Whakaoti i te {l}{2x-6y=34}{8x-3y=-11}

$$\left\{ \begin{array} { l } { 2 x - 6 y = 34 } \\ { 8 x - 3 y = - 11 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-4$$

$$y=-7$$

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.

$$2x-6y=34,8x-3y=-11$$

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

$$2x-6y=34$$

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

$$2x=6y+34$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{1}{2}\left(6y+34\right)$$

Whakareatia \frac{1}{2} ki te 6y+34.

$$x=3y+17$$

Whakakapia te 3y+17 mō te x ki tērā atu whārite, 8x-3y=-11.

$$8\left(3y+17\right)-3y=-11$$

Whakareatia 8 ki te 3y+17.

$$24y+136-3y=-11$$

Tāpiri 24y ki te -3y.

$$21y+136=-11$$

Me tango 136 mai i ngā taha e rua o te whārite.

$$21y=-147$$

Whakawehea ngā taha e rua ki te 21.

$$y=-7$$

Whakaurua te -7 mō y ki x=3y+17. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=3\left(-7\right)+17$$

Whakareatia 3 ki te -7.

$$x=-21+17$$

Tāpiri 17 ki te -21.

$$x=-4$$

Kua oti te pūnaha te whakatau.

$$x=-4,y=-7$$

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.

$$2x-6y=34,8x-3y=-11$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&-6\\8&-3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}34\\-11\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}2&-6\\8&-3\end{matrix}\right).

$$inverse(\left(\begin{matrix}2&-6\\8&-3\end{matrix}\right))\left(\begin{matrix}2&-6\\8&-3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-6\\8&-3\end{matrix}\right))\left(\begin{matrix}34\\-11\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-6\\8&-3\end{matrix}\right))\left(\begin{matrix}34\\-11\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-6\\8&-3\end{matrix}\right))\left(\begin{matrix}34\\-11\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{2\left(-3\right)-\left(-6\times 8\right)}&-\frac{-6}{2\left(-3\right)-\left(-6\times 8\right)}\\-\frac{8}{2\left(-3\right)-\left(-6\times 8\right)}&\frac{2}{2\left(-3\right)-\left(-6\times 8\right)}\end{matrix}\right)\left(\begin{matrix}34\\-11\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{14}&\frac{1}{7}\\-\frac{4}{21}&\frac{1}{21}\end{matrix}\right)\left(\begin{matrix}34\\-11\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{14}\times 34+\frac{1}{7}\left(-11\right)\\-\frac{4}{21}\times 34+\frac{1}{21}\left(-11\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-4\\-7\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-4,y=-7$$

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.

$$2x-6y=34,8x-3y=-11$$

Kia ōrite ai a 2x me 8x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 8 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 2.

$$8\times 2x+8\left(-6\right)y=8\times 34,2\times 8x+2\left(-3\right)y=2\left(-11\right)$$

Whakarūnātia.

$$16x-48y=272,16x-6y=-22$$

Me tango 16x-6y=-22 mai i 16x-48y=272 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$16x-16x-48y+6y=272+22$$

Tāpiri 16x ki te -16x. Ka whakakore atu ngā kupu 16x me -16x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-48y+6y=272+22$$

Tāpiri -48y ki te 6y.

$$-42y=272+22$$

Tāpiri 272 ki te 22.

$$-42y=294$$

Whakawehea ngā taha e rua ki te -42.

$$y=-7$$

Whakaurua te -7 mō y ki 8x-3y=-11. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$8x-3\left(-7\right)=-11$$

Whakareatia -3 ki te -7.

$$8x+21=-11$$

Me tango 21 mai i ngā taha e rua o te whārite.

$$8x=-32$$

Whakawehea ngā taha e rua ki te 8.

$$x=-4$$

Kua oti te pūnaha te whakatau.

$$x=-4,y=-7$$