Whakaoti i te {l}{1c+1V=16500}{2c+3V=40500}

$$\left. \begin{array} { l } { 1 c + 1 V = 16500 } \\ { 2 c + 3 V = 40500 } \end{array} \right.$$

Whakaoti mō c, V

$$c=9000$$

$$V=7500$$

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.

$$c+V=16500,2c+3V=40500$$

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

$$c+V=16500$$

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

$$c=-V+16500$$

Whakakapia te -V+16500 mō te c ki tērā atu whārite, 2c+3V=40500.

$$2\left(-V+16500\right)+3V=40500$$

Whakareatia 2 ki te -V+16500.

$$-2V+33000+3V=40500$$

Tāpiri -2V ki te 3V.

$$V+33000=40500$$

Me tango 33000 mai i ngā taha e rua o te whārite.

$$V=7500$$

Whakaurua te 7500 mō V ki c=-V+16500. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō c hāngai tonu.

$$c=-7500+16500$$

Tāpiri 16500 ki te -7500.

$$c=9000$$

Kua oti te pūnaha te whakatau.

$$c=9000,V=7500$$

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.

$$c+V=16500,2c+3V=40500$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&1\\2&3\end{matrix}\right)\left(\begin{matrix}c\\V\end{matrix}\right)=\left(\begin{matrix}16500\\40500\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}1&1\\2&3\end{matrix}\right))\left(\begin{matrix}1&1\\2&3\end{matrix}\right)\left(\begin{matrix}c\\V\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\2&3\end{matrix}\right))\left(\begin{matrix}16500\\40500\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}c\\V\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\2&3\end{matrix}\right))\left(\begin{matrix}16500\\40500\end{matrix}\right)$$

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

$$\left(\begin{matrix}c\\V\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\2&3\end{matrix}\right))\left(\begin{matrix}16500\\40500\end{matrix}\right)$$

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.

$$\left(\begin{matrix}c\\V\end{matrix}\right)=\left(\begin{matrix}\frac{3}{3-2}&-\frac{1}{3-2}\\-\frac{2}{3-2}&\frac{1}{3-2}\end{matrix}\right)\left(\begin{matrix}16500\\40500\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}c\\V\end{matrix}\right)=\left(\begin{matrix}3&-1\\-2&1\end{matrix}\right)\left(\begin{matrix}16500\\40500\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}c\\V\end{matrix}\right)=\left(\begin{matrix}3\times 16500-40500\\-2\times 16500+40500\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}c\\V\end{matrix}\right)=\left(\begin{matrix}9000\\7500\end{matrix}\right)$$

Tangohia ngā huānga poukapa c me V.

$$c=9000,V=7500$$

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.

$$c+V=16500,2c+3V=40500$$

Kia ōrite ai a c me 2c, 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 1.

$$2c+2V=2\times 16500,2c+3V=40500$$

Whakarūnātia.

$$2c+2V=33000,2c+3V=40500$$

Me tango 2c+3V=40500 mai i 2c+2V=33000 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$2c-2c+2V-3V=33000-40500$$

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

$$2V-3V=33000-40500$$

Tāpiri 2V ki te -3V.

$$-V=33000-40500$$

Tāpiri 33000 ki te -40500.

$$-V=-7500$$

Whakawehea ngā taha e rua ki te -1.

$$V=7500$$

Whakaurua te 7500 mō V ki 2c+3V=40500. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō c hāngai tonu.

$$2c+3\times 7500=40500$$

Whakareatia 3 ki te 7500.

$$2c+22500=40500$$

Me tango 22500 mai i ngā taha e rua o te whārite.

$$2c=18000$$

Whakawehea ngā taha e rua ki te 2.

$$c=9000$$

Kua oti te pūnaha te whakatau.

$$c=9000,V=7500$$