Whakaoti i te {l}{a+b=3}{a-b=7}

$$\left\{ \begin{array} { l } { a + b = 3 } \\ { a - b = 7 } \end{array} \right.$$

Whakaoti mō a, b

$$a=5$$

$$b=-2$$

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.

$$a+b=3,a-b=7$$

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

$$a+b=3$$

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

$$a=-b+3$$

Whakakapia te -b+3 mō te a ki tērā atu whārite, a-b=7.

$$-b+3-b=7$$

Tāpiri -b ki te -b.

$$-2b+3=7$$

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

$$-2b=4$$

Whakawehea ngā taha e rua ki te -2.

$$b=-2$$

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

$$a=-\left(-2\right)+3$$

Whakareatia -1 ki te -2.

$$a=2+3$$

Tāpiri 3 ki te 2.

$$a=5$$

Kua oti te pūnaha te whakatau.

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

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.

$$a+b=3,a-b=7$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&1\\1&-1\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}3\\7\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}1&1\\1&-1\end{matrix}\right))\left(\begin{matrix}1&1\\1&-1\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\1&-1\end{matrix}\right))\left(\begin{matrix}3\\7\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\1&-1\end{matrix}\right))\left(\begin{matrix}3\\7\end{matrix}\right)$$

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

$$\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\1&-1\end{matrix}\right))\left(\begin{matrix}3\\7\end{matrix}\right)$$

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.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{-1-1}&-\frac{1}{-1-1}\\-\frac{1}{-1-1}&\frac{1}{-1-1}\end{matrix}\right)\left(\begin{matrix}3\\7\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2}&\frac{1}{2}\\\frac{1}{2}&-\frac{1}{2}\end{matrix}\right)\left(\begin{matrix}3\\7\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2}\times 3+\frac{1}{2}\times 7\\\frac{1}{2}\times 3-\frac{1}{2}\times 7\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa a me b.

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

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.

$$a+b=3,a-b=7$$

Me tango a-b=7 mai i a+b=3 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$a-a+b+b=3-7$$

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.

$$b+b=3-7$$

Tāpiri b ki te b.

$$2b=3-7$$

Tāpiri 3 ki te -7.

$$2b=-4$$

Whakawehea ngā taha e rua ki te 2.

$$b=-2$$

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

$$a-\left(-2\right)=7$$

Whakareatia -1 ki te -2.

$$a+2=7$$

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

$$a=5$$

Kua oti te pūnaha te whakatau.

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