Whakaoti i te {l}{m+2n=5}{n-2m+2=7}

$$\left. \begin{array} { l } { m + 2 n = 5 } \\ { n - 2 m + 2 = 7 } \end{array} \right.$$

Whakaoti mō m, n

$$m=-1$$

$$n=3$$

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.

$$m+2n=5,-2m+n+2=7$$

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

$$m+2n=5$$

Me tango 2n mai i ngā taha e rua o te whārite.

$$m=-2n+5$$

Whakakapia te -2n+5 mō te m ki tērā atu whārite, -2m+n+2=7.

$$-2\left(-2n+5\right)+n+2=7$$

Whakareatia -2 ki te -2n+5.

$$4n-10+n+2=7$$

Tāpiri 4n ki te n.

$$5n-10+2=7$$

Tāpiri -10 ki te 2.

$$5n-8=7$$

Me tāpiri 8 ki ngā taha e rua o te whārite.

$$5n=15$$

Whakawehea ngā taha e rua ki te 5.

$$n=3$$

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

$$m=-2\times 3+5$$

Whakareatia -2 ki te 3.

$$m=-6+5$$

Tāpiri 5 ki te -6.

$$m=-1$$

Kua oti te pūnaha te whakatau.

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

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.

$$m+2n=5,-2m+n+2=7$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa m me n.

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

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.

$$m+2n=5,-2m+n+2=7$$

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.

$$-2m-2\times 2n=-2\times 5,-2m+n+2=7$$

Whakarūnātia.

$$-2m-4n=-10,-2m+n+2=7$$

Me tango -2m+n+2=7 mai i -2m-4n=-10 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-2m+2m-4n-n-2=-10-7$$

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.

$$-4n-n-2=-10-7$$

Tāpiri -4n ki te -n.

$$-5n-2=-10-7$$

Tāpiri -10 ki te -7.

$$-5n-2=-17$$

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

$$-5n=-15$$

Whakawehea ngā taha e rua ki te -5.

$$n=3$$

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

$$-2m+3+2=7$$

Tāpiri 3 ki te 2.

$$-2m+5=7$$

Me tango 5 mai i ngā taha e rua o te whārite.

$$-2m=2$$

Whakawehea ngā taha e rua ki te -2.

$$m=-1$$

Kua oti te pūnaha te whakatau.

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