Whakaoti i te {l}{(a+2012)-2(b-2013)=3}{3(a+2012)+4(b-2013)=5}

$$\left\{ \begin{array} { l } { ( a + 2012 ) - 2 ( b - 2013 ) = 3 } \\ { 3 ( a + 2012 ) + 4 ( b - 2013 ) = 5 } \end{array} \right.$$

Whakaoti mō a, b

$$a = -\frac{10049}{5} = -2009\frac{4}{5} = -2009.8$$

$$b = \frac{10063}{5} = 2012\frac{3}{5} = 2012.6$$

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.

$$a-2\left(b-2013\right)+2012=3,3\left(a+2012\right)+4\left(b-2013\right)=5$$

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

$$a-2\left(b-2013\right)+2012=3$$

Whakareatia -2 ki te b-2013.

$$a-2b+4026+2012=3$$

Tāpiri 4026 ki te 2012.

$$a-2b+6038=3$$

Me tango 6038 mai i ngā taha e rua o te whārite.

$$a-2b=-6035$$

Me tāpiri 2b ki ngā taha e rua o te whārite.

$$a=2b-6035$$

Whakakapia te 2b-6035 mō te a ki tērā atu whārite, 3\left(a+2012\right)+4\left(b-2013\right)=5.

$$3\left(2b-6035+2012\right)+4\left(b-2013\right)=5$$

Tāpiri -6035 ki te 2012.

$$3\left(2b-4023\right)+4\left(b-2013\right)=5$$

Whakareatia 3 ki te 2b-4023.

$$6b-12069+4\left(b-2013\right)=5$$

Whakareatia 4 ki te b-2013.

$$6b-12069+4b-8052=5$$

Tāpiri 6b ki te 4b.

$$10b-12069-8052=5$$

Tāpiri -12069 ki te -8052.

$$10b-20121=5$$

Me tāpiri 20121 ki ngā taha e rua o te whārite.

$$10b=20126$$

Whakawehea ngā taha e rua ki te 10.

$$b=\frac{10063}{5}$$

Whakaurua te \frac{10063}{5} mō b ki a=2b-6035. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō a hāngai tonu.

$$a=2\times \frac{10063}{5}-6035$$

Whakareatia 2 ki te \frac{10063}{5}.

$$a=\frac{20126}{5}-6035$$

Tāpiri -6035 ki te \frac{20126}{5}.

$$a=-\frac{10049}{5}$$

Kua oti te pūnaha te whakatau.

$$a=-\frac{10049}{5},b=\frac{10063}{5}$$

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.

$$a-2\left(b-2013\right)+2012=3,3\left(a+2012\right)+4\left(b-2013\right)=5$$

Whakarūnātia te whārite tuatahi ki te āhua tānga ngahuru.

$$a-2\left(b-2013\right)+2012=3$$

Whakareatia -2 ki te b-2013.

$$a-2b+4026+2012=3$$

Tāpiri 4026 ki te 2012.

$$a-2b+6038=3$$

Me tango 6038 mai i ngā taha e rua o te whārite.

$$a-2b=-6035$$

Whakarūnātia te whārite tuarua ki te āhua tānga ngahuru.

$$3\left(a+2012\right)+4\left(b-2013\right)=5$$

Whakareatia 3 ki te a+2012.

$$3a+6036+4\left(b-2013\right)=5$$

Whakareatia 4 ki te b-2013.

$$3a+6036+4b-8052=5$$

Tāpiri 6036 ki te -8052.

$$3a+4b-2016=5$$

Me tāpiri 2016 ki ngā taha e rua o te whārite.

$$3a+4b=2021$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&-2\\3&4\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}-6035\\2021\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&-2\\3&4\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&-2\\3&4\end{matrix}\right))\left(\begin{matrix}1&-2\\3&4\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}1&-2\\3&4\end{matrix}\right))\left(\begin{matrix}-6035\\2021\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}1&-2\\3&4\end{matrix}\right))\left(\begin{matrix}-6035\\2021\end{matrix}\right)$$

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

$$\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}1&-2\\3&4\end{matrix}\right))\left(\begin{matrix}-6035\\2021\end{matrix}\right)$$

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.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}\frac{4}{4-\left(-2\times 3\right)}&-\frac{-2}{4-\left(-2\times 3\right)}\\-\frac{3}{4-\left(-2\times 3\right)}&\frac{1}{4-\left(-2\times 3\right)}\end{matrix}\right)\left(\begin{matrix}-6035\\2021\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}\frac{2}{5}&\frac{1}{5}\\-\frac{3}{10}&\frac{1}{10}\end{matrix}\right)\left(\begin{matrix}-6035\\2021\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}\frac{2}{5}\left(-6035\right)+\frac{1}{5}\times 2021\\-\frac{3}{10}\left(-6035\right)+\frac{1}{10}\times 2021\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}-\frac{10049}{5}\\\frac{10063}{5}\end{matrix}\right)$$

Tangohia ngā huānga poukapa a me b.

$$a=-\frac{10049}{5},b=\frac{10063}{5}$$