Whakaoti i te {l}{25c+22T=152000}{11c+12T=75000}

$$\left. \begin{array} { l } { 25 c + 22 T = 152000 } \\ { 11 c + 12 T = 75000 } \end{array} \right.$$

Whakaoti mō c, T

$$c=3000$$

$$T=3500$$

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.

$$25c+22T=152000,11c+12T=75000$$

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

$$25c+22T=152000$$

Me tango 22T mai i ngā taha e rua o te whārite.

$$25c=-22T+152000$$

Whakawehea ngā taha e rua ki te 25.

$$c=\frac{1}{25}\left(-22T+152000\right)$$

Whakareatia \frac{1}{25} ki te -22T+152000.

$$c=-\frac{22}{25}T+6080$$

Whakakapia te -\frac{22T}{25}+6080 mō te c ki tērā atu whārite, 11c+12T=75000.

$$11\left(-\frac{22}{25}T+6080\right)+12T=75000$$

Whakareatia 11 ki te -\frac{22T}{25}+6080.

$$-\frac{242}{25}T+66880+12T=75000$$

Tāpiri -\frac{242T}{25} ki te 12T.

$$\frac{58}{25}T+66880=75000$$

Me tango 66880 mai i ngā taha e rua o te whārite.

$$\frac{58}{25}T=8120$$

Whakawehea ngā taha e rua o te whārite ki te \frac{58}{25}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$T=3500$$

Whakaurua te 3500 mō T ki c=-\frac{22}{25}T+6080. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō c hāngai tonu.

$$c=-\frac{22}{25}\times 3500+6080$$

Whakareatia -\frac{22}{25} ki te 3500.

$$c=-3080+6080$$

Tāpiri 6080 ki te -3080.

$$c=3000$$

Kua oti te pūnaha te whakatau.

$$c=3000,T=3500$$

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.

$$25c+22T=152000,11c+12T=75000$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}25&22\\11&12\end{matrix}\right)\left(\begin{matrix}c\\T\end{matrix}\right)=\left(\begin{matrix}152000\\75000\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}25&22\\11&12\end{matrix}\right).

$$inverse(\left(\begin{matrix}25&22\\11&12\end{matrix}\right))\left(\begin{matrix}25&22\\11&12\end{matrix}\right)\left(\begin{matrix}c\\T\end{matrix}\right)=inverse(\left(\begin{matrix}25&22\\11&12\end{matrix}\right))\left(\begin{matrix}152000\\75000\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}c\\T\end{matrix}\right)=inverse(\left(\begin{matrix}25&22\\11&12\end{matrix}\right))\left(\begin{matrix}152000\\75000\end{matrix}\right)$$

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

$$\left(\begin{matrix}c\\T\end{matrix}\right)=inverse(\left(\begin{matrix}25&22\\11&12\end{matrix}\right))\left(\begin{matrix}152000\\75000\end{matrix}\right)$$

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.

$$\left(\begin{matrix}c\\T\end{matrix}\right)=\left(\begin{matrix}\frac{12}{25\times 12-22\times 11}&-\frac{22}{25\times 12-22\times 11}\\-\frac{11}{25\times 12-22\times 11}&\frac{25}{25\times 12-22\times 11}\end{matrix}\right)\left(\begin{matrix}152000\\75000\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}c\\T\end{matrix}\right)=\left(\begin{matrix}\frac{6}{29}&-\frac{11}{29}\\-\frac{11}{58}&\frac{25}{58}\end{matrix}\right)\left(\begin{matrix}152000\\75000\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}c\\T\end{matrix}\right)=\left(\begin{matrix}\frac{6}{29}\times 152000-\frac{11}{29}\times 75000\\-\frac{11}{58}\times 152000+\frac{25}{58}\times 75000\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}c\\T\end{matrix}\right)=\left(\begin{matrix}3000\\3500\end{matrix}\right)$$

Tangohia ngā huānga poukapa c me T.

$$c=3000,T=3500$$

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.

$$25c+22T=152000,11c+12T=75000$$

Kia ōrite ai a 25c me 11c, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 11 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 25.

$$11\times 25c+11\times 22T=11\times 152000,25\times 11c+25\times 12T=25\times 75000$$

Whakarūnātia.

$$275c+242T=1672000,275c+300T=1875000$$

Me tango 275c+300T=1875000 mai i 275c+242T=1672000 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$275c-275c+242T-300T=1672000-1875000$$

Tāpiri 275c ki te -275c. Ka whakakore atu ngā kupu 275c me -275c, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$242T-300T=1672000-1875000$$

Tāpiri 242T ki te -300T.

$$-58T=1672000-1875000$$

Tāpiri 1672000 ki te -1875000.

$$-58T=-203000$$

Whakawehea ngā taha e rua ki te -58.

$$T=3500$$

Whakaurua te 3500 mō T ki 11c+12T=75000. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō c hāngai tonu.

$$11c+12\times 3500=75000$$

Whakareatia 12 ki te 3500.

$$11c+42000=75000$$

Me tango 42000 mai i ngā taha e rua o te whārite.

$$11c=33000$$

Whakawehea ngā taha e rua ki te 11.

$$c=3000$$

Kua oti te pūnaha te whakatau.

$$c=3000,T=3500$$