Whakaoti i te {l}{-3a=4a+2b-3}{-b/2a=1}

$$\left\{ \begin{array} { l } { - 3 a = 4 a + 2 b - 3 } \\ { - \frac { b } { 2 a } = 1 } \end{array} \right.$$

Whakaoti mō a, b

$$a=1$$

$$b=-2$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Tangohia te 4a mai i ngā taha e rua.

$$-3a-4a=2b-3$$

Pahekotia te -3a me -4a, ka -7a.

$$-7a=2b-3$$

Whakawehea ngā taha e rua ki te -7.

$$a=-\frac{1}{7}\left(2b-3\right)$$

Whakareatia -\frac{1}{7} ki te 2b-3.

$$a=-\frac{2}{7}b+\frac{3}{7}$$

Whakakapia te \frac{-2b+3}{7} mō te a ki tērā atu whārite, -2a-b=0.

$$-2\left(-\frac{2}{7}b+\frac{3}{7}\right)-b=0$$

Whakareatia -2 ki te \frac{-2b+3}{7}.

$$\frac{4}{7}b-\frac{6}{7}-b=0$$

Tāpiri \frac{4b}{7} ki te -b.

$$-\frac{3}{7}b-\frac{6}{7}=0$$

Me tāpiri \frac{6}{7} ki ngā taha e rua o te whārite.

$$-\frac{3}{7}b=\frac{6}{7}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{3}{7}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$b=-2$$

Whakaurua te -2 mō b ki a=-\frac{2}{7}b+\frac{3}{7}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō a hāngai tonu.

$$a=-\frac{2}{7}\left(-2\right)+\frac{3}{7}$$

Whakareatia -\frac{2}{7} ki te -2.

$$a=\frac{4+3}{7}$$

Tāpiri \frac{3}{7} ki te \frac{4}{7} 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.

$$a=1$$

Kua oti te pūnaha te whakatau.

$$a=1,b=-2$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Tangohia te 4a mai i ngā taha e rua.

$$-3a-4a=2b-3$$

Pahekotia te -3a me -4a, ka -7a.

$$-7a=2b-3$$

Tangohia te 2b mai i ngā taha e rua.

$$-7a-2b=-3$$

Whakaarohia te whārite tuarua. Tē taea kia ōrite te tāupe a ki 0 nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te 2a.

$$-b=2a$$

Tangohia te 2a mai i ngā taha e rua.

$$-b-2a=0$$

Tuhia ngā whārite ki te tānga ngahuru ka whakamahi i ngā poukapa hei whakaoti i te pūnaha o ngā whārite.

$$-7a-2b=-3,-2a-b=0$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-7&-2\\-2&-1\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}-3\\0\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}-7&-2\\-2&-1\end{matrix}\right).

$$inverse(\left(\begin{matrix}-7&-2\\-2&-1\end{matrix}\right))\left(\begin{matrix}-7&-2\\-2&-1\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}-7&-2\\-2&-1\end{matrix}\right))\left(\begin{matrix}-3\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}-7&-2\\-2&-1\end{matrix}\right))\left(\begin{matrix}-3\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}-7&-2\\-2&-1\end{matrix}\right))\left(\begin{matrix}-3\\0\end{matrix}\right)$$

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.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{-7\left(-1\right)-\left(-2\left(-2\right)\right)}&-\frac{-2}{-7\left(-1\right)-\left(-2\left(-2\right)\right)}\\-\frac{-2}{-7\left(-1\right)-\left(-2\left(-2\right)\right)}&-\frac{7}{-7\left(-1\right)-\left(-2\left(-2\right)\right)}\end{matrix}\right)\left(\begin{matrix}-3\\0\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{3}&\frac{2}{3}\\\frac{2}{3}&-\frac{7}{3}\end{matrix}\right)\left(\begin{matrix}-3\\0\end{matrix}\right)$$

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa a me b.

$$a=1,b=-2$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Tangohia te 4a mai i ngā taha e rua.

$$-3a-4a=2b-3$$

Pahekotia te -3a me -4a, ka -7a.

$$-7a=2b-3$$

Tangohia te 2b mai i ngā taha e rua.

$$-7a-2b=-3$$

Whakaarohia te whārite tuarua. Tē taea kia ōrite te tāupe a ki 0 nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te 2a.

$$-b=2a$$

Tangohia te 2a mai i ngā taha e rua.

$$-b-2a=0$$

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.

$$-7a-2b=-3,-2a-b=0$$

Kia ōrite ai a -7a 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 -7.

$$-2\left(-7\right)a-2\left(-2\right)b=-2\left(-3\right),-7\left(-2\right)a-7\left(-1\right)b=0$$

Whakarūnātia.

$$14a+4b=6,14a+7b=0$$

Me tango 14a+7b=0 mai i 14a+4b=6 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$14a-14a+4b-7b=6$$

Tāpiri 14a ki te -14a. Ka whakakore atu ngā kupu 14a me -14a, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$4b-7b=6$$

Tāpiri 4b ki te -7b.

$$-3b=6$$

Whakawehea ngā taha e rua ki te -3.

$$b=-2$$

Whakaurua te -2 mō b ki -2a-b=0. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō a hāngai tonu.

$$-2a-\left(-2\right)=0$$

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

$$-2a=-2$$

Whakawehea ngā taha e rua ki te -2.

$$a=1$$

Kua oti te pūnaha te whakatau.

$$a=1,b=-2$$