Whakaoti i te {l}{a+3b=6}{a-6b=12}

$$\left\{ \begin{array} { l } { a + 3 b = 6 } \\ { a - 6 b = 12 } \end{array} \right.$$

Whakaoti mō a, b

$$a=8$$

$$b=-\frac{2}{3}\approx -0.666666667$$

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.

$$a+3b=6,a-6b=12$$

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

$$a+3b=6$$

Me tango 3b mai i ngā taha e rua o te whārite.

$$a=-3b+6$$

Whakakapia te -3b+6 mō te a ki tērā atu whārite, a-6b=12.

$$-3b+6-6b=12$$

Tāpiri -3b ki te -6b.

$$-9b+6=12$$

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

$$-9b=6$$

Whakawehea ngā taha e rua ki te -9.

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

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

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

Whakareatia -3 ki te -\frac{2}{3}.

$$a=2+6$$

Tāpiri 6 ki te 2.

$$a=8$$

Kua oti te pūnaha te whakatau.

$$a=8,b=-\frac{2}{3}$$

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.

$$a+3b=6,a-6b=12$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&3\\1&-6\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}6\\12\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&3\\1&-6\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&3\\1&-6\end{matrix}\right))\left(\begin{matrix}1&3\\1&-6\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}1&3\\1&-6\end{matrix}\right))\left(\begin{matrix}6\\12\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}1&3\\1&-6\end{matrix}\right))\left(\begin{matrix}6\\12\end{matrix}\right)$$

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

$$\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}1&3\\1&-6\end{matrix}\right))\left(\begin{matrix}6\\12\end{matrix}\right)$$

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.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}-\frac{6}{-6-3}&-\frac{3}{-6-3}\\-\frac{1}{-6-3}&\frac{1}{-6-3}\end{matrix}\right)\left(\begin{matrix}6\\12\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}\frac{2}{3}&\frac{1}{3}\\\frac{1}{9}&-\frac{1}{9}\end{matrix}\right)\left(\begin{matrix}6\\12\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}\frac{2}{3}\times 6+\frac{1}{3}\times 12\\\frac{1}{9}\times 6-\frac{1}{9}\times 12\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}8\\-\frac{2}{3}\end{matrix}\right)$$

Tangohia ngā huānga poukapa a me b.

$$a=8,b=-\frac{2}{3}$$

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.

$$a+3b=6,a-6b=12$$

Me tango a-6b=12 mai i a+3b=6 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$a-a+3b+6b=6-12$$

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

$$3b+6b=6-12$$

Tāpiri 3b ki te 6b.

$$9b=6-12$$

Tāpiri 6 ki te -12.

$$9b=-6$$

Whakawehea ngā taha e rua ki te 9.

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

Whakaurua te -\frac{2}{3} mō b ki a-6b=12. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō a hāngai tonu.

$$a-6\left(-\frac{2}{3}\right)=12$$

Whakareatia -6 ki te -\frac{2}{3}.

$$a+4=12$$

Me tango 4 mai i ngā taha e rua o te whārite.

$$a=8$$

Kua oti te pūnaha te whakatau.

$$a=8,b=-\frac{2}{3}$$