Whakaoti i te {l}{x/36-y=21}{x-y/20=16}

$$\left. \begin{array} { l } { \frac { x } { 36 } - y = 21 } \\ { x - \frac { y } { 20 } = 16 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{10764}{719} = 14\frac{698}{719} \approx 14.970792768$$

$$y = -\frac{14800}{719} = -20\frac{420}{719} \approx -20.584144645$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Whakareatia ngā taha e rua o te whārite ki te 36.

$$x-36y=756$$

Whakaarohia te whārite tuarua. Whakareatia ngā taha e rua o te whārite ki te 20.

$$20x-y=320$$

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.

$$x-36y=756,20x-y=320$$

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

$$x-36y=756$$

Me tāpiri 36y ki ngā taha e rua o te whārite.

$$x=36y+756$$

Whakakapia te 756+36y mō te x ki tērā atu whārite, 20x-y=320.

$$20\left(36y+756\right)-y=320$$

Whakareatia 20 ki te 756+36y.

$$720y+15120-y=320$$

Tāpiri 720y ki te -y.

$$719y+15120=320$$

Me tango 15120 mai i ngā taha e rua o te whārite.

$$719y=-14800$$

Whakawehea ngā taha e rua ki te 719.

$$y=-\frac{14800}{719}$$

Whakaurua te -\frac{14800}{719} mō y ki x=36y+756. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=36\left(-\frac{14800}{719}\right)+756$$

Whakareatia 36 ki te -\frac{14800}{719}.

$$x=-\frac{532800}{719}+756$$

Tāpiri 756 ki te -\frac{532800}{719}.

$$x=\frac{10764}{719}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{10764}{719},y=-\frac{14800}{719}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Whakareatia ngā taha e rua o te whārite ki te 36.

$$x-36y=756$$

Whakaarohia te whārite tuarua. Whakareatia ngā taha e rua o te whārite ki te 20.

$$20x-y=320$$

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

$$x-36y=756,20x-y=320$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&-36\\20&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}756\\320\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&-36\\20&-1\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&-36\\20&-1\end{matrix}\right))\left(\begin{matrix}1&-36\\20&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-36\\20&-1\end{matrix}\right))\left(\begin{matrix}756\\320\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-36\\20&-1\end{matrix}\right))\left(\begin{matrix}756\\320\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-36\\20&-1\end{matrix}\right))\left(\begin{matrix}756\\320\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{-1-\left(-36\times 20\right)}&-\frac{-36}{-1-\left(-36\times 20\right)}\\-\frac{20}{-1-\left(-36\times 20\right)}&\frac{1}{-1-\left(-36\times 20\right)}\end{matrix}\right)\left(\begin{matrix}756\\320\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{719}&\frac{36}{719}\\-\frac{20}{719}&\frac{1}{719}\end{matrix}\right)\left(\begin{matrix}756\\320\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{719}\times 756+\frac{36}{719}\times 320\\-\frac{20}{719}\times 756+\frac{1}{719}\times 320\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{10764}{719}\\-\frac{14800}{719}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{10764}{719},y=-\frac{14800}{719}$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Whakareatia ngā taha e rua o te whārite ki te 36.

$$x-36y=756$$

Whakaarohia te whārite tuarua. Whakareatia ngā taha e rua o te whārite ki te 20.

$$20x-y=320$$

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.

$$x-36y=756,20x-y=320$$

Kia ōrite ai a x me 20x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 20 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 1.

$$20x+20\left(-36\right)y=20\times 756,20x-y=320$$

Whakarūnātia.

$$20x-720y=15120,20x-y=320$$

Me tango 20x-y=320 mai i 20x-720y=15120 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$20x-20x-720y+y=15120-320$$

Tāpiri 20x ki te -20x. Ka whakakore atu ngā kupu 20x me -20x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-720y+y=15120-320$$

Tāpiri -720y ki te y.

$$-719y=15120-320$$

Tāpiri 15120 ki te -320.

$$-719y=14800$$

Whakawehea ngā taha e rua ki te -719.

$$y=-\frac{14800}{719}$$

Whakaurua te -\frac{14800}{719} mō y ki 20x-y=320. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$20x-\left(-\frac{14800}{719}\right)=320$$

Me tango \frac{14800}{719} mai i ngā taha e rua o te whārite.

$$20x=\frac{215280}{719}$$

Whakawehea ngā taha e rua ki te 20.

$$x=\frac{10764}{719}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{10764}{719},y=-\frac{14800}{719}$$