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

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

Whakaoti mō x, y

$$x=9$$

$$y=-5$$

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.

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

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

$$-x-3y=6$$

Me tāpiri 3y ki ngā taha e rua o te whārite.

$$-x=3y+6$$

Whakawehea ngā taha e rua ki te -1.

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

Whakareatia -1 ki te 6+3y.

$$x=-3y-6$$

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

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

Whakareatia 2 ki te -3y-6.

$$-6y-12+3y=3$$

Tāpiri -6y ki te 3y.

$$-3y-12=3$$

Me tāpiri 12 ki ngā taha e rua o te whārite.

$$-3y=15$$

Whakawehea ngā taha e rua ki te -3.

$$y=-5$$

Whakaurua te -5 mō y ki x=-3y-6. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-3\left(-5\right)-6$$

Whakareatia -3 ki te -5.

$$x=15-6$$

Tāpiri -6 ki te 15.

$$x=9$$

Kua oti te pūnaha te whakatau.

$$x=9,y=-5$$

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.

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}9\\-5\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=9,y=-5$$

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.

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

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.

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

Whakarūnātia.

$$-2x-6y=12,-2x-3y=-3$$

Me tango -2x-3y=-3 mai i -2x-6y=12 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-2x+2x-6y+3y=12+3$$

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.

$$-6y+3y=12+3$$

Tāpiri -6y ki te 3y.

$$-3y=12+3$$

Tāpiri 12 ki te 3.

$$-3y=15$$

Whakawehea ngā taha e rua ki te -3.

$$y=-5$$

Whakaurua te -5 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.

$$2x+3\left(-5\right)=3$$

Whakareatia 3 ki te -5.

$$2x-15=3$$

Me tāpiri 15 ki ngā taha e rua o te whārite.

$$2x=18$$

Whakawehea ngā taha e rua ki te 2.

$$x=9$$

Kua oti te pūnaha te whakatau.

$$x=9,y=-5$$