Whakaoti i te {r}{x-7y=-11}{5x+2y=-18}

$$\left. \begin{array} { r } { x - 7 y = - 11 } \\ { 5 x + 2 y = - 18 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-4$$

$$y=1$$

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.

$$x-7y=-11,5x+2y=-18$$

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

$$x-7y=-11$$

Me tāpiri 7y ki ngā taha e rua o te whārite.

$$x=7y-11$$

Whakakapia te 7y-11 mō te x ki tērā atu whārite, 5x+2y=-18.

$$5\left(7y-11\right)+2y=-18$$

Whakareatia 5 ki te 7y-11.

$$35y-55+2y=-18$$

Tāpiri 35y ki te 2y.

$$37y-55=-18$$

Me tāpiri 55 ki ngā taha e rua o te whārite.

$$37y=37$$

Whakawehea ngā taha e rua ki te 37.

$$y=1$$

Whakaurua te 1 mō y ki x=7y-11. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=7-11$$

Tāpiri -11 ki te 7.

$$x=-4$$

Kua oti te pūnaha te whakatau.

$$x=-4,y=1$$

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.

$$x-7y=-11,5x+2y=-18$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&-7\\5&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-11\\-18\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&-7\\5&2\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&-7\\5&2\end{matrix}\right))\left(\begin{matrix}1&-7\\5&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-7\\5&2\end{matrix}\right))\left(\begin{matrix}-11\\-18\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-7\\5&2\end{matrix}\right))\left(\begin{matrix}-11\\-18\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-7\\5&2\end{matrix}\right))\left(\begin{matrix}-11\\-18\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{2-\left(-7\times 5\right)}&-\frac{-7}{2-\left(-7\times 5\right)}\\-\frac{5}{2-\left(-7\times 5\right)}&\frac{1}{2-\left(-7\times 5\right)}\end{matrix}\right)\left(\begin{matrix}-11\\-18\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{37}&\frac{7}{37}\\-\frac{5}{37}&\frac{1}{37}\end{matrix}\right)\left(\begin{matrix}-11\\-18\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{37}\left(-11\right)+\frac{7}{37}\left(-18\right)\\-\frac{5}{37}\left(-11\right)+\frac{1}{37}\left(-18\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-4\\1\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-4,y=1$$

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.

$$x-7y=-11,5x+2y=-18$$

Kia ōrite ai a x me 5x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 5 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 1.

$$5x+5\left(-7\right)y=5\left(-11\right),5x+2y=-18$$

Whakarūnātia.

$$5x-35y=-55,5x+2y=-18$$

Me tango 5x+2y=-18 mai i 5x-35y=-55 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$5x-5x-35y-2y=-55+18$$

Tāpiri 5x ki te -5x. Ka whakakore atu ngā kupu 5x me -5x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-35y-2y=-55+18$$

Tāpiri -35y ki te -2y.

$$-37y=-55+18$$

Tāpiri -55 ki te 18.

$$-37y=-37$$

Whakawehea ngā taha e rua ki te -37.

$$y=1$$

Whakaurua te 1 mō y ki 5x+2y=-18. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$5x+2=-18$$

Me tango 2 mai i ngā taha e rua o te whārite.

$$5x=-20$$

Whakawehea ngā taha e rua ki te 5.

$$x=-4$$

Kua oti te pūnaha te whakatau.

$$x=-4,y=1$$