Whakaoti i te {l}{6s+7t=47}{8s+3t=12}

$$\left. \begin{array} { l } { 6 s + 7 t = 47 } \\ { 8 s + 3 t = 12 } \end{array} \right.$$

Whakaoti mō s, t

$$t=8$$

$$s = -\frac{3}{2} = -1\frac{1}{2} = -1.5$$

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.

$$6s+7t=47,8s+3t=12$$

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

$$6s+7t=47$$

Me tango 7t mai i ngā taha e rua o te whārite.

$$6s=-7t+47$$

Whakawehea ngā taha e rua ki te 6.

$$s=\frac{1}{6}\left(-7t+47\right)$$

Whakareatia \frac{1}{6} ki te -7t+47.

$$s=-\frac{7}{6}t+\frac{47}{6}$$

Whakakapia te \frac{-7t+47}{6} mō te s ki tērā atu whārite, 8s+3t=12.

$$8\left(-\frac{7}{6}t+\frac{47}{6}\right)+3t=12$$

Whakareatia 8 ki te \frac{-7t+47}{6}.

$$-\frac{28}{3}t+\frac{188}{3}+3t=12$$

Tāpiri -\frac{28t}{3} ki te 3t.

$$-\frac{19}{3}t+\frac{188}{3}=12$$

Me tango \frac{188}{3} mai i ngā taha e rua o te whārite.

$$-\frac{19}{3}t=-\frac{152}{3}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{19}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$t=8$$

Whakaurua te 8 mō t ki s=-\frac{7}{6}t+\frac{47}{6}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō s hāngai tonu.

$$s=-\frac{7}{6}\times 8+\frac{47}{6}$$

Whakareatia -\frac{7}{6} ki te 8.

$$s=-\frac{28}{3}+\frac{47}{6}$$

Tāpiri \frac{47}{6} ki te -\frac{28}{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.

$$s=-\frac{3}{2}$$

Kua oti te pūnaha te whakatau.

$$s=-\frac{3}{2},t=8$$

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.

$$6s+7t=47,8s+3t=12$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}6&7\\8&3\end{matrix}\right)\left(\begin{matrix}s\\t\end{matrix}\right)=\left(\begin{matrix}47\\12\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}6&7\\8&3\end{matrix}\right).

$$inverse(\left(\begin{matrix}6&7\\8&3\end{matrix}\right))\left(\begin{matrix}6&7\\8&3\end{matrix}\right)\left(\begin{matrix}s\\t\end{matrix}\right)=inverse(\left(\begin{matrix}6&7\\8&3\end{matrix}\right))\left(\begin{matrix}47\\12\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}s\\t\end{matrix}\right)=inverse(\left(\begin{matrix}6&7\\8&3\end{matrix}\right))\left(\begin{matrix}47\\12\end{matrix}\right)$$

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

$$\left(\begin{matrix}s\\t\end{matrix}\right)=inverse(\left(\begin{matrix}6&7\\8&3\end{matrix}\right))\left(\begin{matrix}47\\12\end{matrix}\right)$$

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.

$$\left(\begin{matrix}s\\t\end{matrix}\right)=\left(\begin{matrix}\frac{3}{6\times 3-7\times 8}&-\frac{7}{6\times 3-7\times 8}\\-\frac{8}{6\times 3-7\times 8}&\frac{6}{6\times 3-7\times 8}\end{matrix}\right)\left(\begin{matrix}47\\12\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}s\\t\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{38}&\frac{7}{38}\\\frac{4}{19}&-\frac{3}{19}\end{matrix}\right)\left(\begin{matrix}47\\12\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}s\\t\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{38}\times 47+\frac{7}{38}\times 12\\\frac{4}{19}\times 47-\frac{3}{19}\times 12\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}s\\t\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{2}\\8\end{matrix}\right)$$

Tangohia ngā huānga poukapa s me t.

$$s=-\frac{3}{2},t=8$$

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.

$$6s+7t=47,8s+3t=12$$

Kia ōrite ai a 6s me 8s, 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 6.

$$8\times 6s+8\times 7t=8\times 47,6\times 8s+6\times 3t=6\times 12$$

Whakarūnātia.

$$48s+56t=376,48s+18t=72$$

Me tango 48s+18t=72 mai i 48s+56t=376 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$48s-48s+56t-18t=376-72$$

Tāpiri 48s ki te -48s. Ka whakakore atu ngā kupu 48s me -48s, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$56t-18t=376-72$$

Tāpiri 56t ki te -18t.

$$38t=376-72$$

Tāpiri 376 ki te -72.

$$38t=304$$

Whakawehea ngā taha e rua ki te 38.

$$t=8$$

Whakaurua te 8 mō t ki 8s+3t=12. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō s hāngai tonu.

$$8s+3\times 8=12$$

Whakareatia 3 ki te 8.

$$8s+24=12$$

Me tango 24 mai i ngā taha e rua o te whārite.

$$8s=-12$$

Whakawehea ngā taha e rua ki te 8.

$$s=-\frac{3}{2}$$

Kua oti te pūnaha te whakatau.

$$s=-\frac{3}{2},t=8$$