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

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

Whakaoti mō m, n

$$m=\frac{4}{5}=0.8$$

$$n=\frac{1}{5}=0.2$$

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.

$$m+n=1,-3m+2n=-2$$

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

$$m+n=1$$

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

$$m=-n+1$$

Whakakapia te -n+1 mō te m ki tērā atu whārite, -3m+2n=-2.

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

Whakareatia -3 ki te -n+1.

$$3n-3+2n=-2$$

Tāpiri 3n ki te 2n.

$$5n-3=-2$$

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

$$5n=1$$

Whakawehea ngā taha e rua ki te 5.

$$n=\frac{1}{5}$$

Whakaurua te \frac{1}{5} mō n ki m=-n+1. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō m hāngai tonu.

$$m=-\frac{1}{5}+1$$

Tāpiri 1 ki te -\frac{1}{5}.

$$m=\frac{4}{5}$$

Kua oti te pūnaha te whakatau.

$$m=\frac{4}{5},n=\frac{1}{5}$$

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.

$$m+n=1,-3m+2n=-2$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}m\\n\end{matrix}\right)=\left(\begin{matrix}\frac{2}{5}&-\frac{1}{5}\\\frac{3}{5}&\frac{1}{5}\end{matrix}\right)\left(\begin{matrix}1\\-2\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}m\\n\end{matrix}\right)=\left(\begin{matrix}\frac{2}{5}-\frac{1}{5}\left(-2\right)\\\frac{3}{5}+\frac{1}{5}\left(-2\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}m\\n\end{matrix}\right)=\left(\begin{matrix}\frac{4}{5}\\\frac{1}{5}\end{matrix}\right)$$

Tangohia ngā huānga poukapa m me n.

$$m=\frac{4}{5},n=\frac{1}{5}$$

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.

$$m+n=1,-3m+2n=-2$$

Kia ōrite ai a m me -3m, 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.

$$-3m-3n=-3,-3m+2n=-2$$

Me tango -3m+2n=-2 mai i -3m-3n=-3 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-3m+3m-3n-2n=-3+2$$

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

$$-3n-2n=-3+2$$

Tāpiri -3n ki te -2n.

$$-5n=-3+2$$

Tāpiri -3 ki te 2.

$$-5n=-1$$

Whakawehea ngā taha e rua ki te -5.

$$n=\frac{1}{5}$$

Whakaurua te \frac{1}{5} mō n ki -3m+2n=-2. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō m hāngai tonu.

$$-3m+2\times \frac{1}{5}=-2$$

Whakareatia 2 ki te \frac{1}{5}.

$$-3m+\frac{2}{5}=-2$$

Me tango \frac{2}{5} mai i ngā taha e rua o te whārite.

$$-3m=-\frac{12}{5}$$

Whakawehea ngā taha e rua ki te -3.

$$m=\frac{4}{5}$$

Kua oti te pūnaha te whakatau.

$$m=\frac{4}{5},n=\frac{1}{5}$$