Whakaoti i te {l}{x-2y=10}{2x+3y=-8}

$$\left\{ \begin{array} { l } { x - 2 y = 10 } \\ { 2 x + 3 y = - 8 } \end{array} \right.$$

Whakaoti mō x, y

$$x=2$$

$$y=-4$$

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.

$$x-2y=10,2x+3y=-8$$

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

$$x-2y=10$$

Me tāpiri 2y ki ngā taha e rua o te whārite.

$$x=2y+10$$

Whakakapia te 10+2y mō te x ki tērā atu whārite, 2x+3y=-8.

$$2\left(2y+10\right)+3y=-8$$

Whakareatia 2 ki te 10+2y.

$$4y+20+3y=-8$$

Tāpiri 4y ki te 3y.

$$7y+20=-8$$

Me tango 20 mai i ngā taha e rua o te whārite.

$$7y=-28$$

Whakawehea ngā taha e rua ki te 7.

$$y=-4$$

Whakaurua te -4 mō y ki x=2y+10. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=2\left(-4\right)+10$$

Whakareatia 2 ki te -4.

$$x=-8+10$$

Tāpiri 10 ki te -8.

$$x=2$$

Kua oti te pūnaha te whakatau.

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

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.

$$x-2y=10,2x+3y=-8$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&-2\\2&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}10\\-8\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}1&-2\\2&3\end{matrix}\right))\left(\begin{matrix}1&-2\\2&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-2\\2&3\end{matrix}\right))\left(\begin{matrix}10\\-8\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-2\\2&3\end{matrix}\right))\left(\begin{matrix}10\\-8\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-2\\2&3\end{matrix}\right))\left(\begin{matrix}10\\-8\end{matrix}\right)$$

Mō te poukapa 2\times 2 \left(\begin{matrix}a&b\\c&d\end{matrix}\right), ko te poukapa kōaro ko \left(\begin{matrix}\frac{d}{ad-bc}&\frac{-b}{ad-bc}\\\frac{-c}{ad-bc}&\frac{a}{ad-bc}\end{matrix}\right), kia tuhia anō ai te whārite poukapa hei rapanga whakarea poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{3-\left(-2\times 2\right)}&-\frac{-2}{3-\left(-2\times 2\right)}\\-\frac{2}{3-\left(-2\times 2\right)}&\frac{1}{3-\left(-2\times 2\right)}\end{matrix}\right)\left(\begin{matrix}10\\-8\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{7}&\frac{2}{7}\\-\frac{2}{7}&\frac{1}{7}\end{matrix}\right)\left(\begin{matrix}10\\-8\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{7}\times 10+\frac{2}{7}\left(-8\right)\\-\frac{2}{7}\times 10+\frac{1}{7}\left(-8\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

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

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.

$$x-2y=10,2x+3y=-8$$

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.

$$2x+2\left(-2\right)y=2\times 10,2x+3y=-8$$

Whakarūnātia.

$$2x-4y=20,2x+3y=-8$$

Me tango 2x+3y=-8 mai i 2x-4y=20 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$2x-2x-4y-3y=20+8$$

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.

$$-4y-3y=20+8$$

Tāpiri -4y ki te -3y.

$$-7y=20+8$$

Tāpiri 20 ki te 8.

$$-7y=28$$

Whakawehea ngā taha e rua ki te -7.

$$y=-4$$

Whakaurua te -4 mō y ki 2x+3y=-8. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$2x+3\left(-4\right)=-8$$

Whakareatia 3 ki te -4.

$$2x-12=-8$$

Me tāpiri 12 ki ngā taha e rua o te whārite.

$$2x=4$$

Whakawehea ngā taha e rua ki te 2.

$$x=2$$

Kua oti te pūnaha te whakatau.

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