Whakaoti i te {r}{5p-q=7}{-2p+3q=5}

$$\left\{ \begin{array} { r } { 5 p - q = 7 } \\ { - 2 p + 3 q = 5 } \end{array} \right.$$

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.

$$5p-q=7,-2p+3q=5$$

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

$$5p-q=7$$

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

$$5p=q+7$$

Whakawehea ngā taha e rua ki te 5.

$$p=\frac{1}{5}\left(q+7\right)$$

Whakareatia \frac{1}{5} ki te q+7.

$$p=\frac{1}{5}q+\frac{7}{5}$$

Whakakapia te \frac{7+q}{5} mō te p ki tērā atu whārite, -2p+3q=5.

$$-2\left(\frac{1}{5}q+\frac{7}{5}\right)+3q=5$$

Whakareatia -2 ki te \frac{7+q}{5}.

$$-\frac{2}{5}q-\frac{14}{5}+3q=5$$

Tāpiri -\frac{2q}{5} ki te 3q.

$$\frac{13}{5}q-\frac{14}{5}=5$$

Me tāpiri \frac{14}{5} ki ngā taha e rua o te whārite.

$$\frac{13}{5}q=\frac{39}{5}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{13}{5}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$q=3$$

Whakaurua te 3 mō q ki p=\frac{1}{5}q+\frac{7}{5}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō p hāngai tonu.

$$p=\frac{1}{5}\times 3+\frac{7}{5}$$

Whakareatia \frac{1}{5} ki te 3.

$$p=\frac{3+7}{5}$$

Tāpiri \frac{7}{5} ki te \frac{3}{5} mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$p=2$$

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.

$$5p-q=7,-2p+3q=5$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}5&-1\\-2&3\end{matrix}\right)\left(\begin{matrix}p\\q\end{matrix}\right)=\left(\begin{matrix}7\\5\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}5&-1\\-2&3\end{matrix}\right).

$$inverse(\left(\begin{matrix}5&-1\\-2&3\end{matrix}\right))\left(\begin{matrix}5&-1\\-2&3\end{matrix}\right)\left(\begin{matrix}p\\q\end{matrix}\right)=inverse(\left(\begin{matrix}5&-1\\-2&3\end{matrix}\right))\left(\begin{matrix}7\\5\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}p\\q\end{matrix}\right)=inverse(\left(\begin{matrix}5&-1\\-2&3\end{matrix}\right))\left(\begin{matrix}7\\5\end{matrix}\right)$$

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

$$\left(\begin{matrix}p\\q\end{matrix}\right)=inverse(\left(\begin{matrix}5&-1\\-2&3\end{matrix}\right))\left(\begin{matrix}7\\5\end{matrix}\right)$$

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.

$$\left(\begin{matrix}p\\q\end{matrix}\right)=\left(\begin{matrix}\frac{3}{5\times 3-\left(-\left(-2\right)\right)}&-\frac{-1}{5\times 3-\left(-\left(-2\right)\right)}\\-\frac{-2}{5\times 3-\left(-\left(-2\right)\right)}&\frac{5}{5\times 3-\left(-\left(-2\right)\right)}\end{matrix}\right)\left(\begin{matrix}7\\5\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}p\\q\end{matrix}\right)=\left(\begin{matrix}\frac{3}{13}&\frac{1}{13}\\\frac{2}{13}&\frac{5}{13}\end{matrix}\right)\left(\begin{matrix}7\\5\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}p\\q\end{matrix}\right)=\left(\begin{matrix}\frac{3}{13}\times 7+\frac{1}{13}\times 5\\\frac{2}{13}\times 7+\frac{5}{13}\times 5\end{matrix}\right)$$

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.

$$5p-q=7,-2p+3q=5$$

Kia ōrite ai a 5p me -2p, 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 5.

$$-2\times 5p-2\left(-1\right)q=-2\times 7,5\left(-2\right)p+5\times 3q=5\times 5$$

Whakarūnātia.

$$-10p+2q=-14,-10p+15q=25$$

Me tango -10p+15q=25 mai i -10p+2q=-14 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-10p+10p+2q-15q=-14-25$$

Tāpiri -10p ki te 10p. Ka whakakore atu ngā kupu -10p me 10p, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$2q-15q=-14-25$$

Tāpiri 2q ki te -15q.

$$-13q=-14-25$$

Tāpiri -14 ki te -25.

$$-13q=-39$$

Whakawehea ngā taha e rua ki te -13.

$$q=3$$

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

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

Whakareatia 3 ki te 3.

$$-2p+9=5$$

Me tango 9 mai i ngā taha e rua o te whārite.

$$-2p=-4$$

Whakawehea ngā taha e rua ki te -2.

$$p=2$$

Kua oti te pūnaha te whakatau.

$$p=2,q=3$$