Whakaoti i te {l}{2a+b=5}{a+2b=1}

$$\left. \begin{array} { l } { 2 a + b = 5 } \\ { a + 2 b = 1 } \end{array} \right.$$

Whakaoti mō a, b

$$a=3$$

$$b=-1$$

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.

$$2a+b=5,a+2b=1$$

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

$$2a+b=5$$

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

$$2a=-b+5$$

Whakawehea ngā taha e rua ki te 2.

$$a=\frac{1}{2}\left(-b+5\right)$$

Whakareatia \frac{1}{2} ki te -b+5.

$$a=-\frac{1}{2}b+\frac{5}{2}$$

Whakakapia te \frac{-b+5}{2} mō te a ki tērā atu whārite, a+2b=1.

$$-\frac{1}{2}b+\frac{5}{2}+2b=1$$

Tāpiri -\frac{b}{2} ki te 2b.

$$\frac{3}{2}b+\frac{5}{2}=1$$

Me tango \frac{5}{2} mai i ngā taha e rua o te whārite.

$$\frac{3}{2}b=-\frac{3}{2}$$

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

$$b=-1$$

Whakaurua te -1 mō b ki a=-\frac{1}{2}b+\frac{5}{2}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō a hāngai tonu.

$$a=-\frac{1}{2}\left(-1\right)+\frac{5}{2}$$

Whakareatia -\frac{1}{2} ki te -1.

$$a=\frac{1+5}{2}$$

Tāpiri \frac{5}{2} ki te \frac{1}{2} 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.

$$a=3$$

Kua oti te pūnaha te whakatau.

$$a=3,b=-1$$

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.

$$2a+b=5,a+2b=1$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&1\\1&2\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}5\\1\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}2&1\\1&2\end{matrix}\right))\left(\begin{matrix}2&1\\1&2\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}2&1\\1&2\end{matrix}\right))\left(\begin{matrix}5\\1\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}2&1\\1&2\end{matrix}\right))\left(\begin{matrix}5\\1\end{matrix}\right)$$

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

$$\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}2&1\\1&2\end{matrix}\right))\left(\begin{matrix}5\\1\end{matrix}\right)$$

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.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}\frac{2}{2\times 2-1}&-\frac{1}{2\times 2-1}\\-\frac{1}{2\times 2-1}&\frac{2}{2\times 2-1}\end{matrix}\right)\left(\begin{matrix}5\\1\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}\frac{2}{3}&-\frac{1}{3}\\-\frac{1}{3}&\frac{2}{3}\end{matrix}\right)\left(\begin{matrix}5\\1\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}\frac{2}{3}\times 5-\frac{1}{3}\\-\frac{1}{3}\times 5+\frac{2}{3}\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}3\\-1\end{matrix}\right)$$

Tangohia ngā huānga poukapa a me b.

$$a=3,b=-1$$

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.

$$2a+b=5,a+2b=1$$

Kia ōrite ai a 2a 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 2.

$$2a+b=5,2a+2\times 2b=2$$

Whakarūnātia.

$$2a+b=5,2a+4b=2$$

Me tango 2a+4b=2 mai i 2a+b=5 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$2a-2a+b-4b=5-2$$

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

$$b-4b=5-2$$

Tāpiri b ki te -4b.

$$-3b=5-2$$

Tāpiri 5 ki te -2.

$$-3b=3$$

Whakawehea ngā taha e rua ki te -3.

$$b=-1$$

Whakaurua te -1 mō b ki a+2b=1. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō a hāngai tonu.

$$a+2\left(-1\right)=1$$

Whakareatia 2 ki te -1.

$$a-2=1$$

Me tāpiri 2 ki ngā taha e rua o te whārite.

$$a=3$$

Kua oti te pūnaha te whakatau.

$$a=3,b=-1$$