Whakaoti i te {l}{2a+7b=14}{12(2a+12b)/2=129}

$$\left. \begin{array} { l } { 2 a + 7 b = 14 } \\ { \frac { 12 ( 2 a + 12 b ) } { 2 } = 129 } \end{array} \right.$$

Whakaoti mō a, b

$$a = \frac{7}{4} = 1\frac{3}{4} = 1.75$$

$$b = \frac{3}{2} = 1\frac{1}{2} = 1.5$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuarua. Me whakarea ngā taha e rua ki te 2.

$$12\left(2a+12b\right)=129\times 2$$

Whakamahia te āhuatanga tohatoha hei whakarea te 12 ki te 2a+12b.

$$24a+144b=129\times 2$$

Whakareatia te 129 ki te 2, ka 258.

$$24a+144b=258$$

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.

$$2a+7b=14,24a+144b=258$$

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

$$2a+7b=14$$

Me tango 7b mai i ngā taha e rua o te whārite.

$$2a=-7b+14$$

Whakawehea ngā taha e rua ki te 2.

$$a=\frac{1}{2}\left(-7b+14\right)$$

Whakareatia \frac{1}{2} ki te -7b+14.

$$a=-\frac{7}{2}b+7$$

Whakakapia te -\frac{7b}{2}+7 mō te a ki tērā atu whārite, 24a+144b=258.

$$24\left(-\frac{7}{2}b+7\right)+144b=258$$

Whakareatia 24 ki te -\frac{7b}{2}+7.

$$-84b+168+144b=258$$

Tāpiri -84b ki te 144b.

$$60b+168=258$$

Me tango 168 mai i ngā taha e rua o te whārite.

$$60b=90$$

Whakawehea ngā taha e rua ki te 60.

$$b=\frac{3}{2}$$

Whakaurua te \frac{3}{2} mō b ki a=-\frac{7}{2}b+7. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō a hāngai tonu.

$$a=-\frac{7}{2}\times \frac{3}{2}+7$$

Whakareatia -\frac{7}{2} ki te \frac{3}{2} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$a=-\frac{21}{4}+7$$

Tāpiri 7 ki te -\frac{21}{4}.

$$a=\frac{7}{4}$$

Kua oti te pūnaha te whakatau.

$$a=\frac{7}{4},b=\frac{3}{2}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuarua. Me whakarea ngā taha e rua ki te 2.

$$12\left(2a+12b\right)=129\times 2$$

Whakamahia te āhuatanga tohatoha hei whakarea te 12 ki te 2a+12b.

$$24a+144b=129\times 2$$

Whakareatia te 129 ki te 2, ka 258.

$$24a+144b=258$$

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

$$2a+7b=14,24a+144b=258$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&7\\24&144\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}14\\258\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}2&7\\24&144\end{matrix}\right).

$$inverse(\left(\begin{matrix}2&7\\24&144\end{matrix}\right))\left(\begin{matrix}2&7\\24&144\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}2&7\\24&144\end{matrix}\right))\left(\begin{matrix}14\\258\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}2&7\\24&144\end{matrix}\right))\left(\begin{matrix}14\\258\end{matrix}\right)$$

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

$$\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}2&7\\24&144\end{matrix}\right))\left(\begin{matrix}14\\258\end{matrix}\right)$$

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.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}\frac{144}{2\times 144-7\times 24}&-\frac{7}{2\times 144-7\times 24}\\-\frac{24}{2\times 144-7\times 24}&\frac{2}{2\times 144-7\times 24}\end{matrix}\right)\left(\begin{matrix}14\\258\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}\frac{6}{5}&-\frac{7}{120}\\-\frac{1}{5}&\frac{1}{60}\end{matrix}\right)\left(\begin{matrix}14\\258\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}\frac{6}{5}\times 14-\frac{7}{120}\times 258\\-\frac{1}{5}\times 14+\frac{1}{60}\times 258\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}\frac{7}{4}\\\frac{3}{2}\end{matrix}\right)$$

Tangohia ngā huānga poukapa a me b.

$$a=\frac{7}{4},b=\frac{3}{2}$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuarua. Me whakarea ngā taha e rua ki te 2.

$$12\left(2a+12b\right)=129\times 2$$

Whakamahia te āhuatanga tohatoha hei whakarea te 12 ki te 2a+12b.

$$24a+144b=129\times 2$$

Whakareatia te 129 ki te 2, ka 258.

$$24a+144b=258$$

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.

$$2a+7b=14,24a+144b=258$$

Kia ōrite ai a 2a me 24a, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 24 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 2.

$$24\times 2a+24\times 7b=24\times 14,2\times 24a+2\times 144b=2\times 258$$

Whakarūnātia.

$$48a+168b=336,48a+288b=516$$

Me tango 48a+288b=516 mai i 48a+168b=336 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$48a-48a+168b-288b=336-516$$

Tāpiri 48a ki te -48a. Ka whakakore atu ngā kupu 48a me -48a, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$168b-288b=336-516$$

Tāpiri 168b ki te -288b.

$$-120b=336-516$$

Tāpiri 336 ki te -516.

$$-120b=-180$$

Whakawehea ngā taha e rua ki te -120.

$$b=\frac{3}{2}$$

Whakaurua te \frac{3}{2} mō b ki 24a+144b=258. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō a hāngai tonu.

$$24a+144\times \frac{3}{2}=258$$

Whakareatia 144 ki te \frac{3}{2}.

$$24a+216=258$$

Me tango 216 mai i ngā taha e rua o te whārite.

$$24a=42$$

Whakawehea ngā taha e rua ki te 24.

$$a=\frac{7}{4}$$

Kua oti te pūnaha te whakatau.

$$a=\frac{7}{4},b=\frac{3}{2}$$