Whakaoti i te {l}{2p+3m=8}{p+2m=6}

$$\left\{ \begin{array} { l } { 2 p + 3 m = 8 } \\ { p + 2 m = 6 } \end{array} \right.$$

Whakaoti mō p, m

$$p=-2$$

$$m=4$$

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.

$$2p+3m=8,p+2m=6$$

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

$$2p+3m=8$$

Me tango 3m mai i ngā taha e rua o te whārite.

$$2p=-3m+8$$

Whakawehea ngā taha e rua ki te 2.

$$p=\frac{1}{2}\left(-3m+8\right)$$

Whakareatia \frac{1}{2} ki te -3m+8.

$$p=-\frac{3}{2}m+4$$

Whakakapia te -\frac{3m}{2}+4 mō te p ki tērā atu whārite, p+2m=6.

$$-\frac{3}{2}m+4+2m=6$$

Tāpiri -\frac{3m}{2} ki te 2m.

$$\frac{1}{2}m+4=6$$

Me tango 4 mai i ngā taha e rua o te whārite.

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

Me whakarea ngā taha e rua ki te 2.

$$m=4$$

Whakaurua te 4 mō m ki p=-\frac{3}{2}m+4. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō p hāngai tonu.

$$p=-\frac{3}{2}\times 4+4$$

Whakareatia -\frac{3}{2} ki te 4.

$$p=-6+4$$

Tāpiri 4 ki te -6.

$$p=-2$$

Kua oti te pūnaha te whakatau.

$$p=-2,m=4$$

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.

$$2p+3m=8,p+2m=6$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&3\\1&2\end{matrix}\right)\left(\begin{matrix}p\\m\end{matrix}\right)=\left(\begin{matrix}8\\6\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}2&3\\1&2\end{matrix}\right).

$$inverse(\left(\begin{matrix}2&3\\1&2\end{matrix}\right))\left(\begin{matrix}2&3\\1&2\end{matrix}\right)\left(\begin{matrix}p\\m\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\1&2\end{matrix}\right))\left(\begin{matrix}8\\6\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}p\\m\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\1&2\end{matrix}\right))\left(\begin{matrix}8\\6\end{matrix}\right)$$

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

$$\left(\begin{matrix}p\\m\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\1&2\end{matrix}\right))\left(\begin{matrix}8\\6\end{matrix}\right)$$

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.

$$\left(\begin{matrix}p\\m\end{matrix}\right)=\left(\begin{matrix}\frac{2}{2\times 2-3}&-\frac{3}{2\times 2-3}\\-\frac{1}{2\times 2-3}&\frac{2}{2\times 2-3}\end{matrix}\right)\left(\begin{matrix}8\\6\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}p\\m\end{matrix}\right)=\left(\begin{matrix}2&-3\\-1&2\end{matrix}\right)\left(\begin{matrix}8\\6\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}p\\m\end{matrix}\right)=\left(\begin{matrix}2\times 8-3\times 6\\-8+2\times 6\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa p me m.

$$p=-2,m=4$$

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.

$$2p+3m=8,p+2m=6$$

Kia ōrite ai a 2p me p, 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 2.

$$2p+3m=8,2p+2\times 2m=2\times 6$$

Whakarūnātia.

$$2p+3m=8,2p+4m=12$$

Me tango 2p+4m=12 mai i 2p+3m=8 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$2p-2p+3m-4m=8-12$$

Tāpiri 2p ki te -2p. Ka whakakore atu ngā kupu 2p me -2p, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$3m-4m=8-12$$

Tāpiri 3m ki te -4m.

$$-m=8-12$$

Tāpiri 8 ki te -12.

$$-m=-4$$

Whakawehea ngā taha e rua ki te -1.

$$m=4$$

Whakaurua te 4 mō m ki p+2m=6. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō p hāngai tonu.

$$p+2\times 4=6$$

Whakareatia 2 ki te 4.

$$p+8=6$$

Me tango 8 mai i ngā taha e rua o te whārite.

$$p=-2$$

Kua oti te pūnaha te whakatau.

$$p=-2,m=4$$