Whakaoti i te {r}{a+b=20}{6a+2b=20}

$$\left. \begin{array} { r } { a + b = 20 } \\ { 6 a + 2 b = 20 } \end{array} \right.$$

Whakaoti mō a, b

$$a=-5$$

$$b=25$$

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.

$$a+b=20,6a+2b=20$$

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

$$a+b=20$$

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

$$a=-b+20$$

Whakakapia te -b+20 mō te a ki tērā atu whārite, 6a+2b=20.

$$6\left(-b+20\right)+2b=20$$

Whakareatia 6 ki te -b+20.

$$-6b+120+2b=20$$

Tāpiri -6b ki te 2b.

$$-4b+120=20$$

Me tango 120 mai i ngā taha e rua o te whārite.

$$-4b=-100$$

Whakawehea ngā taha e rua ki te -4.

$$b=25$$

Whakaurua te 25 mō b ki a=-b+20. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō a hāngai tonu.

$$a=-25+20$$

Tāpiri 20 ki te -25.

$$a=-5$$

Kua oti te pūnaha te whakatau.

$$a=-5,b=25$$

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.

$$a+b=20,6a+2b=20$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&1\\6&2\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}20\\20\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&1\\6&2\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&1\\6&2\end{matrix}\right))\left(\begin{matrix}1&1\\6&2\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\6&2\end{matrix}\right))\left(\begin{matrix}20\\20\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\6&2\end{matrix}\right))\left(\begin{matrix}20\\20\end{matrix}\right)$$

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

$$\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\6&2\end{matrix}\right))\left(\begin{matrix}20\\20\end{matrix}\right)$$

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.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}\frac{2}{2-6}&-\frac{1}{2-6}\\-\frac{6}{2-6}&\frac{1}{2-6}\end{matrix}\right)\left(\begin{matrix}20\\20\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{2}&\frac{1}{4}\\\frac{3}{2}&-\frac{1}{4}\end{matrix}\right)\left(\begin{matrix}20\\20\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{2}\times 20+\frac{1}{4}\times 20\\\frac{3}{2}\times 20-\frac{1}{4}\times 20\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}-5\\25\end{matrix}\right)$$

Tangohia ngā huānga poukapa a me b.

$$a=-5,b=25$$

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.

$$a+b=20,6a+2b=20$$

Kia ōrite ai a a me 6a, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 6 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 1.

$$6a+6b=6\times 20,6a+2b=20$$

Whakarūnātia.

$$6a+6b=120,6a+2b=20$$

Me tango 6a+2b=20 mai i 6a+6b=120 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$6a-6a+6b-2b=120-20$$

Tāpiri 6a ki te -6a. Ka whakakore atu ngā kupu 6a me -6a, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$6b-2b=120-20$$

Tāpiri 6b ki te -2b.

$$4b=120-20$$

Tāpiri 120 ki te -20.

$$4b=100$$

Whakawehea ngā taha e rua ki te 4.

$$b=25$$

Whakaurua te 25 mō b ki 6a+2b=20. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō a hāngai tonu.

$$6a+2\times 25=20$$

Whakareatia 2 ki te 25.

$$6a+50=20$$

Me tango 50 mai i ngā taha e rua o te whārite.

$$6a=-30$$

Whakawehea ngā taha e rua ki te 6.

$$a=-5$$

Kua oti te pūnaha te whakatau.

$$a=-5,b=25$$