Whakaoti i te {l}{6m-5n=-9}{4m+3n=13(5)}

$$\left. \begin{array} { l } { 6 m - 5 n = - 9 } \\ { 4 m + 3 n = 13 ( 5 ) } \end{array} \right.$$

Whakaoti mō m, n

$$m = \frac{149}{19} = 7\frac{16}{19} \approx 7.842105263$$

$$n = \frac{213}{19} = 11\frac{4}{19} \approx 11.210526316$$

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.

$$6m-5n=-9,4m+3n=65$$

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

$$6m-5n=-9$$

Me tāpiri 5n ki ngā taha e rua o te whārite.

$$6m=5n-9$$

Whakawehea ngā taha e rua ki te 6.

$$m=\frac{1}{6}\left(5n-9\right)$$

Whakareatia \frac{1}{6} ki te 5n-9.

$$m=\frac{5}{6}n-\frac{3}{2}$$

Whakakapia te \frac{5n}{6}-\frac{3}{2} mō te m ki tērā atu whārite, 4m+3n=65.

$$4\left(\frac{5}{6}n-\frac{3}{2}\right)+3n=65$$

Whakareatia 4 ki te \frac{5n}{6}-\frac{3}{2}.

$$\frac{10}{3}n-6+3n=65$$

Tāpiri \frac{10n}{3} ki te 3n.

$$\frac{19}{3}n-6=65$$

Me tāpiri 6 ki ngā taha e rua o te whārite.

$$\frac{19}{3}n=71$$

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.

$$n=\frac{213}{19}$$

Whakaurua te \frac{213}{19} mō n ki m=\frac{5}{6}n-\frac{3}{2}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō m hāngai tonu.

$$m=\frac{5}{6}\times \frac{213}{19}-\frac{3}{2}$$

Whakareatia \frac{5}{6} ki te \frac{213}{19} 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.

$$m=\frac{355}{38}-\frac{3}{2}$$

Tāpiri -\frac{3}{2} ki te \frac{355}{38} 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.

$$m=\frac{149}{19}$$

Kua oti te pūnaha te whakatau.

$$m=\frac{149}{19},n=\frac{213}{19}$$

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.

$$6m-5n=-9,4m+3n=65$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}6&-5\\4&3\end{matrix}\right)\left(\begin{matrix}m\\n\end{matrix}\right)=\left(\begin{matrix}-9\\65\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}6&-5\\4&3\end{matrix}\right).

$$inverse(\left(\begin{matrix}6&-5\\4&3\end{matrix}\right))\left(\begin{matrix}6&-5\\4&3\end{matrix}\right)\left(\begin{matrix}m\\n\end{matrix}\right)=inverse(\left(\begin{matrix}6&-5\\4&3\end{matrix}\right))\left(\begin{matrix}-9\\65\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}m\\n\end{matrix}\right)=inverse(\left(\begin{matrix}6&-5\\4&3\end{matrix}\right))\left(\begin{matrix}-9\\65\end{matrix}\right)$$

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

$$\left(\begin{matrix}m\\n\end{matrix}\right)=inverse(\left(\begin{matrix}6&-5\\4&3\end{matrix}\right))\left(\begin{matrix}-9\\65\end{matrix}\right)$$

Mō te poukapa 2\times 2 \left(\begin{matrix}a&b\\c&d\end{matrix}\right), ko te poukapa kōaro ko \left(\begin{matrix}\frac{d}{ad-bc}&\frac{-b}{ad-bc}\\\frac{-c}{ad-bc}&\frac{a}{ad-bc}\end{matrix}\right), kia tuhia anō ai te whārite poukapa hei rapanga whakarea poukapa.

$$\left(\begin{matrix}m\\n\end{matrix}\right)=\left(\begin{matrix}\frac{3}{6\times 3-\left(-5\times 4\right)}&-\frac{-5}{6\times 3-\left(-5\times 4\right)}\\-\frac{4}{6\times 3-\left(-5\times 4\right)}&\frac{6}{6\times 3-\left(-5\times 4\right)}\end{matrix}\right)\left(\begin{matrix}-9\\65\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}m\\n\end{matrix}\right)=\left(\begin{matrix}\frac{3}{38}&\frac{5}{38}\\-\frac{2}{19}&\frac{3}{19}\end{matrix}\right)\left(\begin{matrix}-9\\65\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}m\\n\end{matrix}\right)=\left(\begin{matrix}\frac{3}{38}\left(-9\right)+\frac{5}{38}\times 65\\-\frac{2}{19}\left(-9\right)+\frac{3}{19}\times 65\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}m\\n\end{matrix}\right)=\left(\begin{matrix}\frac{149}{19}\\\frac{213}{19}\end{matrix}\right)$$

Tangohia ngā huānga poukapa m me n.

$$m=\frac{149}{19},n=\frac{213}{19}$$

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.

$$6m-5n=-9,4m+3n=65$$

Kia ōrite ai a 6m me 4m, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 4 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 6.

$$4\times 6m+4\left(-5\right)n=4\left(-9\right),6\times 4m+6\times 3n=6\times 65$$

Whakarūnātia.

$$24m-20n=-36,24m+18n=390$$

Me tango 24m+18n=390 mai i 24m-20n=-36 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$24m-24m-20n-18n=-36-390$$

Tāpiri 24m ki te -24m. Ka whakakore atu ngā kupu 24m me -24m, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-20n-18n=-36-390$$

Tāpiri -20n ki te -18n.

$$-38n=-36-390$$

Tāpiri -36 ki te -390.

$$-38n=-426$$

Whakawehea ngā taha e rua ki te -38.

$$n=\frac{213}{19}$$

Whakaurua te \frac{213}{19} mō n ki 4m+3n=65. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō m hāngai tonu.

$$4m+3\times \frac{213}{19}=65$$

Whakareatia 3 ki te \frac{213}{19}.

$$4m+\frac{639}{19}=65$$

Me tango \frac{639}{19} mai i ngā taha e rua o te whārite.

$$4m=\frac{596}{19}$$

Whakawehea ngā taha e rua ki te 4.

$$m=\frac{149}{19}$$

Kua oti te pūnaha te whakatau.

$$m=\frac{149}{19},n=\frac{213}{19}$$