Whakaoti i te {l}{a-b+2=0}{9a+3b+2=-2}

$$\left\{ \begin{array} { l } { a - b + 2 = 0 } \\ { 9 a + 3 b + 2 = - 2 } \end{array} \right.$$

Whakaoti mō a, b

$$a=-\frac{5}{6}\approx -0.833333333$$

$$b = \frac{7}{6} = 1\frac{1}{6} \approx 1.166666667$$

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.

$$a-b+2=0,9a+3b+2=-2$$

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

$$a-b+2=0$$

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

$$a-b=-2$$

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

$$a=b-2$$

Whakakapia te b-2 mō te a ki tērā atu whārite, 9a+3b+2=-2.

$$9\left(b-2\right)+3b+2=-2$$

Whakareatia 9 ki te b-2.

$$9b-18+3b+2=-2$$

Tāpiri 9b ki te 3b.

$$12b-18+2=-2$$

Tāpiri -18 ki te 2.

$$12b-16=-2$$

Me tāpiri 16 ki ngā taha e rua o te whārite.

$$12b=14$$

Whakawehea ngā taha e rua ki te 12.

$$b=\frac{7}{6}$$

Whakaurua te \frac{7}{6} mō b ki a=b-2. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō a hāngai tonu.

$$a=\frac{7}{6}-2$$

Tāpiri -2 ki te \frac{7}{6}.

$$a=-\frac{5}{6}$$

Kua oti te pūnaha te whakatau.

$$a=-\frac{5}{6},b=\frac{7}{6}$$

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.

$$a-b+2=0,9a+3b+2=-2$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&-1\\9&3\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}-2\\-4\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&-1\\9&3\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&-1\\9&3\end{matrix}\right))\left(\begin{matrix}1&-1\\9&3\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}1&-1\\9&3\end{matrix}\right))\left(\begin{matrix}-2\\-4\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}1&-1\\9&3\end{matrix}\right))\left(\begin{matrix}-2\\-4\end{matrix}\right)$$

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

$$\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}1&-1\\9&3\end{matrix}\right))\left(\begin{matrix}-2\\-4\end{matrix}\right)$$

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.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}\frac{3}{3-\left(-9\right)}&-\frac{-1}{3-\left(-9\right)}\\-\frac{9}{3-\left(-9\right)}&\frac{1}{3-\left(-9\right)}\end{matrix}\right)\left(\begin{matrix}-2\\-4\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}\frac{1}{4}&\frac{1}{12}\\-\frac{3}{4}&\frac{1}{12}\end{matrix}\right)\left(\begin{matrix}-2\\-4\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}\frac{1}{4}\left(-2\right)+\frac{1}{12}\left(-4\right)\\-\frac{3}{4}\left(-2\right)+\frac{1}{12}\left(-4\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}-\frac{5}{6}\\\frac{7}{6}\end{matrix}\right)$$

Tangohia ngā huānga poukapa a me b.

$$a=-\frac{5}{6},b=\frac{7}{6}$$

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.

$$a-b+2=0,9a+3b+2=-2$$

Kia ōrite ai a a me 9a, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 9 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 1.

$$9a+9\left(-1\right)b+9\times 2=0,9a+3b+2=-2$$

Whakarūnātia.

$$9a-9b+18=0,9a+3b+2=-2$$

Me tango 9a+3b+2=-2 mai i 9a-9b+18=0 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$9a-9a-9b-3b+18-2=2$$

Tāpiri 9a ki te -9a. Ka whakakore atu ngā kupu 9a me -9a, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-9b-3b+18-2=2$$

Tāpiri -9b ki te -3b.

$$-12b+18-2=2$$

Tāpiri 18 ki te -2.

$$-12b+16=2$$

Me tango 16 mai i ngā taha e rua o te whārite.

$$-12b=-14$$

Whakawehea ngā taha e rua ki te -12.

$$b=\frac{7}{6}$$

Whakaurua te \frac{7}{6} mō b ki 9a+3b+2=-2. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō a hāngai tonu.

$$9a+3\times \frac{7}{6}+2=-2$$

Whakareatia 3 ki te \frac{7}{6}.

$$9a+\frac{7}{2}+2=-2$$

Tāpiri \frac{7}{2} ki te 2.

$$9a+\frac{11}{2}=-2$$

Me tango \frac{11}{2} mai i ngā taha e rua o te whārite.

$$9a=-\frac{15}{2}$$

Whakawehea ngā taha e rua ki te 9.

$$a=-\frac{5}{6}$$

Kua oti te pūnaha te whakatau.

$$a=-\frac{5}{6},b=\frac{7}{6}$$