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

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

Whakaoti mō a, b

$$a=4$$

$$b=5$$

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.

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

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

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

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

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

Whakawehea ngā taha e rua ki te -2.

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

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

$$a=b-1$$

Whakakapia te b-1 mō te a ki tērā atu whārite, 3a-2b=2.

$$3\left(b-1\right)-2b=2$$

Whakareatia 3 ki te b-1.

$$3b-3-2b=2$$

Tāpiri 3b ki te -2b.

$$b-3=2$$

Me tāpiri 3 ki ngā taha e rua o te whārite.

$$b=5$$

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

$$a=5-1$$

Tāpiri -1 ki te 5.

$$a=4$$

Kua oti te pūnaha te whakatau.

$$a=4,b=5$$

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.

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}4\\5\end{matrix}\right)$$

Tangohia ngā huānga poukapa a me b.

$$a=4,b=5$$

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.

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

Kia ōrite ai a -2a me 3a, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 3 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te -2.

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

Whakarūnātia.

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

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

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

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.

$$6b-4b=6+4$$

Tāpiri 6b ki te -4b.

$$2b=6+4$$

Tāpiri 6 ki te 4.

$$2b=10$$

Whakawehea ngā taha e rua ki te 2.

$$b=5$$

Whakaurua te 5 mō b ki 3a-2b=2. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō a hāngai tonu.

$$3a-2\times 5=2$$

Whakareatia -2 ki te 5.

$$3a-10=2$$

Me tāpiri 10 ki ngā taha e rua o te whārite.

$$3a=12$$

Whakawehea ngā taha e rua ki te 3.

$$a=4$$

Kua oti te pūnaha te whakatau.

$$a=4,b=5$$