Whakaoti i te {l}{-2a-b+8=0}{-2b+a+1=0}

$$\left\{ \begin{array} { l } { - 2 a - b + 8 = 0 } \\ { - 2 b + a + 1 = 0 } \end{array} \right.$$

Whakaoti mō a, b

$$a=3$$

$$b=2$$

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.

$$-2a-b+8=0,a-2b+1=0$$

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

$$-2a-b+8=0$$

Me tango 8 mai i ngā taha e rua o te whārite.

$$-2a-b=-8$$

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

$$-2a=b-8$$

Whakawehea ngā taha e rua ki te -2.

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

Whakareatia -\frac{1}{2} ki te b-8.

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

Whakakapia te -\frac{b}{2}+4 mō te a ki tērā atu whārite, a-2b+1=0.

$$-\frac{1}{2}b+4-2b+1=0$$

Tāpiri -\frac{b}{2} ki te -2b.

$$-\frac{5}{2}b+4+1=0$$

Tāpiri 4 ki te 1.

$$-\frac{5}{2}b+5=0$$

Me tango 5 mai i ngā taha e rua o te whārite.

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

Whakawehea ngā taha e rua o te whārite ki te -\frac{5}{2}, 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{1}{2}b+4. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō a hāngai tonu.

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

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

$$a=-1+4$$

Tāpiri 4 ki te -1.

$$a=3$$

Kua oti te pūnaha te whakatau.

$$a=3,b=2$$

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.

$$-2a-b+8=0,a-2b+1=0$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{5}&\frac{1}{5}\\-\frac{1}{5}&-\frac{2}{5}\end{matrix}\right)\left(\begin{matrix}-8\\-1\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{5}\left(-8\right)+\frac{1}{5}\left(-1\right)\\-\frac{1}{5}\left(-8\right)-\frac{2}{5}\left(-1\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa a me b.

$$a=3,b=2$$

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.

$$-2a-b+8=0,a-2b+1=0$$

Kia ōrite ai a -2a 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 -2.

$$-2a-b+8=0,-2a-2\left(-2\right)b-2=0$$

Whakarūnātia.

$$-2a-b+8=0,-2a+4b-2=0$$

Me tango -2a+4b-2=0 mai i -2a-b+8=0 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-2a+2a-b-4b+8+2=0$$

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

$$-b-4b+8+2=0$$

Tāpiri -b ki te -4b.

$$-5b+8+2=0$$

Tāpiri 8 ki te 2.

$$-5b+10=0$$

Me tango 10 mai i ngā taha e rua o te whārite.

$$-5b=-10$$

Whakawehea ngā taha e rua ki te -5.

$$b=2$$

Whakaurua te 2 mō b ki a-2b+1=0. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō a hāngai tonu.

$$a-2\times 2+1=0$$

Whakareatia -2 ki te 2.

$$a-4+1=0$$

Tāpiri -4 ki te 1.

$$a-3=0$$

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

$$a=3$$

Kua oti te pūnaha te whakatau.

$$a=3,b=2$$