Whakaoti i te {l}{-9x-y=-3}{-8x+2y=-20}

$$\left. \begin{array} { l } { - 9 x - y = - 3 } \\ { - 8 x + 2 y = - 20 } \end{array} \right.$$

Whakaoti mō x, y

$$x=1$$

$$y=-6$$

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.

$$-9x-y=-3,-8x+2y=-20$$

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

$$-9x-y=-3$$

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

$$-9x=y-3$$

Whakawehea ngā taha e rua ki te -9.

$$x=-\frac{1}{9}\left(y-3\right)$$

Whakareatia -\frac{1}{9} ki te y-3.

$$x=-\frac{1}{9}y+\frac{1}{3}$$

Whakakapia te -\frac{y}{9}+\frac{1}{3} mō te x ki tērā atu whārite, -8x+2y=-20.

$$-8\left(-\frac{1}{9}y+\frac{1}{3}\right)+2y=-20$$

Whakareatia -8 ki te -\frac{y}{9}+\frac{1}{3}.

$$\frac{8}{9}y-\frac{8}{3}+2y=-20$$

Tāpiri \frac{8y}{9} ki te 2y.

$$\frac{26}{9}y-\frac{8}{3}=-20$$

Me tāpiri \frac{8}{3} ki ngā taha e rua o te whārite.

$$\frac{26}{9}y=-\frac{52}{3}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{26}{9}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-6$$

Whakaurua te -6 mō y ki x=-\frac{1}{9}y+\frac{1}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{1}{9}\left(-6\right)+\frac{1}{3}$$

Whakareatia -\frac{1}{9} ki te -6.

$$x=\frac{2+1}{3}$$

Tāpiri \frac{1}{3} ki te \frac{2}{3} mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=1$$

Kua oti te pūnaha te whakatau.

$$x=1,y=-6$$

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.

$$-9x-y=-3,-8x+2y=-20$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-9&-1\\-8&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-3\\-20\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}-9&-1\\-8&2\end{matrix}\right).

$$inverse(\left(\begin{matrix}-9&-1\\-8&2\end{matrix}\right))\left(\begin{matrix}-9&-1\\-8&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-9&-1\\-8&2\end{matrix}\right))\left(\begin{matrix}-3\\-20\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-9&-1\\-8&2\end{matrix}\right))\left(\begin{matrix}-3\\-20\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-9&-1\\-8&2\end{matrix}\right))\left(\begin{matrix}-3\\-20\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{-9\times 2-\left(-\left(-8\right)\right)}&-\frac{-1}{-9\times 2-\left(-\left(-8\right)\right)}\\-\frac{-8}{-9\times 2-\left(-\left(-8\right)\right)}&-\frac{9}{-9\times 2-\left(-\left(-8\right)\right)}\end{matrix}\right)\left(\begin{matrix}-3\\-20\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{13}&-\frac{1}{26}\\-\frac{4}{13}&\frac{9}{26}\end{matrix}\right)\left(\begin{matrix}-3\\-20\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{13}\left(-3\right)-\frac{1}{26}\left(-20\right)\\-\frac{4}{13}\left(-3\right)+\frac{9}{26}\left(-20\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}1\\-6\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=1,y=-6$$

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.

$$-9x-y=-3,-8x+2y=-20$$

Kia ōrite ai a -9x 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 -9.

$$-8\left(-9\right)x-8\left(-1\right)y=-8\left(-3\right),-9\left(-8\right)x-9\times 2y=-9\left(-20\right)$$

Whakarūnātia.

$$72x+8y=24,72x-18y=180$$

Me tango 72x-18y=180 mai i 72x+8y=24 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$72x-72x+8y+18y=24-180$$

Tāpiri 72x ki te -72x. Ka whakakore atu ngā kupu 72x me -72x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$8y+18y=24-180$$

Tāpiri 8y ki te 18y.

$$26y=24-180$$

Tāpiri 24 ki te -180.

$$26y=-156$$

Whakawehea ngā taha e rua ki te 26.

$$y=-6$$

Whakaurua te -6 mō y ki -8x+2y=-20. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-8x+2\left(-6\right)=-20$$

Whakareatia 2 ki te -6.

$$-8x-12=-20$$

Me tāpiri 12 ki ngā taha e rua o te whārite.

$$-8x=-8$$

Whakawehea ngā taha e rua ki te -8.

$$x=1$$

Kua oti te pūnaha te whakatau.

$$x=1,y=-6$$