Whakaoti i te {l}{x+2y=4+3}{-x-y=278-1}

$$\left. \begin{array} { l } { x + 2 y = 4 + 3 } \\ { - x - y = 278 - 1 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-561$$

$$y=284$$

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.

$$x+2y=7,-x-y=277$$

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

$$x+2y=7$$

Me tango 2y mai i ngā taha e rua o te whārite.

$$x=-2y+7$$

Whakakapia te -2y+7 mō te x ki tērā atu whārite, -x-y=277.

$$-\left(-2y+7\right)-y=277$$

Whakareatia -1 ki te -2y+7.

$$2y-7-y=277$$

Tāpiri 2y ki te -y.

$$y-7=277$$

Me tāpiri 7 ki ngā taha e rua o te whārite.

$$y=284$$

Whakaurua te 284 mō y ki x=-2y+7. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-2\times 284+7$$

Whakareatia -2 ki te 284.

$$x=-568+7$$

Tāpiri 7 ki te -568.

$$x=-561$$

Kua oti te pūnaha te whakatau.

$$x=-561,y=284$$

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.

$$x+2y=7,-x-y=277$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&2\\-1&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}7\\277\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}1&2\\-1&-1\end{matrix}\right))\left(\begin{matrix}1&2\\-1&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&2\\-1&-1\end{matrix}\right))\left(\begin{matrix}7\\277\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&2\\-1&-1\end{matrix}\right))\left(\begin{matrix}7\\277\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&2\\-1&-1\end{matrix}\right))\left(\begin{matrix}7\\277\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{-1-2\left(-1\right)}&-\frac{2}{-1-2\left(-1\right)}\\-\frac{-1}{-1-2\left(-1\right)}&\frac{1}{-1-2\left(-1\right)}\end{matrix}\right)\left(\begin{matrix}7\\277\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-1&-2\\1&1\end{matrix}\right)\left(\begin{matrix}7\\277\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-7-2\times 277\\7+277\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-561\\284\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-561,y=284$$

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.

$$x+2y=7,-x-y=277$$

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

$$-x-2y=-7,-x-y=277$$

Me tango -x-y=277 mai i -x-2y=-7 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-x+x-2y+y=-7-277$$

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

$$-2y+y=-7-277$$

Tāpiri -2y ki te y.

$$-y=-7-277$$

Tāpiri -7 ki te -277.

$$-y=-284$$

Whakawehea ngā taha e rua ki te -1.

$$y=284$$

Whakaurua te 284 mō y ki -x-y=277. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-x-284=277$$

Me tāpiri 284 ki ngā taha e rua o te whārite.

$$-x=561$$

Whakawehea ngā taha e rua ki te -1.

$$x=-561$$

Kua oti te pūnaha te whakatau.

$$x=-561,y=284$$