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

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

Whakaoti mō x, y

$$x=-1$$

$$y=2$$

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.

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

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

$$-x+y=3$$

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

$$-x=-y+3$$

Whakawehea ngā taha e rua ki te -1.

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

Whakareatia -1 ki te -y+3.

$$x=y-3$$

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

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

Whakareatia 2 ki te y-3.

$$2y-6+2y=2$$

Tāpiri 2y ki te 2y.

$$4y-6=2$$

Me tāpiri 6 ki ngā taha e rua o te whārite.

$$4y=8$$

Whakawehea ngā taha e rua ki te 4.

$$y=2$$

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

$$x=2-3$$

Tāpiri -3 ki te 2.

$$x=-1$$

Kua oti te pūnaha te whakatau.

$$x=-1,y=2$$

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.

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{2}&\frac{1}{4}\\\frac{1}{2}&\frac{1}{4}\end{matrix}\right)\left(\begin{matrix}3\\2\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{2}\times 3+\frac{1}{4}\times 2\\\frac{1}{2}\times 3+\frac{1}{4}\times 2\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=-1,y=2$$

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.

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

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.

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

Whakarūnātia.

$$-2x+2y=6,-2x-2y=-2$$

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

$$-2x+2x+2y+2y=6+2$$

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.

$$2y+2y=6+2$$

Tāpiri 2y ki te 2y.

$$4y=6+2$$

Tāpiri 6 ki te 2.

$$4y=8$$

Whakawehea ngā taha e rua ki te 4.

$$y=2$$

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

$$2x+2\times 2=2$$

Whakareatia 2 ki te 2.

$$2x+4=2$$

Me tango 4 mai i ngā taha e rua o te whārite.

$$2x=-2$$

Whakawehea ngā taha e rua ki te 2.

$$x=-1$$

Kua oti te pūnaha te whakatau.

$$x=-1,y=2$$