Whakaoti i te {l}{x+2y=-6}{-8x-5y=-18}

$$\left. \begin{array} { l } { x + 2 y = - 6 } \\ { - 8 x - 5 y = - 18 } \end{array} \right.$$

Whakaoti mō x, y

$$x=6$$

$$y=-6$$

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.

$$x+2y=-6,-8x-5y=-18$$

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

$$x+2y=-6$$

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

$$x=-2y-6$$

Whakakapia te -2y-6 mō te x ki tērā atu whārite, -8x-5y=-18.

$$-8\left(-2y-6\right)-5y=-18$$

Whakareatia -8 ki te -2y-6.

$$16y+48-5y=-18$$

Tāpiri 16y ki te -5y.

$$11y+48=-18$$

Me tango 48 mai i ngā taha e rua o te whārite.

$$11y=-66$$

Whakawehea ngā taha e rua ki te 11.

$$y=-6$$

Whakaurua te -6 mō y ki x=-2y-6. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-2\left(-6\right)-6$$

Whakareatia -2 ki te -6.

$$x=12-6$$

Tāpiri -6 ki te 12.

$$x=6$$

Kua oti te pūnaha te whakatau.

$$x=6,y=-6$$

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.

$$x+2y=-6,-8x-5y=-18$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&2\\-8&-5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-6\\-18\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&2\\-8&-5\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&2\\-8&-5\end{matrix}\right))\left(\begin{matrix}1&2\\-8&-5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&2\\-8&-5\end{matrix}\right))\left(\begin{matrix}-6\\-18\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&2\\-8&-5\end{matrix}\right))\left(\begin{matrix}-6\\-18\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&2\\-8&-5\end{matrix}\right))\left(\begin{matrix}-6\\-18\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{5}{-5-2\left(-8\right)}&-\frac{2}{-5-2\left(-8\right)}\\-\frac{-8}{-5-2\left(-8\right)}&\frac{1}{-5-2\left(-8\right)}\end{matrix}\right)\left(\begin{matrix}-6\\-18\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{5}{11}&-\frac{2}{11}\\\frac{8}{11}&\frac{1}{11}\end{matrix}\right)\left(\begin{matrix}-6\\-18\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{5}{11}\left(-6\right)-\frac{2}{11}\left(-18\right)\\\frac{8}{11}\left(-6\right)+\frac{1}{11}\left(-18\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}6\\-6\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=6,y=-6$$

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.

$$x+2y=-6,-8x-5y=-18$$

Kia ōrite ai a x me -8x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te -8 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 1.

$$-8x-8\times 2y=-8\left(-6\right),-8x-5y=-18$$

Whakarūnātia.

$$-8x-16y=48,-8x-5y=-18$$

Me tango -8x-5y=-18 mai i -8x-16y=48 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-8x+8x-16y+5y=48+18$$

Tāpiri -8x ki te 8x. Ka whakakore atu ngā kupu -8x me 8x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-16y+5y=48+18$$

Tāpiri -16y ki te 5y.

$$-11y=48+18$$

Tāpiri 48 ki te 18.

$$-11y=66$$

Whakawehea ngā taha e rua ki te -11.

$$y=-6$$

Whakaurua te -6 mō y ki -8x-5y=-18. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-8x-5\left(-6\right)=-18$$

Whakareatia -5 ki te -6.

$$-8x+30=-18$$

Me tango 30 mai i ngā taha e rua o te whārite.

$$-8x=-48$$

Whakawehea ngā taha e rua ki te -8.

$$x=6$$

Kua oti te pūnaha te whakatau.

$$x=6,y=-6$$