Whakaoti i te {l}{2/3A+B=400}{A+4/5B=460}

$$\left. \begin{array} { l } { \frac { 2 } { 3 } A + B = 400 } \\ { A + \frac { 4 } { 5 } B = 460 } \end{array} \right.$$

Whakaoti mō A, B

$$A=300$$

$$B=200$$

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.

$$\frac{2}{3}A+B=400,A+\frac{4}{5}B=460$$

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

$$\frac{2}{3}A+B=400$$

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

$$\frac{2}{3}A=-B+400$$

Whakawehea ngā taha e rua o te whārite ki te \frac{2}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$A=\frac{3}{2}\left(-B+400\right)$$

Whakareatia \frac{3}{2} ki te -B+400.

$$A=-\frac{3}{2}B+600$$

Whakakapia te -\frac{3B}{2}+600 mō te A ki tērā atu whārite, A+\frac{4}{5}B=460.

$$-\frac{3}{2}B+600+\frac{4}{5}B=460$$

Tāpiri -\frac{3B}{2} ki te \frac{4B}{5}.

$$-\frac{7}{10}B+600=460$$

Me tango 600 mai i ngā taha e rua o te whārite.

$$-\frac{7}{10}B=-140$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{7}{10}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$B=200$$

Whakaurua te 200 mō B ki A=-\frac{3}{2}B+600. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō A hāngai tonu.

$$A=-\frac{3}{2}\times 200+600$$

Whakareatia -\frac{3}{2} ki te 200.

$$A=-300+600$$

Tāpiri 600 ki te -300.

$$A=300$$

Kua oti te pūnaha te whakatau.

$$A=300,B=200$$

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.

$$\frac{2}{3}A+B=400,A+\frac{4}{5}B=460$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}\frac{2}{3}&1\\1&\frac{4}{5}\end{matrix}\right)\left(\begin{matrix}A\\B\end{matrix}\right)=\left(\begin{matrix}400\\460\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}\frac{2}{3}&1\\1&\frac{4}{5}\end{matrix}\right).

$$inverse(\left(\begin{matrix}\frac{2}{3}&1\\1&\frac{4}{5}\end{matrix}\right))\left(\begin{matrix}\frac{2}{3}&1\\1&\frac{4}{5}\end{matrix}\right)\left(\begin{matrix}A\\B\end{matrix}\right)=inverse(\left(\begin{matrix}\frac{2}{3}&1\\1&\frac{4}{5}\end{matrix}\right))\left(\begin{matrix}400\\460\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}A\\B\end{matrix}\right)=inverse(\left(\begin{matrix}\frac{2}{3}&1\\1&\frac{4}{5}\end{matrix}\right))\left(\begin{matrix}400\\460\end{matrix}\right)$$

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

$$\left(\begin{matrix}A\\B\end{matrix}\right)=inverse(\left(\begin{matrix}\frac{2}{3}&1\\1&\frac{4}{5}\end{matrix}\right))\left(\begin{matrix}400\\460\end{matrix}\right)$$

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.

$$\left(\begin{matrix}A\\B\end{matrix}\right)=\left(\begin{matrix}\frac{\frac{4}{5}}{\frac{2}{3}\times \frac{4}{5}-1}&-\frac{1}{\frac{2}{3}\times \frac{4}{5}-1}\\-\frac{1}{\frac{2}{3}\times \frac{4}{5}-1}&\frac{\frac{2}{3}}{\frac{2}{3}\times \frac{4}{5}-1}\end{matrix}\right)\left(\begin{matrix}400\\460\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}A\\B\end{matrix}\right)=\left(\begin{matrix}-\frac{12}{7}&\frac{15}{7}\\\frac{15}{7}&-\frac{10}{7}\end{matrix}\right)\left(\begin{matrix}400\\460\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}A\\B\end{matrix}\right)=\left(\begin{matrix}-\frac{12}{7}\times 400+\frac{15}{7}\times 460\\\frac{15}{7}\times 400-\frac{10}{7}\times 460\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}A\\B\end{matrix}\right)=\left(\begin{matrix}300\\200\end{matrix}\right)$$

Tangohia ngā huānga poukapa A me B.

$$A=300,B=200$$

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.

$$\frac{2}{3}A+B=400,A+\frac{4}{5}B=460$$

Kia ōrite ai a \frac{2A}{3} me A, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 1 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te \frac{2}{3}.

$$\frac{2}{3}A+B=400,\frac{2}{3}A+\frac{2}{3}\times \frac{4}{5}B=\frac{2}{3}\times 460$$

Whakarūnātia.

$$\frac{2}{3}A+B=400,\frac{2}{3}A+\frac{8}{15}B=\frac{920}{3}$$

Me tango \frac{2}{3}A+\frac{8}{15}B=\frac{920}{3} mai i \frac{2}{3}A+B=400 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$\frac{2}{3}A-\frac{2}{3}A+B-\frac{8}{15}B=400-\frac{920}{3}$$

Tāpiri \frac{2A}{3} ki te -\frac{2A}{3}. Ka whakakore atu ngā kupu \frac{2A}{3} me -\frac{2A}{3}, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$B-\frac{8}{15}B=400-\frac{920}{3}$$

Tāpiri B ki te -\frac{8B}{15}.

$$\frac{7}{15}B=400-\frac{920}{3}$$

Tāpiri 400 ki te -\frac{920}{3}.

$$\frac{7}{15}B=\frac{280}{3}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{7}{15}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$B=200$$

Whakaurua te 200 mō B ki A+\frac{4}{5}B=460. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō A hāngai tonu.

$$A+\frac{4}{5}\times 200=460$$

Whakareatia \frac{4}{5} ki te 200.

$$A+160=460$$

Me tango 160 mai i ngā taha e rua o te whārite.

$$A=300$$

Kua oti te pūnaha te whakatau.

$$A=300,B=200$$