Whakaoti i te {l}{a+4b=8}{3b=5-a}

$$\left\{ \begin{array} { l } { a + 4 b = 8 } \\ { 3 b = 5 - a } \end{array} \right.$$

Whakaoti mō a, b

$$a=-4$$

$$b=3$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuarua. Me tāpiri te a ki ngā taha e rua.

$$3b+a=5$$

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.

$$a+4b=8,a+3b=5$$

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

$$a+4b=8$$

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

$$a=-4b+8$$

Whakakapia te -4b+8 mō te a ki tērā atu whārite, a+3b=5.

$$-4b+8+3b=5$$

Tāpiri -4b ki te 3b.

$$-b+8=5$$

Me tango 8 mai i ngā taha e rua o te whārite.

$$-b=-3$$

Whakawehea ngā taha e rua ki te -1.

$$b=3$$

Whakaurua te 3 mō b ki a=-4b+8. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō a hāngai tonu.

$$a=-4\times 3+8$$

Whakareatia -4 ki te 3.

$$a=-12+8$$

Tāpiri 8 ki te -12.

$$a=-4$$

Kua oti te pūnaha te whakatau.

$$a=-4,b=3$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuarua. Me tāpiri te a ki ngā taha e rua.

$$3b+a=5$$

Tuhia ngā whārite ki te tānga ngahuru ka whakamahi i ngā poukapa hei whakaoti i te pūnaha o ngā whārite.

$$a+4b=8,a+3b=5$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&4\\1&3\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}8\\5\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&4\\1&3\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&4\\1&3\end{matrix}\right))\left(\begin{matrix}1&4\\1&3\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}1&4\\1&3\end{matrix}\right))\left(\begin{matrix}8\\5\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}1&4\\1&3\end{matrix}\right))\left(\begin{matrix}8\\5\end{matrix}\right)$$

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

$$\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}1&4\\1&3\end{matrix}\right))\left(\begin{matrix}8\\5\end{matrix}\right)$$

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.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}\frac{3}{3-4}&-\frac{4}{3-4}\\-\frac{1}{3-4}&\frac{1}{3-4}\end{matrix}\right)\left(\begin{matrix}8\\5\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}-3&4\\1&-1\end{matrix}\right)\left(\begin{matrix}8\\5\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}-3\times 8+4\times 5\\8-5\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa a me b.

$$a=-4,b=3$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuarua. Me tāpiri te a ki ngā taha e rua.

$$3b+a=5$$

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.

$$a+4b=8,a+3b=5$$

Me tango a+3b=5 mai i a+4b=8 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$a-a+4b-3b=8-5$$

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.

$$4b-3b=8-5$$

Tāpiri 4b ki te -3b.

$$b=8-5$$

Tāpiri 8 ki te -5.

$$b=3$$

Whakaurua te 3 mō b ki a+3b=5. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō a hāngai tonu.

$$a+3\times 3=5$$

Whakareatia 3 ki te 3.

$$a+9=5$$

Me tango 9 mai i ngā taha e rua o te whārite.

$$a=-4$$

Kua oti te pūnaha te whakatau.

$$a=-4,b=3$$