Whakaoti i te {c}{3/2a+b=1}{a+b/2=7}

$$\left. \begin{array} { c } { \frac { 3 } { 2 } a + b = 1 } \\ { a + \frac { b } { 2 } = 7 } \end{array} \right.$$

Whakaoti mō a, b

$$a=26$$

$$b=-38$$

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.

$$\frac{3}{2}a+b=1,a+\frac{1}{2}b=7$$

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

$$\frac{3}{2}a+b=1$$

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

$$\frac{3}{2}a=-b+1$$

Whakawehea ngā taha e rua o te whārite ki te \frac{3}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$a=\frac{2}{3}\left(-b+1\right)$$

Whakareatia \frac{2}{3} ki te -b+1.

$$a=-\frac{2}{3}b+\frac{2}{3}$$

Whakakapia te \frac{-2b+2}{3} mō te a ki tērā atu whārite, a+\frac{1}{2}b=7.

$$-\frac{2}{3}b+\frac{2}{3}+\frac{1}{2}b=7$$

Tāpiri -\frac{2b}{3} ki te \frac{b}{2}.

$$-\frac{1}{6}b+\frac{2}{3}=7$$

Me tango \frac{2}{3} mai i ngā taha e rua o te whārite.

$$-\frac{1}{6}b=\frac{19}{3}$$

Me whakarea ngā taha e rua ki te -6.

$$b=-38$$

Whakaurua te -38 mō b ki a=-\frac{2}{3}b+\frac{2}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō a hāngai tonu.

$$a=-\frac{2}{3}\left(-38\right)+\frac{2}{3}$$

Whakareatia -\frac{2}{3} ki te -38.

$$a=\frac{76+2}{3}$$

Tāpiri \frac{2}{3} ki te \frac{76}{3} mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$a=26$$

Kua oti te pūnaha te whakatau.

$$a=26,b=-38$$

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.

$$\frac{3}{2}a+b=1,a+\frac{1}{2}b=7$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}\frac{3}{2}&1\\1&\frac{1}{2}\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}1\\7\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}\frac{3}{2}&1\\1&\frac{1}{2}\end{matrix}\right))\left(\begin{matrix}\frac{3}{2}&1\\1&\frac{1}{2}\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}\frac{3}{2}&1\\1&\frac{1}{2}\end{matrix}\right))\left(\begin{matrix}1\\7\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}\frac{3}{2}&1\\1&\frac{1}{2}\end{matrix}\right))\left(\begin{matrix}1\\7\end{matrix}\right)$$

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

$$\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}\frac{3}{2}&1\\1&\frac{1}{2}\end{matrix}\right))\left(\begin{matrix}1\\7\end{matrix}\right)$$

Mō te poukapa 2\times 2 \left(\begin{matrix}a&b\\c&d\end{matrix}\right), ko te poukapa kōaro ko \left(\begin{matrix}\frac{d}{ad-bc}&\frac{-b}{ad-bc}\\\frac{-c}{ad-bc}&\frac{a}{ad-bc}\end{matrix}\right), kia tuhia anō ai te whārite poukapa hei rapanga whakarea poukapa.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}\frac{\frac{1}{2}}{\frac{3}{2}\times \frac{1}{2}-1}&-\frac{1}{\frac{3}{2}\times \frac{1}{2}-1}\\-\frac{1}{\frac{3}{2}\times \frac{1}{2}-1}&\frac{\frac{3}{2}}{\frac{3}{2}\times \frac{1}{2}-1}\end{matrix}\right)\left(\begin{matrix}1\\7\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}-2&4\\4&-6\end{matrix}\right)\left(\begin{matrix}1\\7\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}-2+4\times 7\\4-6\times 7\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}26\\-38\end{matrix}\right)$$

Tangohia ngā huānga poukapa a me b.

$$a=26,b=-38$$

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.

$$\frac{3}{2}a+b=1,a+\frac{1}{2}b=7$$

Kia ōrite ai a \frac{3a}{2} me a, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 1 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te \frac{3}{2}.

$$\frac{3}{2}a+b=1,\frac{3}{2}a+\frac{3}{2}\times \frac{1}{2}b=\frac{3}{2}\times 7$$

Whakarūnātia.

$$\frac{3}{2}a+b=1,\frac{3}{2}a+\frac{3}{4}b=\frac{21}{2}$$

Me tango \frac{3}{2}a+\frac{3}{4}b=\frac{21}{2} mai i \frac{3}{2}a+b=1 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$\frac{3}{2}a-\frac{3}{2}a+b-\frac{3}{4}b=1-\frac{21}{2}$$

Tāpiri \frac{3a}{2} ki te -\frac{3a}{2}. Ka whakakore atu ngā kupu \frac{3a}{2} me -\frac{3a}{2}, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$b-\frac{3}{4}b=1-\frac{21}{2}$$

Tāpiri b ki te -\frac{3b}{4}.

$$\frac{1}{4}b=1-\frac{21}{2}$$

Tāpiri 1 ki te -\frac{21}{2}.

$$\frac{1}{4}b=-\frac{19}{2}$$

Me whakarea ngā taha e rua ki te 4.

$$b=-38$$

Whakaurua te -38 mō b ki a+\frac{1}{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.

$$a+\frac{1}{2}\left(-38\right)=7$$

Whakareatia \frac{1}{2} ki te -38.

$$a-19=7$$

Me tāpiri 19 ki ngā taha e rua o te whārite.

$$a=26$$

Kua oti te pūnaha te whakatau.

$$a=26,b=-38$$