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

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

Whakaoti mō a, b

$$a=\frac{2}{13}\approx 0.153846154$$

$$b=\frac{10}{13}\approx 0.769230769$$

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.

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

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

$$3a+2b=2$$

Me tango 2b mai i ngā taha e rua o te whārite.

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

Whakawehea ngā taha e rua ki te 3.

$$a=\frac{1}{3}\left(-2b+2\right)$$

Whakareatia \frac{1}{3} ki te -2b+2.

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

Whakakapia te \frac{-2b+2}{3} mō te a ki tērā atu whārite, -2a+3b=2.

$$-2\left(-\frac{2}{3}b+\frac{2}{3}\right)+3b=2$$

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

$$\frac{4}{3}b-\frac{4}{3}+3b=2$$

Tāpiri \frac{4b}{3} ki te 3b.

$$\frac{13}{3}b-\frac{4}{3}=2$$

Me tāpiri \frac{4}{3} ki ngā taha e rua o te whārite.

$$\frac{13}{3}b=\frac{10}{3}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{13}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$b=\frac{10}{13}$$

Whakaurua te \frac{10}{13} mō b ki a=-\frac{2}{3}b+\frac{2}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō a hāngai tonu.

$$a=-\frac{2}{3}\times \frac{10}{13}+\frac{2}{3}$$

Whakareatia -\frac{2}{3} ki te \frac{10}{13} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$a=-\frac{20}{39}+\frac{2}{3}$$

Tāpiri \frac{2}{3} ki te -\frac{20}{39} 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.

$$a=\frac{2}{13}$$

Kua oti te pūnaha te whakatau.

$$a=\frac{2}{13},b=\frac{10}{13}$$

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.

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}\frac{3}{13}&-\frac{2}{13}\\\frac{2}{13}&\frac{3}{13}\end{matrix}\right)\left(\begin{matrix}2\\2\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}\frac{3}{13}\times 2-\frac{2}{13}\times 2\\\frac{2}{13}\times 2+\frac{3}{13}\times 2\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}\frac{2}{13}\\\frac{10}{13}\end{matrix}\right)$$

Tangohia ngā huānga poukapa a me b.

$$a=\frac{2}{13},b=\frac{10}{13}$$

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.

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

Kia ōrite ai a 3a 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 3.

$$-2\times 3a-2\times 2b=-2\times 2,3\left(-2\right)a+3\times 3b=3\times 2$$

Whakarūnātia.

$$-6a-4b=-4,-6a+9b=6$$

Me tango -6a+9b=6 mai i -6a-4b=-4 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-6a+6a-4b-9b=-4-6$$

Tāpiri -6a ki te 6a. Ka whakakore atu ngā kupu -6a me 6a, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-4b-9b=-4-6$$

Tāpiri -4b ki te -9b.

$$-13b=-4-6$$

Tāpiri -4 ki te -6.

$$-13b=-10$$

Whakawehea ngā taha e rua ki te -13.

$$b=\frac{10}{13}$$

Whakaurua te \frac{10}{13} mō b ki -2a+3b=2. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō a hāngai tonu.

$$-2a+3\times \frac{10}{13}=2$$

Whakareatia 3 ki te \frac{10}{13}.

$$-2a+\frac{30}{13}=2$$

Me tango \frac{30}{13} mai i ngā taha e rua o te whārite.

$$-2a=-\frac{4}{13}$$

Whakawehea ngā taha e rua ki te -2.

$$a=\frac{2}{13}$$

Kua oti te pūnaha te whakatau.

$$a=\frac{2}{13},b=\frac{10}{13}$$