Whakaoti i te {l}{a+2b=5}{a-2b=-3}

$$\left. \begin{array} { l } { a + 2 b = 5 } \\ { a - 2 b = - 3 } \end{array} \right.$$

Whakaoti mō a, b

$$a=1$$

$$b=2$$

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.

$$a+2b=5,a-2b=-3$$

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

$$a+2b=5$$

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

$$a=-2b+5$$

Whakakapia te -2b+5 mō te a ki tērā atu whārite, a-2b=-3.

$$-2b+5-2b=-3$$

Tāpiri -2b ki te -2b.

$$-4b+5=-3$$

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

$$-4b=-8$$

Whakawehea ngā taha e rua ki te -4.

$$b=2$$

Whakaurua te 2 mō b ki a=-2b+5. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō a hāngai tonu.

$$a=-2\times 2+5$$

Whakareatia -2 ki te 2.

$$a=-4+5$$

Tāpiri 5 ki te -4.

$$a=1$$

Kua oti te pūnaha te whakatau.

$$a=1,b=2$$

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.

$$a+2b=5,a-2b=-3$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&2\\1&-2\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}5\\-3\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}1&2\\1&-2\end{matrix}\right))\left(\begin{matrix}1&2\\1&-2\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}1&2\\1&-2\end{matrix}\right))\left(\begin{matrix}5\\-3\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}1&2\\1&-2\end{matrix}\right))\left(\begin{matrix}5\\-3\end{matrix}\right)$$

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

$$\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}1&2\\1&-2\end{matrix}\right))\left(\begin{matrix}5\\-3\end{matrix}\right)$$

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.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{-2-2}&-\frac{2}{-2-2}\\-\frac{1}{-2-2}&\frac{1}{-2-2}\end{matrix}\right)\left(\begin{matrix}5\\-3\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2}&\frac{1}{2}\\\frac{1}{4}&-\frac{1}{4}\end{matrix}\right)\left(\begin{matrix}5\\-3\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2}\times 5+\frac{1}{2}\left(-3\right)\\\frac{1}{4}\times 5-\frac{1}{4}\left(-3\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa a me b.

$$a=1,b=2$$

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.

$$a+2b=5,a-2b=-3$$

Me tango a-2b=-3 mai i a+2b=5 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$a-a+2b+2b=5+3$$

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.

$$2b+2b=5+3$$

Tāpiri 2b ki te 2b.

$$4b=5+3$$

Tāpiri 5 ki te 3.

$$4b=8$$

Whakawehea ngā taha e rua ki te 4.

$$b=2$$

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

$$a-2\times 2=-3$$

Whakareatia -2 ki te 2.

$$a-4=-3$$

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

$$a=1$$

Kua oti te pūnaha te whakatau.

$$a=1,b=2$$