Whakaoti i te {l}{m+n=6}{2m-2n=6}

$$\left\{ \begin{array} { l } { m + n = 6 } \\ { 2 m - 2 n = 6 } \end{array} \right.$$

Whakaoti mō m, n

$$m = \frac{9}{2} = 4\frac{1}{2} = 4.5$$

$$n = \frac{3}{2} = 1\frac{1}{2} = 1.5$$

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.

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

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

$$m+n=6$$

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

$$m=-n+6$$

Whakakapia te -n+6 mō te m ki tērā atu whārite, 2m-2n=6.

$$2\left(-n+6\right)-2n=6$$

Whakareatia 2 ki te -n+6.

$$-2n+12-2n=6$$

Tāpiri -2n ki te -2n.

$$-4n+12=6$$

Me tango 12 mai i ngā taha e rua o te whārite.

$$-4n=-6$$

Whakawehea ngā taha e rua ki te -4.

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

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

$$m=-\frac{3}{2}+6$$

Tāpiri 6 ki te -\frac{3}{2}.

$$m=\frac{9}{2}$$

Kua oti te pūnaha te whakatau.

$$m=\frac{9}{2},n=\frac{3}{2}$$

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.

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&1\\2&-2\end{matrix}\right)\left(\begin{matrix}m\\n\end{matrix}\right)=\left(\begin{matrix}6\\6\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}1&1\\2&-2\end{matrix}\right))\left(\begin{matrix}1&1\\2&-2\end{matrix}\right)\left(\begin{matrix}m\\n\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\2&-2\end{matrix}\right))\left(\begin{matrix}6\\6\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}m\\n\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\2&-2\end{matrix}\right))\left(\begin{matrix}6\\6\end{matrix}\right)$$

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

$$\left(\begin{matrix}m\\n\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\2&-2\end{matrix}\right))\left(\begin{matrix}6\\6\end{matrix}\right)$$

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.

$$\left(\begin{matrix}m\\n\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{-2-2}&-\frac{1}{-2-2}\\-\frac{2}{-2-2}&\frac{1}{-2-2}\end{matrix}\right)\left(\begin{matrix}6\\6\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}m\\n\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2}&\frac{1}{4}\\\frac{1}{2}&-\frac{1}{4}\end{matrix}\right)\left(\begin{matrix}6\\6\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}m\\n\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2}\times 6+\frac{1}{4}\times 6\\\frac{1}{2}\times 6-\frac{1}{4}\times 6\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}m\\n\end{matrix}\right)=\left(\begin{matrix}\frac{9}{2}\\\frac{3}{2}\end{matrix}\right)$$

Tangohia ngā huānga poukapa m me n.

$$m=\frac{9}{2},n=\frac{3}{2}$$

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.

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

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

$$2m+2n=2\times 6,2m-2n=6$$

Whakarūnātia.

$$2m+2n=12,2m-2n=6$$

Me tango 2m-2n=6 mai i 2m+2n=12 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$2m-2m+2n+2n=12-6$$

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.

$$2n+2n=12-6$$

Tāpiri 2n ki te 2n.

$$4n=12-6$$

Tāpiri 12 ki te -6.

$$4n=6$$

Whakawehea ngā taha e rua ki te 4.

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

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

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

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

$$2m-3=6$$

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

$$2m=9$$

Whakawehea ngā taha e rua ki te 2.

$$m=\frac{9}{2}$$

Kua oti te pūnaha te whakatau.

$$m=\frac{9}{2},n=\frac{3}{2}$$