Whakaoti i te {l}{50a+40b=40050}{60a+52b=5000}

$$\left\{ \begin{array} { l } { 50 a + 40 b = 40050 } \\ { 60 a + 52 b = 5000 } \end{array} \right.$$

Whakaoti mō a, b

$$a=9413$$

$$b=-10765$$

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.

$$50a+40b=40050,60a+52b=5000$$

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

$$50a+40b=40050$$

Me tango 40b mai i ngā taha e rua o te whārite.

$$50a=-40b+40050$$

Whakawehea ngā taha e rua ki te 50.

$$a=\frac{1}{50}\left(-40b+40050\right)$$

Whakareatia \frac{1}{50} ki te -40b+40050.

$$a=-\frac{4}{5}b+801$$

Whakakapia te -\frac{4b}{5}+801 mō te a ki tērā atu whārite, 60a+52b=5000.

$$60\left(-\frac{4}{5}b+801\right)+52b=5000$$

Whakareatia 60 ki te -\frac{4b}{5}+801.

$$-48b+48060+52b=5000$$

Tāpiri -48b ki te 52b.

$$4b+48060=5000$$

Me tango 48060 mai i ngā taha e rua o te whārite.

$$4b=-43060$$

Whakawehea ngā taha e rua ki te 4.

$$b=-10765$$

Whakaurua te -10765 mō b ki a=-\frac{4}{5}b+801. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō a hāngai tonu.

$$a=-\frac{4}{5}\left(-10765\right)+801$$

Whakareatia -\frac{4}{5} ki te -10765.

$$a=8612+801$$

Tāpiri 801 ki te 8612.

$$a=9413$$

Kua oti te pūnaha te whakatau.

$$a=9413,b=-10765$$

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.

$$50a+40b=40050,60a+52b=5000$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}50&40\\60&52\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}40050\\5000\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}50&40\\60&52\end{matrix}\right).

$$inverse(\left(\begin{matrix}50&40\\60&52\end{matrix}\right))\left(\begin{matrix}50&40\\60&52\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}50&40\\60&52\end{matrix}\right))\left(\begin{matrix}40050\\5000\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}50&40\\60&52\end{matrix}\right))\left(\begin{matrix}40050\\5000\end{matrix}\right)$$

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

$$\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}50&40\\60&52\end{matrix}\right))\left(\begin{matrix}40050\\5000\end{matrix}\right)$$

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.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}\frac{52}{50\times 52-40\times 60}&-\frac{40}{50\times 52-40\times 60}\\-\frac{60}{50\times 52-40\times 60}&\frac{50}{50\times 52-40\times 60}\end{matrix}\right)\left(\begin{matrix}40050\\5000\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}\frac{13}{50}&-\frac{1}{5}\\-\frac{3}{10}&\frac{1}{4}\end{matrix}\right)\left(\begin{matrix}40050\\5000\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}\frac{13}{50}\times 40050-\frac{1}{5}\times 5000\\-\frac{3}{10}\times 40050+\frac{1}{4}\times 5000\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}9413\\-10765\end{matrix}\right)$$

Tangohia ngā huānga poukapa a me b.

$$a=9413,b=-10765$$

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.

$$50a+40b=40050,60a+52b=5000$$

Kia ōrite ai a 50a me 60a, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 60 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 50.

$$60\times 50a+60\times 40b=60\times 40050,50\times 60a+50\times 52b=50\times 5000$$

Whakarūnātia.

$$3000a+2400b=2403000,3000a+2600b=250000$$

Me tango 3000a+2600b=250000 mai i 3000a+2400b=2403000 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$3000a-3000a+2400b-2600b=2403000-250000$$

Tāpiri 3000a ki te -3000a. Ka whakakore atu ngā kupu 3000a me -3000a, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$2400b-2600b=2403000-250000$$

Tāpiri 2400b ki te -2600b.

$$-200b=2403000-250000$$

Tāpiri 2403000 ki te -250000.

$$-200b=2153000$$

Whakawehea ngā taha e rua ki te -200.

$$b=-10765$$

Whakaurua te -10765 mō b ki 60a+52b=5000. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō a hāngai tonu.

$$60a+52\left(-10765\right)=5000$$

Whakareatia 52 ki te -10765.

$$60a-559780=5000$$

Me tāpiri 559780 ki ngā taha e rua o te whārite.

$$60a=564780$$

Whakawehea ngā taha e rua ki te 60.

$$a=9413$$

Kua oti te pūnaha te whakatau.

$$a=9413,b=-10765$$