Whakaoti i te {l}{9x-3y=27}{9x-10y=27}

$$\left. \begin{array} { l } { 9 x - 3 y = 27 } \\ { 9 x - 10 y = 27 } \end{array} \right.$$

Whakaoti mō x, y

$$x=3$$

$$y=0$$

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.

$$9x-3y=27,9x-10y=27$$

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

$$9x-3y=27$$

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

$$9x=3y+27$$

Whakawehea ngā taha e rua ki te 9.

$$x=\frac{1}{9}\left(3y+27\right)$$

Whakareatia \frac{1}{9} ki te 27+3y.

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

Whakakapia te \frac{y}{3}+3 mō te x ki tērā atu whārite, 9x-10y=27.

$$9\left(\frac{1}{3}y+3\right)-10y=27$$

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

$$3y+27-10y=27$$

Tāpiri 3y ki te -10y.

$$-7y+27=27$$

Me tango 27 mai i ngā taha e rua o te whārite.

$$-7y=0$$

Whakawehea ngā taha e rua ki te -7.

$$y=0$$

Whakaurua te 0 mō y ki x=\frac{1}{3}y+3. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=3$$

Kua oti te pūnaha te whakatau.

$$x=3,y=0$$

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.

$$9x-3y=27,9x-10y=27$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}9&-3\\9&-10\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}27\\27\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}9&-3\\9&-10\end{matrix}\right).

$$inverse(\left(\begin{matrix}9&-3\\9&-10\end{matrix}\right))\left(\begin{matrix}9&-3\\9&-10\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}9&-3\\9&-10\end{matrix}\right))\left(\begin{matrix}27\\27\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}9&-3\\9&-10\end{matrix}\right))\left(\begin{matrix}27\\27\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}9&-3\\9&-10\end{matrix}\right))\left(\begin{matrix}27\\27\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{10}{9\left(-10\right)-\left(-3\times 9\right)}&-\frac{-3}{9\left(-10\right)-\left(-3\times 9\right)}\\-\frac{9}{9\left(-10\right)-\left(-3\times 9\right)}&\frac{9}{9\left(-10\right)-\left(-3\times 9\right)}\end{matrix}\right)\left(\begin{matrix}27\\27\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{10}{63}&-\frac{1}{21}\\\frac{1}{7}&-\frac{1}{7}\end{matrix}\right)\left(\begin{matrix}27\\27\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{10}{63}\times 27-\frac{1}{21}\times 27\\\frac{1}{7}\times 27-\frac{1}{7}\times 27\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}3\\0\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=3,y=0$$

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.

$$9x-3y=27,9x-10y=27$$

Me tango 9x-10y=27 mai i 9x-3y=27 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$9x-9x-3y+10y=27-27$$

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.

$$-3y+10y=27-27$$

Tāpiri -3y ki te 10y.

$$7y=27-27$$

Tāpiri 27 ki te -27.

$$7y=0$$

Whakawehea ngā taha e rua ki te 7.

$$y=0$$

Whakaurua te 0 mō y ki 9x-10y=27. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$9x=27$$

Whakawehea ngā taha e rua ki te 9.

$$x=3$$

Kua oti te pūnaha te whakatau.

$$x=3,y=0$$