Whakaoti i te {l}{x/33-y=26}{x-y/88=66}

$$\left. \begin{array} { l } { \frac { x } { 33 } - y = 26 } \\ { x - \frac { y } { 88 } = 66 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{190806}{2903} = 65\frac{2111}{2903} \approx 65.727178781$$

$$y = -\frac{69696}{2903} = -24\frac{24}{2903} \approx -24.00826731$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Whakareatia ngā taha e rua o te whārite ki te 33.

$$x-33y=858$$

Whakaarohia te whārite tuarua. Whakareatia ngā taha e rua o te whārite ki te 88.

$$88x-y=5808$$

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.

$$x-33y=858,88x-y=5808$$

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

$$x-33y=858$$

Me tāpiri 33y ki ngā taha e rua o te whārite.

$$x=33y+858$$

Whakakapia te 858+33y mō te x ki tērā atu whārite, 88x-y=5808.

$$88\left(33y+858\right)-y=5808$$

Whakareatia 88 ki te 858+33y.

$$2904y+75504-y=5808$$

Tāpiri 2904y ki te -y.

$$2903y+75504=5808$$

Me tango 75504 mai i ngā taha e rua o te whārite.

$$2903y=-69696$$

Whakawehea ngā taha e rua ki te 2903.

$$y=-\frac{69696}{2903}$$

Whakaurua te -\frac{69696}{2903} mō y ki x=33y+858. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=33\left(-\frac{69696}{2903}\right)+858$$

Whakareatia 33 ki te -\frac{69696}{2903}.

$$x=-\frac{2299968}{2903}+858$$

Tāpiri 858 ki te -\frac{2299968}{2903}.

$$x=\frac{190806}{2903}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{190806}{2903},y=-\frac{69696}{2903}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Whakareatia ngā taha e rua o te whārite ki te 33.

$$x-33y=858$$

Whakaarohia te whārite tuarua. Whakareatia ngā taha e rua o te whārite ki te 88.

$$88x-y=5808$$

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

$$x-33y=858,88x-y=5808$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&-33\\88&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}858\\5808\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&-33\\88&-1\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&-33\\88&-1\end{matrix}\right))\left(\begin{matrix}1&-33\\88&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-33\\88&-1\end{matrix}\right))\left(\begin{matrix}858\\5808\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-33\\88&-1\end{matrix}\right))\left(\begin{matrix}858\\5808\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-33\\88&-1\end{matrix}\right))\left(\begin{matrix}858\\5808\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{-1-\left(-33\times 88\right)}&-\frac{-33}{-1-\left(-33\times 88\right)}\\-\frac{88}{-1-\left(-33\times 88\right)}&\frac{1}{-1-\left(-33\times 88\right)}\end{matrix}\right)\left(\begin{matrix}858\\5808\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{2903}&\frac{33}{2903}\\-\frac{88}{2903}&\frac{1}{2903}\end{matrix}\right)\left(\begin{matrix}858\\5808\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{2903}\times 858+\frac{33}{2903}\times 5808\\-\frac{88}{2903}\times 858+\frac{1}{2903}\times 5808\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{190806}{2903}\\-\frac{69696}{2903}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{190806}{2903},y=-\frac{69696}{2903}$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Whakareatia ngā taha e rua o te whārite ki te 33.

$$x-33y=858$$

Whakaarohia te whārite tuarua. Whakareatia ngā taha e rua o te whārite ki te 88.

$$88x-y=5808$$

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.

$$x-33y=858,88x-y=5808$$

Kia ōrite ai a x me 88x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 88 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 1.

$$88x+88\left(-33\right)y=88\times 858,88x-y=5808$$

Whakarūnātia.

$$88x-2904y=75504,88x-y=5808$$

Me tango 88x-y=5808 mai i 88x-2904y=75504 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$88x-88x-2904y+y=75504-5808$$

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

$$-2904y+y=75504-5808$$

Tāpiri -2904y ki te y.

$$-2903y=75504-5808$$

Tāpiri 75504 ki te -5808.

$$-2903y=69696$$

Whakawehea ngā taha e rua ki te -2903.

$$y=-\frac{69696}{2903}$$

Whakaurua te -\frac{69696}{2903} mō y ki 88x-y=5808. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$88x-\left(-\frac{69696}{2903}\right)=5808$$

Me tango \frac{69696}{2903} mai i ngā taha e rua o te whārite.

$$88x=\frac{16790928}{2903}$$

Whakawehea ngā taha e rua ki te 88.

$$x=\frac{190806}{2903}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{190806}{2903},y=-\frac{69696}{2903}$$