Whakaoti i te {r}{9x-6y=3}{-3x-6y=15}

$$\left\{ \begin{array} { r } { 9 x - 6 y = 3 } \\ { - 3 x - 6 y = 15 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-1$$

$$y=-2$$

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.

$$9x-6y=3,-3x-6y=15$$

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

$$9x-6y=3$$

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

$$9x=6y+3$$

Whakawehea ngā taha e rua ki te 9.

$$x=\frac{1}{9}\left(6y+3\right)$$

Whakareatia \frac{1}{9} ki te 6y+3.

$$x=\frac{2}{3}y+\frac{1}{3}$$

Whakakapia te \frac{2y+1}{3} mō te x ki tērā atu whārite, -3x-6y=15.

$$-3\left(\frac{2}{3}y+\frac{1}{3}\right)-6y=15$$

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

$$-2y-1-6y=15$$

Tāpiri -2y ki te -6y.

$$-8y-1=15$$

Me tāpiri 1 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{1}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

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

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

$$x=\frac{-4+1}{3}$$

Tāpiri \frac{1}{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.

$$x=-1$$

Kua oti te pūnaha te whakatau.

$$x=-1,y=-2$$

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.

$$9x-6y=3,-3x-6y=15$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}9&-6\\-3&-6\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}3\\15\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}9&-6\\-3&-6\end{matrix}\right))\left(\begin{matrix}9&-6\\-3&-6\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}9&-6\\-3&-6\end{matrix}\right))\left(\begin{matrix}3\\15\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}9&-6\\-3&-6\end{matrix}\right))\left(\begin{matrix}3\\15\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}9&-6\\-3&-6\end{matrix}\right))\left(\begin{matrix}3\\15\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{6}{9\left(-6\right)-\left(-6\left(-3\right)\right)}&-\frac{-6}{9\left(-6\right)-\left(-6\left(-3\right)\right)}\\-\frac{-3}{9\left(-6\right)-\left(-6\left(-3\right)\right)}&\frac{9}{9\left(-6\right)-\left(-6\left(-3\right)\right)}\end{matrix}\right)\left(\begin{matrix}3\\15\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{12}&-\frac{1}{12}\\-\frac{1}{24}&-\frac{1}{8}\end{matrix}\right)\left(\begin{matrix}3\\15\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{12}\times 3-\frac{1}{12}\times 15\\-\frac{1}{24}\times 3-\frac{1}{8}\times 15\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=-1,y=-2$$

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.

$$9x-6y=3,-3x-6y=15$$

Me tango -3x-6y=15 mai i 9x-6y=3 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$9x+3x-6y+6y=3-15$$

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.

$$9x+3x=3-15$$

Tāpiri 9x ki te 3x.

$$12x=3-15$$

Tāpiri 3 ki te -15.

$$12x=-12$$

Whakawehea ngā taha e rua ki te 12.

$$x=-1$$

Whakaurua te -1 mō x ki -3x-6y=15. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$-3\left(-1\right)-6y=15$$

Whakareatia -3 ki te -1.

$$3-6y=15$$

Me tango 3 mai i 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.

$$x=-1,y=-2$$