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

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

Whakaoti mō n, m

$$n=2$$

$$m=1$$

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.

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

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

$$2n-3m=1$$

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

$$2n=3m+1$$

Whakawehea ngā taha e rua ki te 2.

$$n=\frac{1}{2}\left(3m+1\right)$$

Whakareatia \frac{1}{2} ki te 3m+1.

$$n=\frac{3}{2}m+\frac{1}{2}$$

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

$$\frac{3}{2}m+\frac{1}{2}+m=3$$

Tāpiri \frac{3m}{2} ki te m.

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

Me tango \frac{1}{2} mai i ngā taha e rua o te whārite.

$$\frac{5}{2}m=\frac{5}{2}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{5}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$m=1$$

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

$$n=\frac{3+1}{2}$$

Tāpiri \frac{1}{2} ki te \frac{3}{2} 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.

$$n=2$$

Kua oti te pūnaha te whakatau.

$$n=2,m=1$$

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.

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa n me m.

$$n=2,m=1$$

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.

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

Kia ōrite ai a 2n me n, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 1 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 2.

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

Whakarūnātia.

$$2n-3m=1,2n+2m=6$$

Me tango 2n+2m=6 mai i 2n-3m=1 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$2n-2n-3m-2m=1-6$$

Tāpiri 2n ki te -2n. Ka whakakore atu ngā kupu 2n me -2n, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-3m-2m=1-6$$

Tāpiri -3m ki te -2m.

$$-5m=1-6$$

Tāpiri 1 ki te -6.

$$-5m=-5$$

Whakawehea ngā taha e rua ki te -5.

$$m=1$$

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

$$n+1=3$$

Me tango 1 mai i ngā taha e rua o te whārite.

$$n=2$$

Kua oti te pūnaha te whakatau.

$$n=2,m=1$$