Whakaoti i te {l}{3y=-9x+9}{-2x+2y=6}

$$\left\{ \begin{array} { l } { 3 y = - 9 x + 9 } \\ { - 2 x + 2 y = 6 } \end{array} \right.$$

Whakaoti mō y, x

$$x=0$$

$$y=3$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Me tāpiri te 9x ki ngā taha e rua.

$$3y+9x=9$$

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.

$$3y+9x=9,2y-2x=6$$

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

$$3y+9x=9$$

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

$$3y=-9x+9$$

Whakawehea ngā taha e rua ki te 3.

$$y=\frac{1}{3}\left(-9x+9\right)$$

Whakareatia \frac{1}{3} ki te -9x+9.

$$y=-3x+3$$

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

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

Whakareatia 2 ki te -3x+3.

$$-6x+6-2x=6$$

Tāpiri -6x ki te -2x.

$$-8x+6=6$$

Me tango 6 mai i ngā taha e rua o te whārite.

$$-8x=0$$

Whakawehea ngā taha e rua ki te -8.

$$x=0$$

Whakaurua te 0 mō x ki y=-3x+3. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y=3$$

Kua oti te pūnaha te whakatau.

$$y=3,x=0$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Me tāpiri te 9x ki ngā taha e rua.

$$3y+9x=9$$

Tuhia ngā whārite ki te tānga ngahuru ka whakamahi i ngā poukapa hei whakaoti i te pūnaha o ngā whārite.

$$3y+9x=9,2y-2x=6$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&9\\2&-2\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}9\\6\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}3&9\\2&-2\end{matrix}\right).

$$inverse(\left(\begin{matrix}3&9\\2&-2\end{matrix}\right))\left(\begin{matrix}3&9\\2&-2\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}3&9\\2&-2\end{matrix}\right))\left(\begin{matrix}9\\6\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}3&9\\2&-2\end{matrix}\right))\left(\begin{matrix}9\\6\end{matrix}\right)$$

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

$$\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}3&9\\2&-2\end{matrix}\right))\left(\begin{matrix}9\\6\end{matrix}\right)$$

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.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{3\left(-2\right)-9\times 2}&-\frac{9}{3\left(-2\right)-9\times 2}\\-\frac{2}{3\left(-2\right)-9\times 2}&\frac{3}{3\left(-2\right)-9\times 2}\end{matrix}\right)\left(\begin{matrix}9\\6\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{1}{12}&\frac{3}{8}\\\frac{1}{12}&-\frac{1}{8}\end{matrix}\right)\left(\begin{matrix}9\\6\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{1}{12}\times 9+\frac{3}{8}\times 6\\\frac{1}{12}\times 9-\frac{1}{8}\times 6\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa y me x.

$$y=3,x=0$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Me tāpiri te 9x ki ngā taha e rua.

$$3y+9x=9$$

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.

$$3y+9x=9,2y-2x=6$$

Kia ōrite ai a 3y me 2y, 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.

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

Whakarūnātia.

$$6y+18x=18,6y-6x=18$$

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

$$6y-6y+18x+6x=18-18$$

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

$$18x+6x=18-18$$

Tāpiri 18x ki te 6x.

$$24x=18-18$$

Tāpiri 18 ki te -18.

$$24x=0$$

Whakawehea ngā taha e rua ki te 24.

$$x=0$$

Whakaurua te 0 mō x ki 2y-2x=6. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$2y=6$$

Whakawehea ngā taha e rua ki te 2.

$$y=3$$

Kua oti te pūnaha te whakatau.

$$y=3,x=0$$