

$$131x=\frac{131\left(10751n-198121\right)}{1703n-47089}$$

Whakawehea ngā taha e rua ki te 131.

$$x=\frac{10751n-198121}{1703n-47089}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{10751n-198121}{1703n-47089},y=-\frac{59856}{1703n-47089}$$