Whakaoti i te {l}{2x-6y=16}{-x+2y=-4}

$$\left. \begin{array} { l } { 2 x - 6 y = 16 } \\ { - x + 2 y = - 4 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-4$$

$$y=-4$$

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.

$$2x-6y=16,-x+2y=-4$$

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

$$2x-6y=16$$

Me tāpiri 6y ki ngā taha e rua o te whārite.

$$2x=6y+16$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{1}{2}\left(6y+16\right)$$

Whakareatia \frac{1}{2} ki te 6y+16.

$$x=3y+8$$

Whakakapia te 3y+8 mō te x ki tērā atu whārite, -x+2y=-4.

$$-\left(3y+8\right)+2y=-4$$

Whakareatia -1 ki te 3y+8.

$$-3y-8+2y=-4$$

Tāpiri -3y ki te 2y.

$$-y-8=-4$$

Me tāpiri 8 ki ngā taha e rua o te whārite.

$$-y=4$$

Whakawehea ngā taha e rua ki te -1.

$$y=-4$$

Whakaurua te -4 mō y ki x=3y+8. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=3\left(-4\right)+8$$

Whakareatia 3 ki te -4.

$$x=-12+8$$

Tāpiri 8 ki te -12.

$$x=-4$$

Kua oti te pūnaha te whakatau.

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

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.

$$2x-6y=16,-x+2y=-4$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&-6\\-1&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}16\\-4\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}2&-6\\-1&2\end{matrix}\right).

$$inverse(\left(\begin{matrix}2&-6\\-1&2\end{matrix}\right))\left(\begin{matrix}2&-6\\-1&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-6\\-1&2\end{matrix}\right))\left(\begin{matrix}16\\-4\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-6\\-1&2\end{matrix}\right))\left(\begin{matrix}16\\-4\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-6\\-1&2\end{matrix}\right))\left(\begin{matrix}16\\-4\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{2\times 2-\left(-6\left(-1\right)\right)}&-\frac{-6}{2\times 2-\left(-6\left(-1\right)\right)}\\-\frac{-1}{2\times 2-\left(-6\left(-1\right)\right)}&\frac{2}{2\times 2-\left(-6\left(-1\right)\right)}\end{matrix}\right)\left(\begin{matrix}16\\-4\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-1&-3\\-\frac{1}{2}&-1\end{matrix}\right)\left(\begin{matrix}16\\-4\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-16-3\left(-4\right)\\-\frac{1}{2}\times 16-\left(-4\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

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

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.

$$2x-6y=16,-x+2y=-4$$

Kia ōrite ai a 2x me -x, 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 2.

$$-2x-\left(-6y\right)=-16,2\left(-1\right)x+2\times 2y=2\left(-4\right)$$

Whakarūnātia.

$$-2x+6y=-16,-2x+4y=-8$$

Me tango -2x+4y=-8 mai i -2x+6y=-16 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-2x+2x+6y-4y=-16+8$$

Tāpiri -2x ki te 2x. Ka whakakore atu ngā kupu -2x me 2x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$6y-4y=-16+8$$

Tāpiri 6y ki te -4y.

$$2y=-16+8$$

Tāpiri -16 ki te 8.

$$2y=-8$$

Whakawehea ngā taha e rua ki te 2.

$$y=-4$$

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

$$-x+2\left(-4\right)=-4$$

Whakareatia 2 ki te -4.

$$-x-8=-4$$

Me tāpiri 8 ki ngā taha e rua o te whārite.

$$-x=4$$

Whakawehea ngā taha e rua ki te -1.

$$x=-4$$

Kua oti te pūnaha te whakatau.

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