Whakaoti i te {l}{1-m+n=0}{4+2m+n=0}

$$\left\{ \begin{array} { l } { 1 - m + n = 0 } \\ { 4 + 2 m + n = 0 } \end{array} \right.$$

Whakaoti mō m, n

$$m=-1$$

$$n=-2$$

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.

$$-m+n+1=0,2m+n+4=0$$

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

$$-m+n+1=0$$

Me tango 1 mai i ngā taha e rua o te whārite.

$$-m+n=-1$$

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

$$-m=-n-1$$

Whakawehea ngā taha e rua ki te -1.

$$m=-\left(-n-1\right)$$

Whakareatia -1 ki te -n-1.

$$m=n+1$$

Whakakapia te n+1 mō te m ki tērā atu whārite, 2m+n+4=0.

$$2\left(n+1\right)+n+4=0$$

Whakareatia 2 ki te n+1.

$$2n+2+n+4=0$$

Tāpiri 2n ki te n.

$$3n+2+4=0$$

Tāpiri 2 ki te 4.

$$3n+6=0$$

Me tango 6 mai i ngā taha e rua o te whārite.

$$3n=-6$$

Whakawehea ngā taha e rua ki te 3.

$$n=-2$$

Whakaurua te -2 mō n ki m=n+1. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō m hāngai tonu.

$$m=-2+1$$

Tāpiri 1 ki te -2.

$$m=-1$$

Kua oti te pūnaha te whakatau.

$$m=-1,n=-2$$

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.

$$-m+n+1=0,2m+n+4=0$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa m me n.

$$m=-1,n=-2$$

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.

$$-m+n+1=0,2m+n+4=0$$

Me tango 2m+n+4=0 mai i -m+n+1=0 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-m-2m+n-n+1-4=0$$

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

$$-m-2m+1-4=0$$

Tāpiri -m ki te -2m.

$$-3m+1-4=0$$

Tāpiri 1 ki te -4.

$$-3m-3=0$$

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

$$-3m=3$$

Whakawehea ngā taha e rua ki te -3.

$$m=-1$$

Whakaurua te -1 mō m ki 2m+n+4=0. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō n hāngai tonu.

$$2\left(-1\right)+n+4=0$$

Whakareatia 2 ki te -1.

$$-2+n+4=0$$

Tāpiri -2 ki te 4.

$$n+2=0$$

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

$$n=-2$$

Kua oti te pūnaha te whakatau.

$$m=-1,n=-2$$