Whakaoti i te {l}{3c-4d=15}{-2c+3d=-12}

$$\left. \begin{array} { l } { 3 c - 4 d = 15 } \\ { - 2 c + 3 d = - 12 } \end{array} \right.$$

Whakaoti mō c, d

$$c=-3$$

$$d=-6$$

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.

$$3c-4d=15,-2c+3d=-12$$

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

$$3c-4d=15$$

Me tāpiri 4d ki ngā taha e rua o te whārite.

$$3c=4d+15$$

Whakawehea ngā taha e rua ki te 3.

$$c=\frac{1}{3}\left(4d+15\right)$$

Whakareatia \frac{1}{3} ki te 4d+15.

$$c=\frac{4}{3}d+5$$

Whakakapia te \frac{4d}{3}+5 mō te c ki tērā atu whārite, -2c+3d=-12.

$$-2\left(\frac{4}{3}d+5\right)+3d=-12$$

Whakareatia -2 ki te \frac{4d}{3}+5.

$$-\frac{8}{3}d-10+3d=-12$$

Tāpiri -\frac{8d}{3} ki te 3d.

$$\frac{1}{3}d-10=-12$$

Me tāpiri 10 ki ngā taha e rua o te whārite.

$$\frac{1}{3}d=-2$$

Me whakarea ngā taha e rua ki te 3.

$$d=-6$$

Whakaurua te -6 mō d ki c=\frac{4}{3}d+5. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō c hāngai tonu.

$$c=\frac{4}{3}\left(-6\right)+5$$

Whakareatia \frac{4}{3} ki te -6.

$$c=-8+5$$

Tāpiri 5 ki te -8.

$$c=-3$$

Kua oti te pūnaha te whakatau.

$$c=-3,d=-6$$

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.

$$3c-4d=15,-2c+3d=-12$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&-4\\-2&3\end{matrix}\right)\left(\begin{matrix}c\\d\end{matrix}\right)=\left(\begin{matrix}15\\-12\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}3&-4\\-2&3\end{matrix}\right).

$$inverse(\left(\begin{matrix}3&-4\\-2&3\end{matrix}\right))\left(\begin{matrix}3&-4\\-2&3\end{matrix}\right)\left(\begin{matrix}c\\d\end{matrix}\right)=inverse(\left(\begin{matrix}3&-4\\-2&3\end{matrix}\right))\left(\begin{matrix}15\\-12\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}c\\d\end{matrix}\right)=inverse(\left(\begin{matrix}3&-4\\-2&3\end{matrix}\right))\left(\begin{matrix}15\\-12\end{matrix}\right)$$

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

$$\left(\begin{matrix}c\\d\end{matrix}\right)=inverse(\left(\begin{matrix}3&-4\\-2&3\end{matrix}\right))\left(\begin{matrix}15\\-12\end{matrix}\right)$$

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.

$$\left(\begin{matrix}c\\d\end{matrix}\right)=\left(\begin{matrix}\frac{3}{3\times 3-\left(-4\left(-2\right)\right)}&-\frac{-4}{3\times 3-\left(-4\left(-2\right)\right)}\\-\frac{-2}{3\times 3-\left(-4\left(-2\right)\right)}&\frac{3}{3\times 3-\left(-4\left(-2\right)\right)}\end{matrix}\right)\left(\begin{matrix}15\\-12\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}c\\d\end{matrix}\right)=\left(\begin{matrix}3&4\\2&3\end{matrix}\right)\left(\begin{matrix}15\\-12\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}c\\d\end{matrix}\right)=\left(\begin{matrix}3\times 15+4\left(-12\right)\\2\times 15+3\left(-12\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}c\\d\end{matrix}\right)=\left(\begin{matrix}-3\\-6\end{matrix}\right)$$

Tangohia ngā huānga poukapa c me d.

$$c=-3,d=-6$$

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.

$$3c-4d=15,-2c+3d=-12$$

Kia ōrite ai a 3c me -2c, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te -2 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 3.

$$-2\times 3c-2\left(-4\right)d=-2\times 15,3\left(-2\right)c+3\times 3d=3\left(-12\right)$$

Whakarūnātia.

$$-6c+8d=-30,-6c+9d=-36$$

Me tango -6c+9d=-36 mai i -6c+8d=-30 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-6c+6c+8d-9d=-30+36$$

Tāpiri -6c ki te 6c. Ka whakakore atu ngā kupu -6c me 6c, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$8d-9d=-30+36$$

Tāpiri 8d ki te -9d.

$$-d=-30+36$$

Tāpiri -30 ki te 36.

$$-d=6$$

Whakawehea ngā taha e rua ki te -1.

$$d=-6$$

Whakaurua te -6 mō d ki -2c+3d=-12. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō c hāngai tonu.

$$-2c+3\left(-6\right)=-12$$

Whakareatia 3 ki te -6.

$$-2c-18=-12$$

Me tāpiri 18 ki ngā taha e rua o te whārite.

$$-2c=6$$

Whakawehea ngā taha e rua ki te -2.

$$c=-3$$

Kua oti te pūnaha te whakatau.

$$c=-3,d=-6$$