Whakaoti i te {l}{9m-2n=3}{4n+m=-1}

$$\left\{ \begin{array} { l } { 9 m - 2 n = 3 } \\ { 4 n + m = - 1 } \end{array} \right.$$

Whakaoti mō m, n

$$m=\frac{5}{19}\approx 0.263157895$$

$$n=-\frac{6}{19}\approx -0.315789474$$

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.

$$9m-2n=3,m+4n=-1$$

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

$$9m-2n=3$$

Me tāpiri 2n ki ngā taha e rua o te whārite.

$$9m=2n+3$$

Whakawehea ngā taha e rua ki te 9.

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

Whakareatia \frac{1}{9} ki te 2n+3.

$$m=\frac{2}{9}n+\frac{1}{3}$$

Whakakapia te \frac{2n}{9}+\frac{1}{3} mō te m ki tērā atu whārite, m+4n=-1.

$$\frac{2}{9}n+\frac{1}{3}+4n=-1$$

Tāpiri \frac{2n}{9} ki te 4n.

$$\frac{38}{9}n+\frac{1}{3}=-1$$

Me tango \frac{1}{3} mai i ngā taha e rua o te whārite.

$$\frac{38}{9}n=-\frac{4}{3}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{38}{9}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

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

Whakaurua te -\frac{6}{19} mō n ki m=\frac{2}{9}n+\frac{1}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō m hāngai tonu.

$$m=\frac{2}{9}\left(-\frac{6}{19}\right)+\frac{1}{3}$$

Whakareatia \frac{2}{9} ki te -\frac{6}{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.

$$m=-\frac{4}{57}+\frac{1}{3}$$

Tāpiri \frac{1}{3} ki te -\frac{4}{57} 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.

$$m=\frac{5}{19}$$

Kua oti te pūnaha te whakatau.

$$m=\frac{5}{19},n=-\frac{6}{19}$$

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.

$$9m-2n=3,m+4n=-1$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}9&-2\\1&4\end{matrix}\right)\left(\begin{matrix}m\\n\end{matrix}\right)=\left(\begin{matrix}3\\-1\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}9&-2\\1&4\end{matrix}\right).

$$inverse(\left(\begin{matrix}9&-2\\1&4\end{matrix}\right))\left(\begin{matrix}9&-2\\1&4\end{matrix}\right)\left(\begin{matrix}m\\n\end{matrix}\right)=inverse(\left(\begin{matrix}9&-2\\1&4\end{matrix}\right))\left(\begin{matrix}3\\-1\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}m\\n\end{matrix}\right)=inverse(\left(\begin{matrix}9&-2\\1&4\end{matrix}\right))\left(\begin{matrix}3\\-1\end{matrix}\right)$$

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

$$\left(\begin{matrix}m\\n\end{matrix}\right)=inverse(\left(\begin{matrix}9&-2\\1&4\end{matrix}\right))\left(\begin{matrix}3\\-1\end{matrix}\right)$$

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.

$$\left(\begin{matrix}m\\n\end{matrix}\right)=\left(\begin{matrix}\frac{4}{9\times 4-\left(-2\right)}&-\frac{-2}{9\times 4-\left(-2\right)}\\-\frac{1}{9\times 4-\left(-2\right)}&\frac{9}{9\times 4-\left(-2\right)}\end{matrix}\right)\left(\begin{matrix}3\\-1\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}m\\n\end{matrix}\right)=\left(\begin{matrix}\frac{2}{19}&\frac{1}{19}\\-\frac{1}{38}&\frac{9}{38}\end{matrix}\right)\left(\begin{matrix}3\\-1\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}m\\n\end{matrix}\right)=\left(\begin{matrix}\frac{2}{19}\times 3+\frac{1}{19}\left(-1\right)\\-\frac{1}{38}\times 3+\frac{9}{38}\left(-1\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}m\\n\end{matrix}\right)=\left(\begin{matrix}\frac{5}{19}\\-\frac{6}{19}\end{matrix}\right)$$

Tangohia ngā huānga poukapa m me n.

$$m=\frac{5}{19},n=-\frac{6}{19}$$

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.

$$9m-2n=3,m+4n=-1$$

Kia ōrite ai a 9m 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 9.

$$9m-2n=3,9m+9\times 4n=9\left(-1\right)$$

Whakarūnātia.

$$9m-2n=3,9m+36n=-9$$

Me tango 9m+36n=-9 mai i 9m-2n=3 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$9m-9m-2n-36n=3+9$$

Tāpiri 9m ki te -9m. Ka whakakore atu ngā kupu 9m me -9m, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-2n-36n=3+9$$

Tāpiri -2n ki te -36n.

$$-38n=3+9$$

Tāpiri 3 ki te 9.

$$-38n=12$$

Whakawehea ngā taha e rua ki te -38.

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

Whakaurua te -\frac{6}{19} mō n ki m+4n=-1. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō m hāngai tonu.

$$m+4\left(-\frac{6}{19}\right)=-1$$

Whakareatia 4 ki te -\frac{6}{19}.

$$m-\frac{24}{19}=-1$$

Me tāpiri \frac{24}{19} ki ngā taha e rua o te whārite.

$$m=\frac{5}{19}$$

Kua oti te pūnaha te whakatau.

$$m=\frac{5}{19},n=-\frac{6}{19}$$