Whakaoti i te {l}{-9x+6y=13}{cx+8y=-12}

$$\left. \begin{array} { l } { - 9 x + 6 y = 13 } \\ { c x + 8 y = - 12 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-\frac{88}{3\left(c+12\right)}$$

$$y=-\frac{108-13c}{6\left(c+12\right)}$$

$$c\neq -12$$

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.

$$-9x+6y=13,cx+8y=-12$$

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

$$-9x+6y=13$$

Me tango 6y mai i ngā taha e rua o te whārite.

$$-9x=-6y+13$$

Whakawehea ngā taha e rua ki te -9.

$$x=-\frac{1}{9}\left(-6y+13\right)$$

Whakareatia -\frac{1}{9} ki te -6y+13.

$$x=\frac{2}{3}y-\frac{13}{9}$$

Whakakapia te \frac{2y}{3}-\frac{13}{9} mō te x ki tērā atu whārite, cx+8y=-12.

$$c\left(\frac{2}{3}y-\frac{13}{9}\right)+8y=-12$$

Whakareatia c ki te \frac{2y}{3}-\frac{13}{9}.

$$\frac{2c}{3}y-\frac{13c}{9}+8y=-12$$

Tāpiri \frac{2cy}{3} ki te 8y.

$$\left(\frac{2c}{3}+8\right)y-\frac{13c}{9}=-12$$

Me tāpiri \frac{13c}{9} ki ngā taha e rua o te whārite.

$$\left(\frac{2c}{3}+8\right)y=\frac{13c}{9}-12$$

Whakawehea ngā taha e rua ki te \frac{2c}{3}+8.

$$y=\frac{13c-108}{6\left(c+12\right)}$$

Whakaurua te \frac{-108+13c}{6\left(c+12\right)} mō y ki x=\frac{2}{3}y-\frac{13}{9}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{2}{3}\times \frac{13c-108}{6\left(c+12\right)}-\frac{13}{9}$$

Whakareatia \frac{2}{3} ki te \frac{-108+13c}{6\left(c+12\right)}.

$$x=\frac{13c-108}{9\left(c+12\right)}-\frac{13}{9}$$

Tāpiri -\frac{13}{9} ki te \frac{-108+13c}{9\left(c+12\right)}.

$$x=-\frac{88}{3\left(c+12\right)}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{88}{3\left(c+12\right)},y=\frac{13c-108}{6\left(c+12\right)}$$

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.

$$-9x+6y=13,cx+8y=-12$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-9&6\\c&8\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}13\\-12\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}-9&6\\c&8\end{matrix}\right).

$$inverse(\left(\begin{matrix}-9&6\\c&8\end{matrix}\right))\left(\begin{matrix}-9&6\\c&8\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-9&6\\c&8\end{matrix}\right))\left(\begin{matrix}13\\-12\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-9&6\\c&8\end{matrix}\right))\left(\begin{matrix}13\\-12\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-9&6\\c&8\end{matrix}\right))\left(\begin{matrix}13\\-12\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{8}{-9\times 8-6c}&-\frac{6}{-9\times 8-6c}\\-\frac{c}{-9\times 8-6c}&-\frac{9}{-9\times 8-6c}\end{matrix}\right)\left(\begin{matrix}13\\-12\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{4}{3\left(c+12\right)}&\frac{1}{c+12}\\\frac{c}{6\left(c+12\right)}&\frac{3}{2\left(c+12\right)}\end{matrix}\right)\left(\begin{matrix}13\\-12\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\left(-\frac{4}{3\left(c+12\right)}\right)\times 13+\frac{1}{c+12}\left(-12\right)\\\frac{c}{6\left(c+12\right)}\times 13+\frac{3}{2\left(c+12\right)}\left(-12\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{88}{3\left(c+12\right)}\\\frac{13c-108}{6\left(c+12\right)}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-\frac{88}{3\left(c+12\right)},y=\frac{13c-108}{6\left(c+12\right)}$$

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.

$$-9x+6y=13,cx+8y=-12$$

Kia ōrite ai a -9x me cx, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te c me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te -9.

$$c\left(-9\right)x+c\times 6y=c\times 13,-9cx-9\times 8y=-9\left(-12\right)$$

Whakarūnātia.

$$\left(-9c\right)x+6cy=13c,\left(-9c\right)x-72y=108$$

Me tango \left(-9c\right)x-72y=108 mai i \left(-9c\right)x+6cy=13c mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$\left(-9c\right)x+9cx+6cy+72y=13c-108$$

Tāpiri -9cx ki te 9cx. Ka whakakore atu ngā kupu -9cx me 9cx, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$6cy+72y=13c-108$$

Tāpiri 6cy ki te 72y.

$$\left(6c+72\right)y=13c-108$$

Whakawehea ngā taha e rua ki te 72+6c.

$$y=\frac{13c-108}{6\left(c+12\right)}$$

Whakaurua te \frac{13c-108}{6\left(c+12\right)} mō y ki cx+8y=-12. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$cx+8\times \frac{13c-108}{6\left(c+12\right)}=-12$$

Whakareatia 8 ki te \frac{13c-108}{6\left(c+12\right)}.

$$cx+\frac{4\left(13c-108\right)}{3\left(c+12\right)}=-12$$

Me tango \frac{4\left(13c-108\right)}{3\left(c+12\right)} mai i ngā taha e rua o te whārite.

$$cx=-\frac{88c}{3\left(c+12\right)}$$

Whakawehea ngā taha e rua ki te c.

$$x=-\frac{88}{3\left(c+12\right)}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{88}{3\left(c+12\right)},y=\frac{13c-108}{6\left(c+12\right)}$$