Whakaoti i te {l}{3*a+c=5}{a-c=7}

$$\left. \begin{array} { l } { 3 \cdot a + c = 5 } \\ { a - c = 7 } \end{array} \right.$$

Whakaoti mō a, c

$$a=3$$

$$c=-4$$

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.

$$3a+c=5,a-c=7$$

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

$$3a+c=5$$

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

$$3a=-c+5$$

Whakawehea ngā taha e rua ki te 3.

$$a=\frac{1}{3}\left(-c+5\right)$$

Whakareatia \frac{1}{3} ki te -c+5.

$$a=-\frac{1}{3}c+\frac{5}{3}$$

Whakakapia te \frac{-c+5}{3} mō te a ki tērā atu whārite, a-c=7.

$$-\frac{1}{3}c+\frac{5}{3}-c=7$$

Tāpiri -\frac{c}{3} ki te -c.

$$-\frac{4}{3}c+\frac{5}{3}=7$$

Me tango \frac{5}{3} mai i ngā taha e rua o te whārite.

$$-\frac{4}{3}c=\frac{16}{3}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{4}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$c=-4$$

Whakaurua te -4 mō c ki a=-\frac{1}{3}c+\frac{5}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō a hāngai tonu.

$$a=-\frac{1}{3}\left(-4\right)+\frac{5}{3}$$

Whakareatia -\frac{1}{3} ki te -4.

$$a=\frac{4+5}{3}$$

Tāpiri \frac{5}{3} ki te \frac{4}{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.

$$a=3$$

Kua oti te pūnaha te whakatau.

$$a=3,c=-4$$

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.

$$3a+c=5,a-c=7$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&1\\1&-1\end{matrix}\right)\left(\begin{matrix}a\\c\end{matrix}\right)=\left(\begin{matrix}5\\7\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}3&1\\1&-1\end{matrix}\right))\left(\begin{matrix}3&1\\1&-1\end{matrix}\right)\left(\begin{matrix}a\\c\end{matrix}\right)=inverse(\left(\begin{matrix}3&1\\1&-1\end{matrix}\right))\left(\begin{matrix}5\\7\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}a\\c\end{matrix}\right)=inverse(\left(\begin{matrix}3&1\\1&-1\end{matrix}\right))\left(\begin{matrix}5\\7\end{matrix}\right)$$

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

$$\left(\begin{matrix}a\\c\end{matrix}\right)=inverse(\left(\begin{matrix}3&1\\1&-1\end{matrix}\right))\left(\begin{matrix}5\\7\end{matrix}\right)$$

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.

$$\left(\begin{matrix}a\\c\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{3\left(-1\right)-1}&-\frac{1}{3\left(-1\right)-1}\\-\frac{1}{3\left(-1\right)-1}&\frac{3}{3\left(-1\right)-1}\end{matrix}\right)\left(\begin{matrix}5\\7\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}a\\c\end{matrix}\right)=\left(\begin{matrix}\frac{1}{4}&\frac{1}{4}\\\frac{1}{4}&-\frac{3}{4}\end{matrix}\right)\left(\begin{matrix}5\\7\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}a\\c\end{matrix}\right)=\left(\begin{matrix}\frac{1}{4}\times 5+\frac{1}{4}\times 7\\\frac{1}{4}\times 5-\frac{3}{4}\times 7\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}a\\c\end{matrix}\right)=\left(\begin{matrix}3\\-4\end{matrix}\right)$$

Tangohia ngā huānga poukapa a me c.

$$a=3,c=-4$$

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.

$$3a+c=5,a-c=7$$

Kia ōrite ai a 3a 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 3.

$$3a+c=5,3a+3\left(-1\right)c=3\times 7$$

Whakarūnātia.

$$3a+c=5,3a-3c=21$$

Me tango 3a-3c=21 mai i 3a+c=5 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$3a-3a+c+3c=5-21$$

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

$$c+3c=5-21$$

Tāpiri c ki te 3c.

$$4c=5-21$$

Tāpiri 5 ki te -21.

$$4c=-16$$

Whakawehea ngā taha e rua ki te 4.

$$c=-4$$

Whakaurua te -4 mō c ki a-c=7. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō a hāngai tonu.

$$a-\left(-4\right)=7$$

Me tango 4 mai i ngā taha e rua o te whārite.

$$a=3$$

Kua oti te pūnaha te whakatau.

$$a=3,c=-4$$