Whakaoti i te {l}{-5a=4a+2b-3}{-b/2a=1}

$$\left\{ \begin{array} { l } { - 5 a = 4 a + 2 b - 3 } \\ { - \frac { b } { 2 a } = 1 } \end{array} \right.$$

Whakaoti mō a, b

$$a=\frac{3}{5}=0.6$$

$$b = -\frac{6}{5} = -1\frac{1}{5} = -1.2$$

Ngā Upane e Whakamahi ana i te Whakakapinga

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

$$-5a-4a=2b-3$$

Pahekotia te -5a me -4a, ka -9a.

$$-9a=2b-3$$

Whakawehea ngā taha e rua ki te -9.

$$a=-\frac{1}{9}\left(2b-3\right)$$

Whakareatia -\frac{1}{9} ki te 2b-3.

$$a=-\frac{2}{9}b+\frac{1}{3}$$

Whakakapia te -\frac{2b}{9}+\frac{1}{3} mō te a ki tērā atu whārite, -2a-b=0.

$$-2\left(-\frac{2}{9}b+\frac{1}{3}\right)-b=0$$

Whakareatia -2 ki te -\frac{2b}{9}+\frac{1}{3}.

$$\frac{4}{9}b-\frac{2}{3}-b=0$$

Tāpiri \frac{4b}{9} ki te -b.

$$-\frac{5}{9}b-\frac{2}{3}=0$$

Me tāpiri \frac{2}{3} ki ngā taha e rua o te whārite.

$$-\frac{5}{9}b=\frac{2}{3}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{5}{9}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$b=-\frac{6}{5}$$

Whakaurua te -\frac{6}{5} mō b ki a=-\frac{2}{9}b+\frac{1}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō a hāngai tonu.

$$a=-\frac{2}{9}\left(-\frac{6}{5}\right)+\frac{1}{3}$$

Whakareatia -\frac{2}{9} ki te -\frac{6}{5} 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.

$$a=\frac{4}{15}+\frac{1}{3}$$

Tāpiri \frac{1}{3} ki te \frac{4}{15} 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.

$$a=\frac{3}{5}$$

Kua oti te pūnaha te whakatau.

$$a=\frac{3}{5},b=-\frac{6}{5}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

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

$$-5a-4a=2b-3$$

Pahekotia te -5a me -4a, ka -9a.

$$-9a=2b-3$$

Tangohia te 2b mai i ngā taha e rua.

$$-9a-2b=-3$$

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.

$$-9a-2b=-3,-2a-b=0$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-9&-2\\-2&-1\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}-3\\0\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}-9&-2\\-2&-1\end{matrix}\right).

$$inverse(\left(\begin{matrix}-9&-2\\-2&-1\end{matrix}\right))\left(\begin{matrix}-9&-2\\-2&-1\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}-9&-2\\-2&-1\end{matrix}\right))\left(\begin{matrix}-3\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}-9&-2\\-2&-1\end{matrix}\right))\left(\begin{matrix}-3\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}-9&-2\\-2&-1\end{matrix}\right))\left(\begin{matrix}-3\\0\end{matrix}\right)$$

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.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{-9\left(-1\right)-\left(-2\left(-2\right)\right)}&-\frac{-2}{-9\left(-1\right)-\left(-2\left(-2\right)\right)}\\-\frac{-2}{-9\left(-1\right)-\left(-2\left(-2\right)\right)}&-\frac{9}{-9\left(-1\right)-\left(-2\left(-2\right)\right)}\end{matrix}\right)\left(\begin{matrix}-3\\0\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{5}&\frac{2}{5}\\\frac{2}{5}&-\frac{9}{5}\end{matrix}\right)\left(\begin{matrix}-3\\0\end{matrix}\right)$$

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}\frac{3}{5}\\-\frac{6}{5}\end{matrix}\right)$$

Tangohia ngā huānga poukapa a me b.

$$a=\frac{3}{5},b=-\frac{6}{5}$$

Ngā Upane e Whakamahi i te Tangohanga

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

$$-5a-4a=2b-3$$

Pahekotia te -5a me -4a, ka -9a.

$$-9a=2b-3$$

Tangohia te 2b mai i ngā taha e rua.

$$-9a-2b=-3$$

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.

$$-9a-2b=-3,-2a-b=0$$

Kia ōrite ai a -9a 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 -9.

$$-2\left(-9\right)a-2\left(-2\right)b=-2\left(-3\right),-9\left(-2\right)a-9\left(-1\right)b=0$$

Whakarūnātia.

$$18a+4b=6,18a+9b=0$$

Me tango 18a+9b=0 mai i 18a+4b=6 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$18a-18a+4b-9b=6$$

Tāpiri 18a ki te -18a. Ka whakakore atu ngā kupu 18a me -18a, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$4b-9b=6$$

Tāpiri 4b ki te -9b.

$$-5b=6$$

Whakawehea ngā taha e rua ki te -5.

$$b=-\frac{6}{5}$$

Whakaurua te -\frac{6}{5} 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.

$$-2a-\left(-\frac{6}{5}\right)=0$$

Me tango \frac{6}{5} mai i ngā taha e rua o te whārite.

$$-2a=-\frac{6}{5}$$

Whakawehea ngā taha e rua ki te -2.

$$a=\frac{3}{5}$$

Kua oti te pūnaha te whakatau.

$$a=\frac{3}{5},b=-\frac{6}{5}$$