Whakaoti i te {l}{-6x-4y=2}{2x+8y=26}

$$\left\{ \begin{array} { l } { - 6 x - 4 y = 2 } \\ { 2 x + 8 y = 26 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-3$$

$$y=4$$

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.

$$-6x-4y=2,2x+8y=26$$

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

$$-6x-4y=2$$

Me tāpiri 4y ki ngā taha e rua o te whārite.

$$-6x=4y+2$$

Whakawehea ngā taha e rua ki te -6.

$$x=-\frac{1}{6}\left(4y+2\right)$$

Whakareatia -\frac{1}{6} ki te 4y+2.

$$x=-\frac{2}{3}y-\frac{1}{3}$$

Whakakapia te \frac{-2y-1}{3} mō te x ki tērā atu whārite, 2x+8y=26.

$$2\left(-\frac{2}{3}y-\frac{1}{3}\right)+8y=26$$

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

$$-\frac{4}{3}y-\frac{2}{3}+8y=26$$

Tāpiri -\frac{4y}{3} ki te 8y.

$$\frac{20}{3}y-\frac{2}{3}=26$$

Me tāpiri \frac{2}{3} ki ngā taha e rua o te whārite.

$$\frac{20}{3}y=\frac{80}{3}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{20}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=4$$

Whakaurua te 4 mō y ki x=-\frac{2}{3}y-\frac{1}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{2}{3}\times 4-\frac{1}{3}$$

Whakareatia -\frac{2}{3} ki te 4.

$$x=\frac{-8-1}{3}$$

Tāpiri -\frac{1}{3} ki te -\frac{8}{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.

$$x=-3$$

Kua oti te pūnaha te whakatau.

$$x=-3,y=4$$

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.

$$-6x-4y=2,2x+8y=26$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-6&-4\\2&8\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}2\\26\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}-6&-4\\2&8\end{matrix}\right).

$$inverse(\left(\begin{matrix}-6&-4\\2&8\end{matrix}\right))\left(\begin{matrix}-6&-4\\2&8\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-6&-4\\2&8\end{matrix}\right))\left(\begin{matrix}2\\26\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-6&-4\\2&8\end{matrix}\right))\left(\begin{matrix}2\\26\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-6&-4\\2&8\end{matrix}\right))\left(\begin{matrix}2\\26\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{8}{-6\times 8-\left(-4\times 2\right)}&-\frac{-4}{-6\times 8-\left(-4\times 2\right)}\\-\frac{2}{-6\times 8-\left(-4\times 2\right)}&-\frac{6}{-6\times 8-\left(-4\times 2\right)}\end{matrix}\right)\left(\begin{matrix}2\\26\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{5}&-\frac{1}{10}\\\frac{1}{20}&\frac{3}{20}\end{matrix}\right)\left(\begin{matrix}2\\26\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{5}\times 2-\frac{1}{10}\times 26\\\frac{1}{20}\times 2+\frac{3}{20}\times 26\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-3\\4\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-3,y=4$$

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.

$$-6x-4y=2,2x+8y=26$$

Kia ōrite ai a -6x me 2x, 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 -6.

$$2\left(-6\right)x+2\left(-4\right)y=2\times 2,-6\times 2x-6\times 8y=-6\times 26$$

Whakarūnātia.

$$-12x-8y=4,-12x-48y=-156$$

Me tango -12x-48y=-156 mai i -12x-8y=4 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-12x+12x-8y+48y=4+156$$

Tāpiri -12x ki te 12x. Ka whakakore atu ngā kupu -12x me 12x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-8y+48y=4+156$$

Tāpiri -8y ki te 48y.

$$40y=4+156$$

Tāpiri 4 ki te 156.

$$40y=160$$

Whakawehea ngā taha e rua ki te 40.

$$y=4$$

Whakaurua te 4 mō y ki 2x+8y=26. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$2x+8\times 4=26$$

Whakareatia 8 ki te 4.

$$2x+32=26$$

Me tango 32 mai i ngā taha e rua o te whārite.

$$2x=-6$$

Whakawehea ngā taha e rua ki te 2.

$$x=-3$$

Kua oti te pūnaha te whakatau.

$$x=-3,y=4$$