Whakaoti i te {l}{a+3b=2}{2a-3b=8}

$$\left\{ \begin{array} { l } { a + 3 b = 2 } \\ { 2 a - 3 b = 8 } \end{array} \right.$$

Whakaoti mō a, b

$$a = \frac{10}{3} = 3\frac{1}{3} \approx 3.333333333$$

$$b=-\frac{4}{9}\approx -0.444444444$$

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.

$$a+3b=2,2a-3b=8$$

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

$$a+3b=2$$

Me tango 3b mai i ngā taha e rua o te whārite.

$$a=-3b+2$$

Whakakapia te -3b+2 mō te a ki tērā atu whārite, 2a-3b=8.

$$2\left(-3b+2\right)-3b=8$$

Whakareatia 2 ki te -3b+2.

$$-6b+4-3b=8$$

Tāpiri -6b ki te -3b.

$$-9b+4=8$$

Me tango 4 mai i ngā taha e rua o te whārite.

$$-9b=4$$

Whakawehea ngā taha e rua ki te -9.

$$b=-\frac{4}{9}$$

Whakaurua te -\frac{4}{9} mō b ki a=-3b+2. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō a hāngai tonu.

$$a=-3\left(-\frac{4}{9}\right)+2$$

Whakareatia -3 ki te -\frac{4}{9}.

$$a=\frac{4}{3}+2$$

Tāpiri 2 ki te \frac{4}{3}.

$$a=\frac{10}{3}$$

Kua oti te pūnaha te whakatau.

$$a=\frac{10}{3},b=-\frac{4}{9}$$

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.

$$a+3b=2,2a-3b=8$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&3\\2&-3\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}2\\8\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}1&3\\2&-3\end{matrix}\right))\left(\begin{matrix}1&3\\2&-3\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}1&3\\2&-3\end{matrix}\right))\left(\begin{matrix}2\\8\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}1&3\\2&-3\end{matrix}\right))\left(\begin{matrix}2\\8\end{matrix}\right)$$

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

$$\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}1&3\\2&-3\end{matrix}\right))\left(\begin{matrix}2\\8\end{matrix}\right)$$

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.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{-3-3\times 2}&-\frac{3}{-3-3\times 2}\\-\frac{2}{-3-3\times 2}&\frac{1}{-3-3\times 2}\end{matrix}\right)\left(\begin{matrix}2\\8\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}\frac{1}{3}&\frac{1}{3}\\\frac{2}{9}&-\frac{1}{9}\end{matrix}\right)\left(\begin{matrix}2\\8\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}\frac{1}{3}\times 2+\frac{1}{3}\times 8\\\frac{2}{9}\times 2-\frac{1}{9}\times 8\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}\frac{10}{3}\\-\frac{4}{9}\end{matrix}\right)$$

Tangohia ngā huānga poukapa a me b.

$$a=\frac{10}{3},b=-\frac{4}{9}$$

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.

$$a+3b=2,2a-3b=8$$

Kia ōrite ai a a me 2a, 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.

$$2a+2\times 3b=2\times 2,2a-3b=8$$

Whakarūnātia.

$$2a+6b=4,2a-3b=8$$

Me tango 2a-3b=8 mai i 2a+6b=4 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$2a-2a+6b+3b=4-8$$

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.

$$6b+3b=4-8$$

Tāpiri 6b ki te 3b.

$$9b=4-8$$

Tāpiri 4 ki te -8.

$$9b=-4$$

Whakawehea ngā taha e rua ki te 9.

$$b=-\frac{4}{9}$$

Whakaurua te -\frac{4}{9} mō b ki 2a-3b=8. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō a hāngai tonu.

$$2a-3\left(-\frac{4}{9}\right)=8$$

Whakareatia -3 ki te -\frac{4}{9}.

$$2a+\frac{4}{3}=8$$

Me tango \frac{4}{3} mai i ngā taha e rua o te whārite.

$$2a=\frac{20}{3}$$

Whakawehea ngā taha e rua ki te 2.

$$a=\frac{10}{3}$$

Kua oti te pūnaha te whakatau.

$$a=\frac{10}{3},b=-\frac{4}{9}$$