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

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

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.

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

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

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

Me tāpiri 2y ki ngā taha e rua o te whārite.

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

Whakawehea ngā taha e rua ki te -3.

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

Whakareatia -\frac{1}{3} ki te 6+2y.

$$x=-\frac{2}{3}y-2$$

Whakakapia te -\frac{2y}{3}-2 mō te x ki tērā atu whārite, 3x+3y=-9.

$$3\left(-\frac{2}{3}y-2\right)+3y=-9$$

Whakareatia 3 ki te -\frac{2y}{3}-2.

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

Tāpiri -2y ki te 3y.

$$y-6=-9$$

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

$$y=-3$$

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

$$x=-\frac{2}{3}\left(-3\right)-2$$

Whakareatia -\frac{2}{3} ki te -3.

$$x=2-2$$

Tāpiri -2 ki te 2.

$$x=0$$

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.

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

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.

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

Kia ōrite ai a -3x me 3x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 3 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te -3.

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

Whakarūnātia.

$$-9x-6y=18,-9x-9y=27$$

Me tango -9x-9y=27 mai i -9x-6y=18 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-9x+9x-6y+9y=18-27$$

Tāpiri -9x ki te 9x. Ka whakakore atu ngā kupu -9x me 9x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-6y+9y=18-27$$

Tāpiri -6y ki te 9y.

$$3y=18-27$$

Tāpiri 18 ki te -27.

$$3y=-9$$

Whakawehea ngā taha e rua ki te 3.

$$y=-3$$

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

$$3x+3\left(-3\right)=-9$$

Whakareatia 3 ki te -3.

$$3x-9=-9$$

Me tāpiri 9 ki ngā taha e rua o te whārite.

$$3x=0$$

Whakawehea ngā taha e rua ki te 3.

$$x=0$$

Kua oti te pūnaha te whakatau.

$$x=0,y=-3$$