Whakaoti i te {l}{9v+2w=7}{3v-8w=-2}

$$\left. \begin{array} { l } { 9 v + 2 w = 7 } \\ { 3 v - 8 w = - 2 } \end{array} \right.$$

Whakaoti mō v, w

$$v=\frac{2}{3}\approx 0.666666667$$

$$w=\frac{1}{2}=0.5$$

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.

$$9v+2w=7,3v-8w=-2$$

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

$$9v+2w=7$$

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

$$9v=-2w+7$$

Whakawehea ngā taha e rua ki te 9.

$$v=\frac{1}{9}\left(-2w+7\right)$$

Whakareatia \frac{1}{9} ki te -2w+7.

$$v=-\frac{2}{9}w+\frac{7}{9}$$

Whakakapia te \frac{-2w+7}{9} mō te v ki tērā atu whārite, 3v-8w=-2.

$$3\left(-\frac{2}{9}w+\frac{7}{9}\right)-8w=-2$$

Whakareatia 3 ki te \frac{-2w+7}{9}.

$$-\frac{2}{3}w+\frac{7}{3}-8w=-2$$

Tāpiri -\frac{2w}{3} ki te -8w.

$$-\frac{26}{3}w+\frac{7}{3}=-2$$

Me tango \frac{7}{3} mai i ngā taha e rua o te whārite.

$$-\frac{26}{3}w=-\frac{13}{3}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{26}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$w=\frac{1}{2}$$

Whakaurua te \frac{1}{2} mō w ki v=-\frac{2}{9}w+\frac{7}{9}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō v hāngai tonu.

$$v=-\frac{2}{9}\times \frac{1}{2}+\frac{7}{9}$$

Whakareatia -\frac{2}{9} ki te \frac{1}{2} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$v=\frac{-1+7}{9}$$

Tāpiri \frac{7}{9} ki te -\frac{1}{9} 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.

$$v=\frac{2}{3}$$

Kua oti te pūnaha te whakatau.

$$v=\frac{2}{3},w=\frac{1}{2}$$

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.

$$9v+2w=7,3v-8w=-2$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}9&2\\3&-8\end{matrix}\right)\left(\begin{matrix}v\\w\end{matrix}\right)=\left(\begin{matrix}7\\-2\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}9&2\\3&-8\end{matrix}\right).

$$inverse(\left(\begin{matrix}9&2\\3&-8\end{matrix}\right))\left(\begin{matrix}9&2\\3&-8\end{matrix}\right)\left(\begin{matrix}v\\w\end{matrix}\right)=inverse(\left(\begin{matrix}9&2\\3&-8\end{matrix}\right))\left(\begin{matrix}7\\-2\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}v\\w\end{matrix}\right)=inverse(\left(\begin{matrix}9&2\\3&-8\end{matrix}\right))\left(\begin{matrix}7\\-2\end{matrix}\right)$$

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

$$\left(\begin{matrix}v\\w\end{matrix}\right)=inverse(\left(\begin{matrix}9&2\\3&-8\end{matrix}\right))\left(\begin{matrix}7\\-2\end{matrix}\right)$$

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.

$$\left(\begin{matrix}v\\w\end{matrix}\right)=\left(\begin{matrix}-\frac{8}{9\left(-8\right)-2\times 3}&-\frac{2}{9\left(-8\right)-2\times 3}\\-\frac{3}{9\left(-8\right)-2\times 3}&\frac{9}{9\left(-8\right)-2\times 3}\end{matrix}\right)\left(\begin{matrix}7\\-2\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}v\\w\end{matrix}\right)=\left(\begin{matrix}\frac{4}{39}&\frac{1}{39}\\\frac{1}{26}&-\frac{3}{26}\end{matrix}\right)\left(\begin{matrix}7\\-2\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}v\\w\end{matrix}\right)=\left(\begin{matrix}\frac{4}{39}\times 7+\frac{1}{39}\left(-2\right)\\\frac{1}{26}\times 7-\frac{3}{26}\left(-2\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}v\\w\end{matrix}\right)=\left(\begin{matrix}\frac{2}{3}\\\frac{1}{2}\end{matrix}\right)$$

Tangohia ngā huānga poukapa v me w.

$$v=\frac{2}{3},w=\frac{1}{2}$$

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.

$$9v+2w=7,3v-8w=-2$$

Kia ōrite ai a 9v 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 9.

$$3\times 9v+3\times 2w=3\times 7,9\times 3v+9\left(-8\right)w=9\left(-2\right)$$

Whakarūnātia.

$$27v+6w=21,27v-72w=-18$$

Me tango 27v-72w=-18 mai i 27v+6w=21 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$27v-27v+6w+72w=21+18$$

Tāpiri 27v ki te -27v. Ka whakakore atu ngā kupu 27v me -27v, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$6w+72w=21+18$$

Tāpiri 6w ki te 72w.

$$78w=21+18$$

Tāpiri 21 ki te 18.

$$78w=39$$

Whakawehea ngā taha e rua ki te 78.

$$w=\frac{1}{2}$$

Whakaurua te \frac{1}{2} mō w ki 3v-8w=-2. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō v hāngai tonu.

$$3v-8\times \frac{1}{2}=-2$$

Whakareatia -8 ki te \frac{1}{2}.

$$3v-4=-2$$

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

$$3v=2$$

Whakawehea ngā taha e rua ki te 3.

$$v=\frac{2}{3}$$

Kua oti te pūnaha te whakatau.

$$v=\frac{2}{3},w=\frac{1}{2}$$