Whakaoti i te {l}{a+b+3=2}{4a+2b+3=3}

$$\left\{ \begin{array} { l } { a + b + 3 = 2 } \\ { 4 a + 2 b + 3 = 3 } \end{array} \right.$$

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.

$$a+b+3=2,4a+2b+3=3$$

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

$$a+b+3=2$$

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

$$a+b=-1$$

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

$$a=-b-1$$

Whakakapia te -b-1 mō te a ki tērā atu whārite, 4a+2b+3=3.

$$4\left(-b-1\right)+2b+3=3$$

Whakareatia 4 ki te -b-1.

$$-4b-4+2b+3=3$$

Tāpiri -4b ki te 2b.

$$-2b-4+3=3$$

Tāpiri -4 ki te 3.

$$-2b-1=3$$

Me tāpiri 1 ki 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-1. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō a hāngai tonu.

$$a=-\left(-2\right)-1$$

Whakareatia -1 ki te -2.

$$a=2-1$$

Tāpiri -1 ki te 2.

$$a=1$$

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.

$$a+b+3=2,4a+2b+3=3$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&1\\4&2\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}-1\\0\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&1\\4&2\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&1\\4&2\end{matrix}\right))\left(\begin{matrix}1&1\\4&2\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\4&2\end{matrix}\right))\left(\begin{matrix}-1\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\4&2\end{matrix}\right))\left(\begin{matrix}-1\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\4&2\end{matrix}\right))\left(\begin{matrix}-1\\0\end{matrix}\right)$$

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.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}\frac{2}{2-4}&-\frac{1}{2-4}\\-\frac{4}{2-4}&\frac{1}{2-4}\end{matrix}\right)\left(\begin{matrix}-1\\0\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

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.

$$a+b+3=2,4a+2b+3=3$$

Kia ōrite ai a a me 4a, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 4 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 1.

$$4a+4b+4\times 3=4\times 2,4a+2b+3=3$$

Whakarūnātia.

$$4a+4b+12=8,4a+2b+3=3$$

Me tango 4a+2b+3=3 mai i 4a+4b+12=8 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$4a-4a+4b-2b+12-3=8-3$$

Tāpiri 4a ki te -4a. Ka whakakore atu ngā kupu 4a me -4a, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$4b-2b+12-3=8-3$$

Tāpiri 4b ki te -2b.

$$2b+12-3=8-3$$

Tāpiri 12 ki te -3.

$$2b+9=8-3$$

Tāpiri 8 ki te -3.

$$2b+9=5$$

Me tango 9 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 4a+2b+3=3. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō a hāngai tonu.

$$4a+2\left(-2\right)+3=3$$

Whakareatia 2 ki te -2.

$$4a-4+3=3$$

Tāpiri -4 ki te 3.

$$4a-1=3$$

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

$$4a=4$$

Whakawehea ngā taha e rua ki te 4.

$$a=1$$

Kua oti te pūnaha te whakatau.

$$a=1,b=-2$$