Whakaoti i te {l}{-9x-6y=6}{3x-6y=-18}

$$\left. \begin{array} { l } { - 9 x - 6 y = 6 } \\ { 3 x - 6 y = - 18 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-2$$

$$y=2$$

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.

$$-9x-6y=6,3x-6y=-18$$

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

$$-9x-6y=6$$

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

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

Whakawehea ngā taha e rua ki te -9.

$$x=-\frac{1}{9}\left(6y+6\right)$$

Whakareatia -\frac{1}{9} ki te 6+6y.

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

Whakakapia te \frac{-2y-2}{3} mō te x ki tērā atu whārite, 3x-6y=-18.

$$3\left(-\frac{2}{3}y-\frac{2}{3}\right)-6y=-18$$

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

$$-2y-2-6y=-18$$

Tāpiri -2y ki te -6y.

$$-8y-2=-18$$

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

$$-8y=-16$$

Whakawehea ngā taha e rua ki te -8.

$$y=2$$

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

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

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

$$x=\frac{-4-2}{3}$$

Tāpiri -\frac{2}{3} ki te -\frac{4}{3} mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=-2$$

Kua oti te pūnaha te whakatau.

$$x=-2,y=2$$

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.

$$-9x-6y=6,3x-6y=-18$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-9&-6\\3&-6\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}6\\-18\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}-9&-6\\3&-6\end{matrix}\right).

$$inverse(\left(\begin{matrix}-9&-6\\3&-6\end{matrix}\right))\left(\begin{matrix}-9&-6\\3&-6\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-9&-6\\3&-6\end{matrix}\right))\left(\begin{matrix}6\\-18\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-9&-6\\3&-6\end{matrix}\right))\left(\begin{matrix}6\\-18\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-9&-6\\3&-6\end{matrix}\right))\left(\begin{matrix}6\\-18\end{matrix}\right)$$

Mō te poukapa 2\times 2 \left(\begin{matrix}a&b\\c&d\end{matrix}\right), ko te poukapa kōaro ko \left(\begin{matrix}\frac{d}{ad-bc}&\frac{-b}{ad-bc}\\\frac{-c}{ad-bc}&\frac{a}{ad-bc}\end{matrix}\right), kia tuhia anō ai te whārite poukapa hei rapanga whakarea poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{6}{-9\left(-6\right)-\left(-6\times 3\right)}&-\frac{-6}{-9\left(-6\right)-\left(-6\times 3\right)}\\-\frac{3}{-9\left(-6\right)-\left(-6\times 3\right)}&-\frac{9}{-9\left(-6\right)-\left(-6\times 3\right)}\end{matrix}\right)\left(\begin{matrix}6\\-18\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{12}&\frac{1}{12}\\-\frac{1}{24}&-\frac{1}{8}\end{matrix}\right)\left(\begin{matrix}6\\-18\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{12}\times 6+\frac{1}{12}\left(-18\right)\\-\frac{1}{24}\times 6-\frac{1}{8}\left(-18\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=-2,y=2$$

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.

$$-9x-6y=6,3x-6y=-18$$

Me tango 3x-6y=-18 mai i -9x-6y=6 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-9x-3x-6y+6y=6+18$$

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.

$$-9x-3x=6+18$$

Tāpiri -9x ki te -3x.

$$-12x=6+18$$

Tāpiri 6 ki te 18.

$$-12x=24$$

Whakawehea ngā taha e rua ki te -12.

$$x=-2$$

Whakaurua te -2 mō x ki 3x-6y=-18. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$3\left(-2\right)-6y=-18$$

Whakareatia 3 ki te -2.

$$-6-6y=-18$$

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

$$-6y=-12$$

Whakawehea ngā taha e rua ki te -6.

$$y=2$$

Kua oti te pūnaha te whakatau.

$$x=-2,y=2$$