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

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

Whakaoti mō x, y

$$x=3$$

$$y=3$$

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.

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

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

$$-x-y=-6$$

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

$$-x=y-6$$

Whakawehea ngā taha e rua ki te -1.

$$x=-\left(y-6\right)$$

Whakareatia -1 ki te y-6.

$$x=-y+6$$

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

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

Whakareatia 2 ki te -y+6.

$$-2y+12-3y=-3$$

Tāpiri -2y ki te -3y.

$$-5y+12=-3$$

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

$$-5y=-15$$

Whakawehea ngā taha e rua ki te -5.

$$y=3$$

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

$$x=-3+6$$

Tāpiri 6 ki te -3.

$$x=3$$

Kua oti te pūnaha te whakatau.

$$x=3,y=3$$

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.

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}-1&-1\\2&-3\end{matrix}\right).

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=3,y=3$$

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.

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

Kia ōrite ai a -x 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 -1.

$$2\left(-1\right)x+2\left(-1\right)y=2\left(-6\right),-2x-\left(-3y\right)=-\left(-3\right)$$

Whakarūnātia.

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

Me tango -2x+3y=3 mai i -2x-2y=-12 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-2x+2x-2y-3y=-12-3$$

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.

$$-2y-3y=-12-3$$

Tāpiri -2y ki te -3y.

$$-5y=-12-3$$

Tāpiri -12 ki te -3.

$$-5y=-15$$

Whakawehea ngā taha e rua ki te -5.

$$y=3$$

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

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

Whakareatia -3 ki te 3.

$$2x-9=-3$$

Me tāpiri 9 ki 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.

$$x=3,y=3$$