Whakaoti i te {l}{12=a+b}{2=6a+b}

$$\left\{ \begin{array} { l } { 12 = a + b } \\ { 2 = 6 a + b } \end{array} \right.$$

Whakaoti mō a, b

$$a=-2$$

$$b=14$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$a+b=12$$

Whakaarohia te whārite tuarua. Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$6a+b=2$$

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.

$$a+b=12,6a+b=2$$

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

$$a+b=12$$

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

$$a=-b+12$$

Whakakapia te -b+12 mō te a ki tērā atu whārite, 6a+b=2.

$$6\left(-b+12\right)+b=2$$

Whakareatia 6 ki te -b+12.

$$-6b+72+b=2$$

Tāpiri -6b ki te b.

$$-5b+72=2$$

Me tango 72 mai i ngā taha e rua o te whārite.

$$-5b=-70$$

Whakawehea ngā taha e rua ki te -5.

$$b=14$$

Whakaurua te 14 mō b ki a=-b+12. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō a hāngai tonu.

$$a=-14+12$$

Tāpiri 12 ki te -14.

$$a=-2$$

Kua oti te pūnaha te whakatau.

$$a=-2,b=14$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$a+b=12$$

Whakaarohia te whārite tuarua. Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$6a+b=2$$

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

$$a+b=12,6a+b=2$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&1\\6&1\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}12\\2\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&1\\6&1\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&1\\6&1\end{matrix}\right))\left(\begin{matrix}1&1\\6&1\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\6&1\end{matrix}\right))\left(\begin{matrix}12\\2\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\6&1\end{matrix}\right))\left(\begin{matrix}12\\2\end{matrix}\right)$$

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

$$\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\6&1\end{matrix}\right))\left(\begin{matrix}12\\2\end{matrix}\right)$$

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.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}\frac{1}{1-6}&-\frac{1}{1-6}\\-\frac{6}{1-6}&\frac{1}{1-6}\end{matrix}\right)\left(\begin{matrix}12\\2\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{5}&\frac{1}{5}\\\frac{6}{5}&-\frac{1}{5}\end{matrix}\right)\left(\begin{matrix}12\\2\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{5}\times 12+\frac{1}{5}\times 2\\\frac{6}{5}\times 12-\frac{1}{5}\times 2\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}-2\\14\end{matrix}\right)$$

Tangohia ngā huānga poukapa a me b.

$$a=-2,b=14$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$a+b=12$$

Whakaarohia te whārite tuarua. Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$6a+b=2$$

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.

$$a+b=12,6a+b=2$$

Me tango 6a+b=2 mai i a+b=12 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$a-6a+b-b=12-2$$

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

$$a-6a=12-2$$

Tāpiri a ki te -6a.

$$-5a=12-2$$

Tāpiri 12 ki te -2.

$$-5a=10$$

Whakawehea ngā taha e rua ki te -5.

$$a=-2$$

Whakaurua te -2 mō a ki 6a+b=2. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō b hāngai tonu.

$$6\left(-2\right)+b=2$$

Whakareatia 6 ki te -2.

$$-12+b=2$$

Me tāpiri 12 ki ngā taha e rua o te whārite.

$$b=14$$

Kua oti te pūnaha te whakatau.

$$a=-2,b=14$$