Whakaoti i te {r}{-v+2d=-10}{d+3v=9}

$$\left. \begin{array} { r } { - v + 2 d = - 10 } \\ { d + 3 v = 9 } \end{array} \right.$$

Whakaoti mō v, d

$$v=4$$

$$d=-3$$

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.

$$-v+2d=-10,3v+d=9$$

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

$$-v+2d=-10$$

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

$$-v=-2d-10$$

Whakawehea ngā taha e rua ki te -1.

$$v=-\left(-2d-10\right)$$

Whakareatia -1 ki te -2d-10.

$$v=2d+10$$

Whakakapia te 10+2d mō te v ki tērā atu whārite, 3v+d=9.

$$3\left(2d+10\right)+d=9$$

Whakareatia 3 ki te 10+2d.

$$6d+30+d=9$$

Tāpiri 6d ki te d.

$$7d+30=9$$

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

$$7d=-21$$

Whakawehea ngā taha e rua ki te 7.

$$d=-3$$

Whakaurua te -3 mō d ki v=2d+10. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō v hāngai tonu.

$$v=2\left(-3\right)+10$$

Whakareatia 2 ki te -3.

$$v=-6+10$$

Tāpiri 10 ki te -6.

$$v=4$$

Kua oti te pūnaha te whakatau.

$$v=4,d=-3$$

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.

$$-v+2d=-10,3v+d=9$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-1&2\\3&1\end{matrix}\right)\left(\begin{matrix}v\\d\end{matrix}\right)=\left(\begin{matrix}-10\\9\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}-1&2\\3&1\end{matrix}\right).

$$inverse(\left(\begin{matrix}-1&2\\3&1\end{matrix}\right))\left(\begin{matrix}-1&2\\3&1\end{matrix}\right)\left(\begin{matrix}v\\d\end{matrix}\right)=inverse(\left(\begin{matrix}-1&2\\3&1\end{matrix}\right))\left(\begin{matrix}-10\\9\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}v\\d\end{matrix}\right)=inverse(\left(\begin{matrix}-1&2\\3&1\end{matrix}\right))\left(\begin{matrix}-10\\9\end{matrix}\right)$$

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

$$\left(\begin{matrix}v\\d\end{matrix}\right)=inverse(\left(\begin{matrix}-1&2\\3&1\end{matrix}\right))\left(\begin{matrix}-10\\9\end{matrix}\right)$$

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.

$$\left(\begin{matrix}v\\d\end{matrix}\right)=\left(\begin{matrix}\frac{1}{-1-2\times 3}&-\frac{2}{-1-2\times 3}\\-\frac{3}{-1-2\times 3}&-\frac{1}{-1-2\times 3}\end{matrix}\right)\left(\begin{matrix}-10\\9\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}v\\d\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{7}&\frac{2}{7}\\\frac{3}{7}&\frac{1}{7}\end{matrix}\right)\left(\begin{matrix}-10\\9\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}v\\d\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{7}\left(-10\right)+\frac{2}{7}\times 9\\\frac{3}{7}\left(-10\right)+\frac{1}{7}\times 9\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}v\\d\end{matrix}\right)=\left(\begin{matrix}4\\-3\end{matrix}\right)$$

Tangohia ngā huānga poukapa v me d.

$$v=4,d=-3$$

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.

$$-v+2d=-10,3v+d=9$$

Kia ōrite ai a -v me 3v, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 3 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te -1.

$$3\left(-1\right)v+3\times 2d=3\left(-10\right),-3v-d=-9$$

Whakarūnātia.

$$-3v+6d=-30,-3v-d=-9$$

Me tango -3v-d=-9 mai i -3v+6d=-30 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-3v+3v+6d+d=-30+9$$

Tāpiri -3v ki te 3v. Ka whakakore atu ngā kupu -3v me 3v, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$6d+d=-30+9$$

Tāpiri 6d ki te d.

$$7d=-30+9$$

Tāpiri -30 ki te 9.

$$7d=-21$$

Whakawehea ngā taha e rua ki te 7.

$$d=-3$$

Whakaurua te -3 mō d ki 3v+d=9. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō v hāngai tonu.

$$3v-3=9$$

Me tāpiri 3 ki ngā taha e rua o te whārite.

$$3v=12$$

Whakawehea ngā taha e rua ki te 3.

$$v=4$$

Kua oti te pūnaha te whakatau.

$$v=4,d=-3$$