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

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

Whakaoti mō x, y

$$x=4$$

$$y=2$$

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.

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

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

$$x-y=2$$

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

$$x=y+2$$

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

$$2\left(y+2\right)+y=10$$

Whakareatia 2 ki te y+2.

$$2y+4+y=10$$

Tāpiri 2y ki te y.

$$3y+4=10$$

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

$$3y=6$$

Whakawehea ngā taha e rua ki te 3.

$$y=2$$

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

$$x=2+2$$

Tāpiri 2 ki te 2.

$$x=4$$

Kua oti te pūnaha te whakatau.

$$x=4,y=2$$

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.

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{1-\left(-2\right)}&-\frac{-1}{1-\left(-2\right)}\\-\frac{2}{1-\left(-2\right)}&\frac{1}{1-\left(-2\right)}\end{matrix}\right)\left(\begin{matrix}2\\10\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{3}\times 2+\frac{1}{3}\times 10\\-\frac{2}{3}\times 2+\frac{1}{3}\times 10\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}4\\2\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=4,y=2$$

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.

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

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.

$$2x+2\left(-1\right)y=2\times 2,2x+y=10$$

Whakarūnātia.

$$2x-2y=4,2x+y=10$$

Me tango 2x+y=10 mai i 2x-2y=4 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$2x-2x-2y-y=4-10$$

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.

$$-2y-y=4-10$$

Tāpiri -2y ki te -y.

$$-3y=4-10$$

Tāpiri 4 ki te -10.

$$-3y=-6$$

Whakawehea ngā taha e rua ki te -3.

$$y=2$$

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

$$2x+2=10$$

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

$$2x=8$$

Whakawehea ngā taha e rua ki te 2.

$$x=4$$

Kua oti te pūnaha te whakatau.

$$x=4,y=2$$