Whakaoti i te {c}{10a+b=10}{-a+b=2}

$$\left. \begin{array} { c } { 10 a + b = 10 } \\ { - a + b = 2 } \end{array} \right.$$

Whakaoti mō a, b

$$a=\frac{8}{11}\approx 0.727272727$$

$$b = \frac{30}{11} = 2\frac{8}{11} \approx 2.727272727$$

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.

$$10a+b=10,-a+b=2$$

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

$$10a+b=10$$

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

$$10a=-b+10$$

Whakawehea ngā taha e rua ki te 10.

$$a=\frac{1}{10}\left(-b+10\right)$$

Whakareatia \frac{1}{10} ki te -b+10.

$$a=-\frac{1}{10}b+1$$

Whakakapia te -\frac{b}{10}+1 mō te a ki tērā atu whārite, -a+b=2.

$$-\left(-\frac{1}{10}b+1\right)+b=2$$

Whakareatia -1 ki te -\frac{b}{10}+1.

$$\frac{1}{10}b-1+b=2$$

Tāpiri \frac{b}{10} ki te b.

$$\frac{11}{10}b-1=2$$

Me tāpiri 1 ki ngā taha e rua o te whārite.

$$\frac{11}{10}b=3$$

Whakawehea ngā taha e rua o te whārite ki te \frac{11}{10}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$b=\frac{30}{11}$$

Whakaurua te \frac{30}{11} mō b ki a=-\frac{1}{10}b+1. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō a hāngai tonu.

$$a=-\frac{1}{10}\times \frac{30}{11}+1$$

Whakareatia -\frac{1}{10} ki te \frac{30}{11} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$a=-\frac{3}{11}+1$$

Tāpiri 1 ki te -\frac{3}{11}.

$$a=\frac{8}{11}$$

Kua oti te pūnaha te whakatau.

$$a=\frac{8}{11},b=\frac{30}{11}$$

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.

$$10a+b=10,-a+b=2$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}10&1\\-1&1\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}10\\2\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}10&1\\-1&1\end{matrix}\right).

$$inverse(\left(\begin{matrix}10&1\\-1&1\end{matrix}\right))\left(\begin{matrix}10&1\\-1&1\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}10&1\\-1&1\end{matrix}\right))\left(\begin{matrix}10\\2\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}10&1\\-1&1\end{matrix}\right))\left(\begin{matrix}10\\2\end{matrix}\right)$$

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

$$\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}10&1\\-1&1\end{matrix}\right))\left(\begin{matrix}10\\2\end{matrix}\right)$$

Mō te poukapa 2\times 2 \left(\begin{matrix}a&b\\c&d\end{matrix}\right), ko te poukapa kōaro ko \left(\begin{matrix}\frac{d}{ad-bc}&\frac{-b}{ad-bc}\\\frac{-c}{ad-bc}&\frac{a}{ad-bc}\end{matrix}\right), kia tuhia anō ai te whārite poukapa hei rapanga whakarea poukapa.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}\frac{1}{10-\left(-1\right)}&-\frac{1}{10-\left(-1\right)}\\-\frac{-1}{10-\left(-1\right)}&\frac{10}{10-\left(-1\right)}\end{matrix}\right)\left(\begin{matrix}10\\2\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}\frac{1}{11}&-\frac{1}{11}\\\frac{1}{11}&\frac{10}{11}\end{matrix}\right)\left(\begin{matrix}10\\2\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}\frac{1}{11}\times 10-\frac{1}{11}\times 2\\\frac{1}{11}\times 10+\frac{10}{11}\times 2\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}\frac{8}{11}\\\frac{30}{11}\end{matrix}\right)$$

Tangohia ngā huānga poukapa a me b.

$$a=\frac{8}{11},b=\frac{30}{11}$$

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.

$$10a+b=10,-a+b=2$$

Me tango -a+b=2 mai i 10a+b=10 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$10a+a+b-b=10-2$$

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.

$$10a+a=10-2$$

Tāpiri 10a ki te a.

$$11a=10-2$$

Tāpiri 10 ki te -2.

$$11a=8$$

Whakawehea ngā taha e rua ki te 11.

$$a=\frac{8}{11}$$

Whakaurua te \frac{8}{11} mō a ki -a+b=2. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō b hāngai tonu.

$$-\frac{8}{11}+b=2$$

Me tāpiri \frac{8}{11} ki ngā taha e rua o te whārite.

$$b=\frac{30}{11}$$

Kua oti te pūnaha te whakatau.

$$a=\frac{8}{11},b=\frac{30}{11}$$