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

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

Whakaoti mō m, n

$$m = \frac{8}{5} = 1\frac{3}{5} = 1.6$$

$$n = \frac{7}{5} = 1\frac{2}{5} = 1.4$$

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.

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

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

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

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

$$2m=3n-1$$

Whakawehea ngā taha e rua ki te 2.

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

Whakareatia \frac{1}{2} ki te 3n-1.

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

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

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

Tāpiri \frac{3n}{2} ki te n.

$$\frac{5}{2}n-\frac{1}{2}=3$$

Me tāpiri \frac{1}{2} ki ngā taha e rua o te whārite.

$$\frac{5}{2}n=\frac{7}{2}$$

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.

$$n=\frac{7}{5}$$

Whakaurua te \frac{7}{5} mō n ki m=\frac{3}{2}n-\frac{1}{2}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō m hāngai tonu.

$$m=\frac{3}{2}\times \frac{7}{5}-\frac{1}{2}$$

Whakareatia \frac{3}{2} ki te \frac{7}{5} 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.

$$m=\frac{21}{10}-\frac{1}{2}$$

Tāpiri -\frac{1}{2} ki te \frac{21}{10} 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.

$$m=\frac{8}{5}$$

Kua oti te pūnaha te whakatau.

$$m=\frac{8}{5},n=\frac{7}{5}$$

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.

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}m\\n\end{matrix}\right)=\left(\begin{matrix}\frac{8}{5}\\\frac{7}{5}\end{matrix}\right)$$

Tangohia ngā huānga poukapa m me n.

$$m=\frac{8}{5},n=\frac{7}{5}$$

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.

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

Kia ōrite ai a 2m me m, 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.

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

Whakarūnātia.

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

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

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

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

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

Tāpiri -3n ki te -2n.

$$-5n=-1-6$$

Tāpiri -1 ki te -6.

$$-5n=-7$$

Whakawehea ngā taha e rua ki te -5.

$$n=\frac{7}{5}$$

Whakaurua te \frac{7}{5} mō n ki m+n=3. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō m hāngai tonu.

$$m+\frac{7}{5}=3$$

Me tango \frac{7}{5} mai i ngā taha e rua o te whārite.

$$m=\frac{8}{5}$$

Kua oti te pūnaha te whakatau.

$$m=\frac{8}{5},n=\frac{7}{5}$$