Whakaoti i te {l}{2y-2x=-40}{3x+2y=10}

$$\left. \begin{array} { l } { 2 y - 2 x = - 40 } \\ { 3 x + 2 y = 10 } \end{array} \right.$$

Whakaoti mō y, x

$$x=10$$

$$y=-10$$

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.

$$2y-2x=-40,2y+3x=10$$

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

$$2y-2x=-40$$

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

$$2y=2x-40$$

Whakawehea ngā taha e rua ki te 2.

$$y=\frac{1}{2}\left(2x-40\right)$$

Whakareatia \frac{1}{2} ki te -40+2x.

$$y=x-20$$

Whakakapia te x-20 mō te y ki tērā atu whārite, 2y+3x=10.

$$2\left(x-20\right)+3x=10$$

Whakareatia 2 ki te x-20.

$$2x-40+3x=10$$

Tāpiri 2x ki te 3x.

$$5x-40=10$$

Me tāpiri 40 ki ngā taha e rua o te whārite.

$$5x=50$$

Whakawehea ngā taha e rua ki te 5.

$$x=10$$

Whakaurua te 10 mō x ki y=x-20. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y=10-20$$

Tāpiri -20 ki te 10.

$$y=-10$$

Kua oti te pūnaha te whakatau.

$$y=-10,x=10$$

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.

$$2y-2x=-40,2y+3x=10$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&-2\\2&3\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-40\\10\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}2&-2\\2&3\end{matrix}\right))\left(\begin{matrix}2&-2\\2&3\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}2&-2\\2&3\end{matrix}\right))\left(\begin{matrix}-40\\10\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}2&-2\\2&3\end{matrix}\right))\left(\begin{matrix}-40\\10\end{matrix}\right)$$

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

$$\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}2&-2\\2&3\end{matrix}\right))\left(\begin{matrix}-40\\10\end{matrix}\right)$$

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.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{3}{2\times 3-\left(-2\times 2\right)}&-\frac{-2}{2\times 3-\left(-2\times 2\right)}\\-\frac{2}{2\times 3-\left(-2\times 2\right)}&\frac{2}{2\times 3-\left(-2\times 2\right)}\end{matrix}\right)\left(\begin{matrix}-40\\10\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{3}{10}&\frac{1}{5}\\-\frac{1}{5}&\frac{1}{5}\end{matrix}\right)\left(\begin{matrix}-40\\10\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{3}{10}\left(-40\right)+\frac{1}{5}\times 10\\-\frac{1}{5}\left(-40\right)+\frac{1}{5}\times 10\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa y me x.

$$y=-10,x=10$$

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.

$$2y-2x=-40,2y+3x=10$$

Me tango 2y+3x=10 mai i 2y-2x=-40 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$2y-2y-2x-3x=-40-10$$

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.

$$-2x-3x=-40-10$$

Tāpiri -2x ki te -3x.

$$-5x=-40-10$$

Tāpiri -40 ki te -10.

$$-5x=-50$$

Whakawehea ngā taha e rua ki te -5.

$$x=10$$

Whakaurua te 10 mō x ki 2y+3x=10. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$2y+3\times 10=10$$

Whakareatia 3 ki te 10.

$$2y+30=10$$

Me tango 30 mai i ngā taha e rua o te whārite.

$$2y=-20$$

Whakawehea ngā taha e rua ki te 2.

$$y=-10$$

Kua oti te pūnaha te whakatau.

$$y=-10,x=10$$