Whakaoti i te {l}{5p+6q=59}{9p+6q=87}

$$\left. \begin{array} { l } { 5 p + 6 q = 59 } \\ { 9 p + 6 q = 87 } \end{array} \right.$$

Whakaoti mō p, q

$$p=7$$

$$q=4$$

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.

$$5p+6q=59,9p+6q=87$$

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

$$5p+6q=59$$

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

$$5p=-6q+59$$

Whakawehea ngā taha e rua ki te 5.

$$p=\frac{1}{5}\left(-6q+59\right)$$

Whakareatia \frac{1}{5} ki te -6q+59.

$$p=-\frac{6}{5}q+\frac{59}{5}$$

Whakakapia te \frac{-6q+59}{5} mō te p ki tērā atu whārite, 9p+6q=87.

$$9\left(-\frac{6}{5}q+\frac{59}{5}\right)+6q=87$$

Whakareatia 9 ki te \frac{-6q+59}{5}.

$$-\frac{54}{5}q+\frac{531}{5}+6q=87$$

Tāpiri -\frac{54q}{5} ki te 6q.

$$-\frac{24}{5}q+\frac{531}{5}=87$$

Me tango \frac{531}{5} mai i ngā taha e rua o te whārite.

$$-\frac{24}{5}q=-\frac{96}{5}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{24}{5}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$q=4$$

Whakaurua te 4 mō q ki p=-\frac{6}{5}q+\frac{59}{5}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō p hāngai tonu.

$$p=-\frac{6}{5}\times 4+\frac{59}{5}$$

Whakareatia -\frac{6}{5} ki te 4.

$$p=\frac{-24+59}{5}$$

Tāpiri \frac{59}{5} ki te -\frac{24}{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.

$$p=7$$

Kua oti te pūnaha te whakatau.

$$p=7,q=4$$

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.

$$5p+6q=59,9p+6q=87$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}5&6\\9&6\end{matrix}\right)\left(\begin{matrix}p\\q\end{matrix}\right)=\left(\begin{matrix}59\\87\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}5&6\\9&6\end{matrix}\right).

$$inverse(\left(\begin{matrix}5&6\\9&6\end{matrix}\right))\left(\begin{matrix}5&6\\9&6\end{matrix}\right)\left(\begin{matrix}p\\q\end{matrix}\right)=inverse(\left(\begin{matrix}5&6\\9&6\end{matrix}\right))\left(\begin{matrix}59\\87\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}p\\q\end{matrix}\right)=inverse(\left(\begin{matrix}5&6\\9&6\end{matrix}\right))\left(\begin{matrix}59\\87\end{matrix}\right)$$

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

$$\left(\begin{matrix}p\\q\end{matrix}\right)=inverse(\left(\begin{matrix}5&6\\9&6\end{matrix}\right))\left(\begin{matrix}59\\87\end{matrix}\right)$$

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.

$$\left(\begin{matrix}p\\q\end{matrix}\right)=\left(\begin{matrix}\frac{6}{5\times 6-6\times 9}&-\frac{6}{5\times 6-6\times 9}\\-\frac{9}{5\times 6-6\times 9}&\frac{5}{5\times 6-6\times 9}\end{matrix}\right)\left(\begin{matrix}59\\87\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}p\\q\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{4}&\frac{1}{4}\\\frac{3}{8}&-\frac{5}{24}\end{matrix}\right)\left(\begin{matrix}59\\87\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}p\\q\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{4}\times 59+\frac{1}{4}\times 87\\\frac{3}{8}\times 59-\frac{5}{24}\times 87\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}p\\q\end{matrix}\right)=\left(\begin{matrix}7\\4\end{matrix}\right)$$

Tangohia ngā huānga poukapa p me q.

$$p=7,q=4$$

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.

$$5p+6q=59,9p+6q=87$$

Me tango 9p+6q=87 mai i 5p+6q=59 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$5p-9p+6q-6q=59-87$$

Tāpiri 6q ki te -6q. Ka whakakore atu ngā kupu 6q me -6q, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$5p-9p=59-87$$

Tāpiri 5p ki te -9p.

$$-4p=59-87$$

Tāpiri 59 ki te -87.

$$-4p=-28$$

Whakawehea ngā taha e rua ki te -4.

$$p=7$$

Whakaurua te 7 mō p ki 9p+6q=87. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō q hāngai tonu.

$$9\times 7+6q=87$$

Whakareatia 9 ki te 7.

$$63+6q=87$$

Me tango 63 mai i ngā taha e rua o te whārite.

$$6q=24$$

Whakawehea ngā taha e rua ki te 6.

$$q=4$$

Kua oti te pūnaha te whakatau.

$$p=7,q=4$$