Whakaoti i te {l}{p+2q=4}{-3p+4q=18}

$$\left. \begin{array} { l } { p + 2 q = 4 } \\ { - 3 p + 4 q = 18 } \end{array} \right.$$

Whakaoti mō p, q

$$p=-2$$

$$q=3$$

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.

$$p+2q=4,-3p+4q=18$$

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

$$p+2q=4$$

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

$$p=-2q+4$$

Whakakapia te -2q+4 mō te p ki tērā atu whārite, -3p+4q=18.

$$-3\left(-2q+4\right)+4q=18$$

Whakareatia -3 ki te -2q+4.

$$6q-12+4q=18$$

Tāpiri 6q ki te 4q.

$$10q-12=18$$

Me tāpiri 12 ki ngā taha e rua o te whārite.

$$10q=30$$

Whakawehea ngā taha e rua ki te 10.

$$q=3$$

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

$$p=-2\times 3+4$$

Whakareatia -2 ki te 3.

$$p=-6+4$$

Tāpiri 4 ki te -6.

$$p=-2$$

Kua oti te pūnaha te whakatau.

$$p=-2,q=3$$

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.

$$p+2q=4,-3p+4q=18$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&2\\-3&4\end{matrix}\right)\left(\begin{matrix}p\\q\end{matrix}\right)=\left(\begin{matrix}4\\18\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&2\\-3&4\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&2\\-3&4\end{matrix}\right))\left(\begin{matrix}1&2\\-3&4\end{matrix}\right)\left(\begin{matrix}p\\q\end{matrix}\right)=inverse(\left(\begin{matrix}1&2\\-3&4\end{matrix}\right))\left(\begin{matrix}4\\18\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}p\\q\end{matrix}\right)=inverse(\left(\begin{matrix}1&2\\-3&4\end{matrix}\right))\left(\begin{matrix}4\\18\end{matrix}\right)$$

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

$$\left(\begin{matrix}p\\q\end{matrix}\right)=inverse(\left(\begin{matrix}1&2\\-3&4\end{matrix}\right))\left(\begin{matrix}4\\18\end{matrix}\right)$$

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.

$$\left(\begin{matrix}p\\q\end{matrix}\right)=\left(\begin{matrix}\frac{4}{4-2\left(-3\right)}&-\frac{2}{4-2\left(-3\right)}\\-\frac{-3}{4-2\left(-3\right)}&\frac{1}{4-2\left(-3\right)}\end{matrix}\right)\left(\begin{matrix}4\\18\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}p\\q\end{matrix}\right)=\left(\begin{matrix}\frac{2}{5}&-\frac{1}{5}\\\frac{3}{10}&\frac{1}{10}\end{matrix}\right)\left(\begin{matrix}4\\18\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}p\\q\end{matrix}\right)=\left(\begin{matrix}\frac{2}{5}\times 4-\frac{1}{5}\times 18\\\frac{3}{10}\times 4+\frac{1}{10}\times 18\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}p\\q\end{matrix}\right)=\left(\begin{matrix}-2\\3\end{matrix}\right)$$

Tangohia ngā huānga poukapa p me q.

$$p=-2,q=3$$

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.

$$p+2q=4,-3p+4q=18$$

Kia ōrite ai a p me -3p, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te -3 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 1.

$$-3p-3\times 2q=-3\times 4,-3p+4q=18$$

Whakarūnātia.

$$-3p-6q=-12,-3p+4q=18$$

Me tango -3p+4q=18 mai i -3p-6q=-12 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-3p+3p-6q-4q=-12-18$$

Tāpiri -3p ki te 3p. Ka whakakore atu ngā kupu -3p me 3p, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-6q-4q=-12-18$$

Tāpiri -6q ki te -4q.

$$-10q=-12-18$$

Tāpiri -12 ki te -18.

$$-10q=-30$$

Whakawehea ngā taha e rua ki te -10.

$$q=3$$

Whakaurua te 3 mō q ki -3p+4q=18. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō p hāngai tonu.

$$-3p+4\times 3=18$$

Whakareatia 4 ki te 3.

$$-3p+12=18$$

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

$$-3p=6$$

Whakawehea ngā taha e rua ki te -3.

$$p=-2$$

Kua oti te pūnaha te whakatau.

$$p=-2,q=3$$