Whakaoti i te {l}{2m+3n=1}{15/9m-2n=1}

$$\left\{ \begin{array} { l } { 2 m + 3 n = 1 } \\ { \frac { 15 } { 9 } m - 2 n = 1 } \end{array} \right.$$

Whakaoti mō m, n

$$m=\frac{5}{9}\approx 0.555555556$$

$$n=-\frac{1}{27}\approx -0.037037037$$

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.

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

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

$$2m+3n=1$$

Me tango 3n mai i 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, \frac{5}{3}m-2n=1.

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

Whakareatia \frac{5}{3} ki te \frac{-3n+1}{2}.

$$-\frac{5}{2}n+\frac{5}{6}-2n=1$$

Tāpiri -\frac{5n}{2} ki te -2n.

$$-\frac{9}{2}n+\frac{5}{6}=1$$

Me tango \frac{5}{6} mai i ngā taha e rua o te whārite.

$$-\frac{9}{2}n=\frac{1}{6}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{9}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$n=-\frac{1}{27}$$

Whakaurua te -\frac{1}{27} 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.

$$m=-\frac{3}{2}\left(-\frac{1}{27}\right)+\frac{1}{2}$$

Whakareatia -\frac{3}{2} ki te -\frac{1}{27} 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.

$$m=\frac{1}{18}+\frac{1}{2}$$

Tāpiri \frac{1}{2} ki te \frac{1}{18} 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.

$$m=\frac{5}{9}$$

Kua oti te pūnaha te whakatau.

$$m=\frac{5}{9},n=-\frac{1}{27}$$

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.

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}2&3\\\frac{5}{3}&-2\end{matrix}\right).

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

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}m\\n\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\\frac{5}{3}&-2\end{matrix}\right))\left(\begin{matrix}1\\1\end{matrix}\right)$$

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

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

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.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}m\\n\end{matrix}\right)=\left(\begin{matrix}\frac{2}{9}&\frac{1}{3}\\\frac{5}{27}&-\frac{2}{9}\end{matrix}\right)\left(\begin{matrix}1\\1\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}m\\n\end{matrix}\right)=\left(\begin{matrix}\frac{2}{9}+\frac{1}{3}\\\frac{5}{27}-\frac{2}{9}\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}m\\n\end{matrix}\right)=\left(\begin{matrix}\frac{5}{9}\\-\frac{1}{27}\end{matrix}\right)$$

Tangohia ngā huānga poukapa m me n.

$$m=\frac{5}{9},n=-\frac{1}{27}$$

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.

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

Kia ōrite ai a 2m me \frac{5m}{3}, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te \frac{5}{3} me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 2.

$$\frac{5}{3}\times 2m+\frac{5}{3}\times 3n=\frac{5}{3},2\times \frac{5}{3}m+2\left(-2\right)n=2$$

Whakarūnātia.

$$\frac{10}{3}m+5n=\frac{5}{3},\frac{10}{3}m-4n=2$$

Me tango \frac{10}{3}m-4n=2 mai i \frac{10}{3}m+5n=\frac{5}{3} mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$\frac{10}{3}m-\frac{10}{3}m+5n+4n=\frac{5}{3}-2$$

Tāpiri \frac{10m}{3} ki te -\frac{10m}{3}. Ka whakakore atu ngā kupu \frac{10m}{3} me -\frac{10m}{3}, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$5n+4n=\frac{5}{3}-2$$

Tāpiri 5n ki te 4n.

$$9n=\frac{5}{3}-2$$

Tāpiri \frac{5}{3} ki te -2.

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

Whakawehea ngā taha e rua ki te 9.

$$n=-\frac{1}{27}$$

Whakaurua te -\frac{1}{27} mō n ki \frac{5}{3}m-2n=1. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō m hāngai tonu.

$$\frac{5}{3}m-2\left(-\frac{1}{27}\right)=1$$

Whakareatia -2 ki te -\frac{1}{27}.

$$\frac{5}{3}m+\frac{2}{27}=1$$

Me tango \frac{2}{27} mai i ngā taha e rua o te whārite.

$$\frac{5}{3}m=\frac{25}{27}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{5}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$m=\frac{5}{9}$$

Kua oti te pūnaha te whakatau.

$$m=\frac{5}{9},n=-\frac{1}{27}$$