Whakaoti i te {c}{-2x-4y=-12}{2x+3y=9}

$$\left\{ \begin{array} { c } { - 2 x - 4 y = - 12 } \\ { 2 x + 3 y = 9 } \end{array} \right.$$

Whakaoti mō x, y

$$x=0$$

$$y=3$$

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.

$$-2x-4y=-12,2x+3y=9$$

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

$$-2x-4y=-12$$

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

$$-2x=4y-12$$

Whakawehea ngā taha e rua ki te -2.

$$x=-\frac{1}{2}\left(4y-12\right)$$

Whakareatia -\frac{1}{2} ki te -12+4y.

$$x=-2y+6$$

Whakakapia te -2y+6 mō te x ki tērā atu whārite, 2x+3y=9.

$$2\left(-2y+6\right)+3y=9$$

Whakareatia 2 ki te -2y+6.

$$-4y+12+3y=9$$

Tāpiri -4y ki te 3y.

$$-y+12=9$$

Me tango 12 mai i ngā taha e rua o te whārite.

$$-y=-3$$

Whakawehea ngā taha e rua ki te -1.

$$y=3$$

Whakaurua te 3 mō y ki x=-2y+6. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-2\times 3+6$$

Whakareatia -2 ki te 3.

$$x=-6+6$$

Tāpiri 6 ki te -6.

$$x=0$$

Kua oti te pūnaha te whakatau.

$$x=0,y=3$$

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.

$$-2x-4y=-12,2x+3y=9$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-2&-4\\2&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-12\\9\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}-2&-4\\2&3\end{matrix}\right))\left(\begin{matrix}-2&-4\\2&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-2&-4\\2&3\end{matrix}\right))\left(\begin{matrix}-12\\9\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-2&-4\\2&3\end{matrix}\right))\left(\begin{matrix}-12\\9\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-2&-4\\2&3\end{matrix}\right))\left(\begin{matrix}-12\\9\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{-2\times 3-\left(-4\times 2\right)}&-\frac{-4}{-2\times 3-\left(-4\times 2\right)}\\-\frac{2}{-2\times 3-\left(-4\times 2\right)}&-\frac{2}{-2\times 3-\left(-4\times 2\right)}\end{matrix}\right)\left(\begin{matrix}-12\\9\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{2}&2\\-1&-1\end{matrix}\right)\left(\begin{matrix}-12\\9\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{2}\left(-12\right)+2\times 9\\-\left(-12\right)-9\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}0\\3\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=0,y=3$$

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.

$$-2x-4y=-12,2x+3y=9$$

Kia ōrite ai a -2x 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 -2.

$$2\left(-2\right)x+2\left(-4\right)y=2\left(-12\right),-2\times 2x-2\times 3y=-2\times 9$$

Whakarūnātia.

$$-4x-8y=-24,-4x-6y=-18$$

Me tango -4x-6y=-18 mai i -4x-8y=-24 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-4x+4x-8y+6y=-24+18$$

Tāpiri -4x ki te 4x. Ka whakakore atu ngā kupu -4x me 4x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-8y+6y=-24+18$$

Tāpiri -8y ki te 6y.

$$-2y=-24+18$$

Tāpiri -24 ki te 18.

$$-2y=-6$$

Whakawehea ngā taha e rua ki te -2.

$$y=3$$

Whakaurua te 3 mō y ki 2x+3y=9. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$2x+3\times 3=9$$

Whakareatia 3 ki te 3.

$$2x+9=9$$

Me tango 9 mai i ngā taha e rua o te whārite.

$$2x=0$$

Whakawehea ngā taha e rua ki te 2.

$$x=0$$

Kua oti te pūnaha te whakatau.

$$x=0,y=3$$