Whakaoti i te {l}{3c+2x=5}{2c+4x=6}

$$\left. \begin{array} { l } { 3 c + 2 x = 5 } \\ { 2 c + 4 x = 6 } \end{array} \right.$$

Whakaoti mō c, x

$$x=1$$

$$c=1$$

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.

$$3c+2x=5,2c+4x=6$$

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

$$3c+2x=5$$

Me tango 2x mai i ngā taha e rua o te whārite.

$$3c=-2x+5$$

Whakawehea ngā taha e rua ki te 3.

$$c=\frac{1}{3}\left(-2x+5\right)$$

Whakareatia \frac{1}{3} ki te -2x+5.

$$c=-\frac{2}{3}x+\frac{5}{3}$$

Whakakapia te \frac{-2x+5}{3} mō te c ki tērā atu whārite, 2c+4x=6.

$$2\left(-\frac{2}{3}x+\frac{5}{3}\right)+4x=6$$

Whakareatia 2 ki te \frac{-2x+5}{3}.

$$-\frac{4}{3}x+\frac{10}{3}+4x=6$$

Tāpiri -\frac{4x}{3} ki te 4x.

$$\frac{8}{3}x+\frac{10}{3}=6$$

Me tango \frac{10}{3} mai i ngā taha e rua o te whārite.

$$\frac{8}{3}x=\frac{8}{3}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{8}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$x=1$$

Whakaurua te 1 mō x ki c=-\frac{2}{3}x+\frac{5}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō c hāngai tonu.

$$c=\frac{-2+5}{3}$$

Tāpiri \frac{5}{3} ki te -\frac{2}{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.

$$c=1$$

Kua oti te pūnaha te whakatau.

$$c=1,x=1$$

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.

$$3c+2x=5,2c+4x=6$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&2\\2&4\end{matrix}\right)\left(\begin{matrix}c\\x\end{matrix}\right)=\left(\begin{matrix}5\\6\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}3&2\\2&4\end{matrix}\right))\left(\begin{matrix}3&2\\2&4\end{matrix}\right)\left(\begin{matrix}c\\x\end{matrix}\right)=inverse(\left(\begin{matrix}3&2\\2&4\end{matrix}\right))\left(\begin{matrix}5\\6\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}c\\x\end{matrix}\right)=inverse(\left(\begin{matrix}3&2\\2&4\end{matrix}\right))\left(\begin{matrix}5\\6\end{matrix}\right)$$

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

$$\left(\begin{matrix}c\\x\end{matrix}\right)=inverse(\left(\begin{matrix}3&2\\2&4\end{matrix}\right))\left(\begin{matrix}5\\6\end{matrix}\right)$$

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.

$$\left(\begin{matrix}c\\x\end{matrix}\right)=\left(\begin{matrix}\frac{4}{3\times 4-2\times 2}&-\frac{2}{3\times 4-2\times 2}\\-\frac{2}{3\times 4-2\times 2}&\frac{3}{3\times 4-2\times 2}\end{matrix}\right)\left(\begin{matrix}5\\6\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}c\\x\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2}&-\frac{1}{4}\\-\frac{1}{4}&\frac{3}{8}\end{matrix}\right)\left(\begin{matrix}5\\6\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}c\\x\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2}\times 5-\frac{1}{4}\times 6\\-\frac{1}{4}\times 5+\frac{3}{8}\times 6\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}c\\x\end{matrix}\right)=\left(\begin{matrix}1\\1\end{matrix}\right)$$

Tangohia ngā huānga poukapa c me x.

$$c=1,x=1$$

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.

$$3c+2x=5,2c+4x=6$$

Kia ōrite ai a 3c me 2c, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 2 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 3.

$$2\times 3c+2\times 2x=2\times 5,3\times 2c+3\times 4x=3\times 6$$

Whakarūnātia.

$$6c+4x=10,6c+12x=18$$

Me tango 6c+12x=18 mai i 6c+4x=10 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$6c-6c+4x-12x=10-18$$

Tāpiri 6c ki te -6c. Ka whakakore atu ngā kupu 6c me -6c, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$4x-12x=10-18$$

Tāpiri 4x ki te -12x.

$$-8x=10-18$$

Tāpiri 10 ki te -18.

$$-8x=-8$$

Whakawehea ngā taha e rua ki te -8.

$$x=1$$

Whakaurua te 1 mō x ki 2c+4x=6. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō c hāngai tonu.

$$2c+4=6$$

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

$$2c=2$$

Whakawehea ngā taha e rua ki te 2.

$$c=1$$

Kua oti te pūnaha te whakatau.

$$c=1,x=1$$