Whakaoti i te {l}{-x-2y=9}{3x-2y=21}

$$\left. \begin{array} { l } { - x - 2 y = 9 } \\ { 3 x - 2 y = 21 } \end{array} \right.$$

Whakaoti mō x, y

$$x=3$$

$$y=-6$$

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.

$$-x-2y=9,3x-2y=21$$

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

$$-x-2y=9$$

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

$$-x=2y+9$$

Whakawehea ngā taha e rua ki te -1.

$$x=-\left(2y+9\right)$$

Whakareatia -1 ki te 2y+9.

$$x=-2y-9$$

Whakakapia te -2y-9 mō te x ki tērā atu whārite, 3x-2y=21.

$$3\left(-2y-9\right)-2y=21$$

Whakareatia 3 ki te -2y-9.

$$-6y-27-2y=21$$

Tāpiri -6y ki te -2y.

$$-8y-27=21$$

Me tāpiri 27 ki ngā taha e rua o te whārite.

$$-8y=48$$

Whakawehea ngā taha e rua ki te -8.

$$y=-6$$

Whakaurua te -6 mō y ki x=-2y-9. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-2\left(-6\right)-9$$

Whakareatia -2 ki te -6.

$$x=12-9$$

Tāpiri -9 ki te 12.

$$x=3$$

Kua oti te pūnaha te whakatau.

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

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.

$$-x-2y=9,3x-2y=21$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-1&-2\\3&-2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}9\\21\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}-1&-2\\3&-2\end{matrix}\right))\left(\begin{matrix}-1&-2\\3&-2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-1&-2\\3&-2\end{matrix}\right))\left(\begin{matrix}9\\21\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-1&-2\\3&-2\end{matrix}\right))\left(\begin{matrix}9\\21\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-1&-2\\3&-2\end{matrix}\right))\left(\begin{matrix}9\\21\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{-\left(-2\right)-\left(-2\times 3\right)}&-\frac{-2}{-\left(-2\right)-\left(-2\times 3\right)}\\-\frac{3}{-\left(-2\right)-\left(-2\times 3\right)}&-\frac{1}{-\left(-2\right)-\left(-2\times 3\right)}\end{matrix}\right)\left(\begin{matrix}9\\21\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{4}&\frac{1}{4}\\-\frac{3}{8}&-\frac{1}{8}\end{matrix}\right)\left(\begin{matrix}9\\21\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{4}\times 9+\frac{1}{4}\times 21\\-\frac{3}{8}\times 9-\frac{1}{8}\times 21\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

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

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.

$$-x-2y=9,3x-2y=21$$

Me tango 3x-2y=21 mai i -x-2y=9 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-x-3x-2y+2y=9-21$$

Tāpiri -2y ki te 2y. Ka whakakore atu ngā kupu -2y me 2y, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-x-3x=9-21$$

Tāpiri -x ki te -3x.

$$-4x=9-21$$

Tāpiri 9 ki te -21.

$$-4x=-12$$

Whakawehea ngā taha e rua ki te -4.

$$x=3$$

Whakaurua te 3 mō x ki 3x-2y=21. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$3\times 3-2y=21$$

Whakareatia 3 ki te 3.

$$9-2y=21$$

Me tango 9 mai i ngā taha e rua o te whārite.

$$-2y=12$$

Whakawehea ngā taha e rua ki te -2.

$$y=-6$$

Kua oti te pūnaha te whakatau.

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