Whakaoti i te {c}{-x+3y=6}{-2x+5y=11}

$$\left. \begin{array} { c } { - x + 3 y = 6 } \\ { - 2 x + 5 y = 11 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-3$$

$$y=1$$

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.

$$-x+3y=6,-2x+5y=11$$

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

$$-x+3y=6$$

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

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

Whakawehea ngā taha e rua ki te -1.

$$x=-\left(-3y+6\right)$$

Whakareatia -1 ki te -3y+6.

$$x=3y-6$$

Whakakapia te -6+3y mō te x ki tērā atu whārite, -2x+5y=11.

$$-2\left(3y-6\right)+5y=11$$

Whakareatia -2 ki te -6+3y.

$$-6y+12+5y=11$$

Tāpiri -6y ki te 5y.

$$-y+12=11$$

Me tango 12 mai i ngā taha e rua o te whārite.

$$-y=-1$$

Whakawehea ngā taha e rua ki te -1.

$$y=1$$

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

$$x=3-6$$

Tāpiri -6 ki te 3.

$$x=-3$$

Kua oti te pūnaha te whakatau.

$$x=-3,y=1$$

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.

$$-x+3y=6,-2x+5y=11$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-1&3\\-2&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}6\\11\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}-1&3\\-2&5\end{matrix}\right))\left(\begin{matrix}-1&3\\-2&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-1&3\\-2&5\end{matrix}\right))\left(\begin{matrix}6\\11\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-1&3\\-2&5\end{matrix}\right))\left(\begin{matrix}6\\11\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-1&3\\-2&5\end{matrix}\right))\left(\begin{matrix}6\\11\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{-5-3\left(-2\right)}&-\frac{3}{-5-3\left(-2\right)}\\-\frac{-2}{-5-3\left(-2\right)}&-\frac{1}{-5-3\left(-2\right)}\end{matrix}\right)\left(\begin{matrix}6\\11\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}5&-3\\2&-1\end{matrix}\right)\left(\begin{matrix}6\\11\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}5\times 6-3\times 11\\2\times 6-11\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=-3,y=1$$

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.

$$-x+3y=6,-2x+5y=11$$

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.

$$-2\left(-1\right)x-2\times 3y=-2\times 6,-\left(-2\right)x-5y=-11$$

Whakarūnātia.

$$2x-6y=-12,2x-5y=-11$$

Me tango 2x-5y=-11 mai i 2x-6y=-12 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$2x-2x-6y+5y=-12+11$$

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.

$$-6y+5y=-12+11$$

Tāpiri -6y ki te 5y.

$$-y=-12+11$$

Tāpiri -12 ki te 11.

$$-y=-1$$

Whakawehea ngā taha e rua ki te -1.

$$y=1$$

Whakaurua te 1 mō y ki -2x+5y=11. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-2x+5=11$$

Me tango 5 mai i ngā taha e rua o te whārite.

$$-2x=6$$

Whakawehea ngā taha e rua ki te -2.

$$x=-3$$

Kua oti te pūnaha te whakatau.

$$x=-3,y=1$$