Whakaoti i te {l}{-x+3y=2}{2x-5y=7}

$$\left\{ \begin{array} { l } { - x + 3 y = 2 } \\ { 2 x - 5 y = 7 } \end{array} \right.$$

Whakaoti mō x, y

$$x=31$$

$$y=11$$

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.

$$-x+3y=2,2x-5y=7$$

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

$$-x+3y=2$$

Me tango 3y mai i ngā taha e rua o te whārite.

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

Whakawehea ngā taha e rua ki te -1.

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

Whakareatia -1 ki te -3y+2.

$$x=3y-2$$

Whakakapia te 3y-2 mō te x ki tērā atu whārite, 2x-5y=7.

$$2\left(3y-2\right)-5y=7$$

Whakareatia 2 ki te 3y-2.

$$6y-4-5y=7$$

Tāpiri 6y ki te -5y.

$$y-4=7$$

Me tāpiri 4 ki ngā taha e rua o te whārite.

$$y=11$$

Whakaurua te 11 mō y ki x=3y-2. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=3\times 11-2$$

Whakareatia 3 ki te 11.

$$x=33-2$$

Tāpiri -2 ki te 33.

$$x=31$$

Kua oti te pūnaha te whakatau.

$$x=31,y=11$$

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.

$$-x+3y=2,2x-5y=7$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-1&3\\2&-5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}2\\7\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}-1&3\\2&-5\end{matrix}\right).

$$inverse(\left(\begin{matrix}-1&3\\2&-5\end{matrix}\right))\left(\begin{matrix}-1&3\\2&-5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-1&3\\2&-5\end{matrix}\right))\left(\begin{matrix}2\\7\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-1&3\\2&-5\end{matrix}\right))\left(\begin{matrix}2\\7\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-1&3\\2&-5\end{matrix}\right))\left(\begin{matrix}2\\7\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{5}{-\left(-5\right)-3\times 2}&-\frac{3}{-\left(-5\right)-3\times 2}\\-\frac{2}{-\left(-5\right)-3\times 2}&-\frac{1}{-\left(-5\right)-3\times 2}\end{matrix}\right)\left(\begin{matrix}2\\7\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}5&3\\2&1\end{matrix}\right)\left(\begin{matrix}2\\7\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}5\times 2+3\times 7\\2\times 2+7\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}31\\11\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=31,y=11$$

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.

$$-x+3y=2,2x-5y=7$$

Kia ōrite ai a -x me 2x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 2 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te -1.

$$2\left(-1\right)x+2\times 3y=2\times 2,-2x-\left(-5y\right)=-7$$

Whakarūnātia.

$$-2x+6y=4,-2x+5y=-7$$

Me tango -2x+5y=-7 mai i -2x+6y=4 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-2x+2x+6y-5y=4+7$$

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

$$6y-5y=4+7$$

Tāpiri 6y ki te -5y.

$$y=4+7$$

Tāpiri 4 ki te 7.

$$y=11$$

Whakaurua te 11 mō y ki 2x-5y=7. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$2x-5\times 11=7$$

Whakareatia -5 ki te 11.

$$2x-55=7$$

Me tāpiri 55 ki ngā taha e rua o te whārite.

$$2x=62$$

Whakawehea ngā taha e rua ki te 2.

$$x=31$$

Kua oti te pūnaha te whakatau.

$$x=31,y=11$$