Whakaoti i te {c}{1/2a+b=-2}{a-2b=8}

$$\left. \begin{array} { c } { \frac { 1 } { 2 } a + b = - 2 } \\ { a - 2 b = 8 } \end{array} \right.$$

Whakaoti mō a, b

$$a=2$$

$$b=-3$$

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.

$$\frac{1}{2}a+b=-2,a-2b=8$$

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

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

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

$$\frac{1}{2}a=-b-2$$

Me whakarea ngā taha e rua ki te 2.

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

Whakareatia 2 ki te -b-2.

$$a=-2b-4$$

Whakakapia te -2b-4 mō te a ki tērā atu whārite, a-2b=8.

$$-2b-4-2b=8$$

Tāpiri -2b ki te -2b.

$$-4b-4=8$$

Me tāpiri 4 ki ngā taha e rua o te whārite.

$$-4b=12$$

Whakawehea ngā taha e rua ki te -4.

$$b=-3$$

Whakaurua te -3 mō b ki a=-2b-4. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō a hāngai tonu.

$$a=-2\left(-3\right)-4$$

Whakareatia -2 ki te -3.

$$a=6-4$$

Tāpiri -4 ki te 6.

$$a=2$$

Kua oti te pūnaha te whakatau.

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

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.

$$\frac{1}{2}a+b=-2,a-2b=8$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}\frac{1}{2}&1\\1&-2\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}-2\\8\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}\frac{1}{2}&1\\1&-2\end{matrix}\right))\left(\begin{matrix}\frac{1}{2}&1\\1&-2\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}\frac{1}{2}&1\\1&-2\end{matrix}\right))\left(\begin{matrix}-2\\8\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}\frac{1}{2}&1\\1&-2\end{matrix}\right))\left(\begin{matrix}-2\\8\end{matrix}\right)$$

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

$$\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}\frac{1}{2}&1\\1&-2\end{matrix}\right))\left(\begin{matrix}-2\\8\end{matrix}\right)$$

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.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{\frac{1}{2}\left(-2\right)-1}&-\frac{1}{\frac{1}{2}\left(-2\right)-1}\\-\frac{1}{\frac{1}{2}\left(-2\right)-1}&\frac{\frac{1}{2}}{\frac{1}{2}\left(-2\right)-1}\end{matrix}\right)\left(\begin{matrix}-2\\8\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}1&\frac{1}{2}\\\frac{1}{2}&-\frac{1}{4}\end{matrix}\right)\left(\begin{matrix}-2\\8\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}-2+\frac{1}{2}\times 8\\\frac{1}{2}\left(-2\right)-\frac{1}{4}\times 8\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa a me b.

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

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.

$$\frac{1}{2}a+b=-2,a-2b=8$$

Kia ōrite ai a \frac{a}{2} me a, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 1 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te \frac{1}{2}.

$$\frac{1}{2}a+b=-2,\frac{1}{2}a+\frac{1}{2}\left(-2\right)b=\frac{1}{2}\times 8$$

Whakarūnātia.

$$\frac{1}{2}a+b=-2,\frac{1}{2}a-b=4$$

Me tango \frac{1}{2}a-b=4 mai i \frac{1}{2}a+b=-2 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$\frac{1}{2}a-\frac{1}{2}a+b+b=-2-4$$

Tāpiri \frac{a}{2} ki te -\frac{a}{2}. Ka whakakore atu ngā kupu \frac{a}{2} me -\frac{a}{2}, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$b+b=-2-4$$

Tāpiri b ki te b.

$$2b=-2-4$$

Tāpiri -2 ki te -4.

$$2b=-6$$

Whakawehea ngā taha e rua ki te 2.

$$b=-3$$

Whakaurua te -3 mō b ki a-2b=8. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō a hāngai tonu.

$$a-2\left(-3\right)=8$$

Whakareatia -2 ki te -3.

$$a+6=8$$

Me tango 6 mai i ngā taha e rua o te whārite.

$$a=2$$

Kua oti te pūnaha te whakatau.

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