Whakaoti i te {l}{9m-13n=22}{2m+3n=-1}

$$\left\{ \begin{array} { l } { 9 m - 13 n = 22 } \\ { 2 m + 3 n = - 1 } \end{array} \right.$$

Whakaoti mō m, n

$$m=1$$

$$n=-1$$

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.

$$9m-13n=22,2m+3n=-1$$

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

$$9m-13n=22$$

Me tāpiri 13n ki ngā taha e rua o te whārite.

$$9m=13n+22$$

Whakawehea ngā taha e rua ki te 9.

$$m=\frac{1}{9}\left(13n+22\right)$$

Whakareatia \frac{1}{9} ki te 13n+22.

$$m=\frac{13}{9}n+\frac{22}{9}$$

Whakakapia te \frac{13n+22}{9} mō te m ki tērā atu whārite, 2m+3n=-1.

$$2\left(\frac{13}{9}n+\frac{22}{9}\right)+3n=-1$$

Whakareatia 2 ki te \frac{13n+22}{9}.

$$\frac{26}{9}n+\frac{44}{9}+3n=-1$$

Tāpiri \frac{26n}{9} ki te 3n.

$$\frac{53}{9}n+\frac{44}{9}=-1$$

Me tango \frac{44}{9} mai i ngā taha e rua o te whārite.

$$\frac{53}{9}n=-\frac{53}{9}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{53}{9}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$n=-1$$

Whakaurua te -1 mō n ki m=\frac{13}{9}n+\frac{22}{9}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō m hāngai tonu.

$$m=\frac{13}{9}\left(-1\right)+\frac{22}{9}$$

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

$$m=\frac{-13+22}{9}$$

Tāpiri \frac{22}{9} ki te -\frac{13}{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.

$$m=1$$

Kua oti te pūnaha te whakatau.

$$m=1,n=-1$$

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.

$$9m-13n=22,2m+3n=-1$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}9&-13\\2&3\end{matrix}\right)\left(\begin{matrix}m\\n\end{matrix}\right)=\left(\begin{matrix}22\\-1\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}9&-13\\2&3\end{matrix}\right).

$$inverse(\left(\begin{matrix}9&-13\\2&3\end{matrix}\right))\left(\begin{matrix}9&-13\\2&3\end{matrix}\right)\left(\begin{matrix}m\\n\end{matrix}\right)=inverse(\left(\begin{matrix}9&-13\\2&3\end{matrix}\right))\left(\begin{matrix}22\\-1\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}m\\n\end{matrix}\right)=inverse(\left(\begin{matrix}9&-13\\2&3\end{matrix}\right))\left(\begin{matrix}22\\-1\end{matrix}\right)$$

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

$$\left(\begin{matrix}m\\n\end{matrix}\right)=inverse(\left(\begin{matrix}9&-13\\2&3\end{matrix}\right))\left(\begin{matrix}22\\-1\end{matrix}\right)$$

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.

$$\left(\begin{matrix}m\\n\end{matrix}\right)=\left(\begin{matrix}\frac{3}{9\times 3-\left(-13\times 2\right)}&-\frac{-13}{9\times 3-\left(-13\times 2\right)}\\-\frac{2}{9\times 3-\left(-13\times 2\right)}&\frac{9}{9\times 3-\left(-13\times 2\right)}\end{matrix}\right)\left(\begin{matrix}22\\-1\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}m\\n\end{matrix}\right)=\left(\begin{matrix}\frac{3}{53}&\frac{13}{53}\\-\frac{2}{53}&\frac{9}{53}\end{matrix}\right)\left(\begin{matrix}22\\-1\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}m\\n\end{matrix}\right)=\left(\begin{matrix}\frac{3}{53}\times 22+\frac{13}{53}\left(-1\right)\\-\frac{2}{53}\times 22+\frac{9}{53}\left(-1\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}m\\n\end{matrix}\right)=\left(\begin{matrix}1\\-1\end{matrix}\right)$$

Tangohia ngā huānga poukapa m me n.

$$m=1,n=-1$$

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.

$$9m-13n=22,2m+3n=-1$$

Kia ōrite ai a 9m me 2m, 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 9.

$$2\times 9m+2\left(-13\right)n=2\times 22,9\times 2m+9\times 3n=9\left(-1\right)$$

Whakarūnātia.

$$18m-26n=44,18m+27n=-9$$

Me tango 18m+27n=-9 mai i 18m-26n=44 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$18m-18m-26n-27n=44+9$$

Tāpiri 18m ki te -18m. Ka whakakore atu ngā kupu 18m me -18m, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-26n-27n=44+9$$

Tāpiri -26n ki te -27n.

$$-53n=44+9$$

Tāpiri 44 ki te 9.

$$-53n=53$$

Whakawehea ngā taha e rua ki te -53.

$$n=-1$$

Whakaurua te -1 mō n ki 2m+3n=-1. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō m hāngai tonu.

$$2m+3\left(-1\right)=-1$$

Whakareatia 3 ki te -1.

$$2m-3=-1$$

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

$$2m=2$$

Whakawehea ngā taha e rua ki te 2.

$$m=1$$

Kua oti te pūnaha te whakatau.

$$m=1,n=-1$$