Whakaoti i te {l}{x-2y=-6}{6x+3y=2}

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

Whakaoti mō x, y

$$x=-\frac{14}{15}\approx -0.933333333$$

$$y = \frac{38}{15} = 2\frac{8}{15} \approx 2.533333333$$

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.

$$x-2y=-6,6x+3y=2$$

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

$$x-2y=-6$$

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

$$x=2y-6$$

Whakakapia te -6+2y mō te x ki tērā atu whārite, 6x+3y=2.

$$6\left(2y-6\right)+3y=2$$

Whakareatia 6 ki te -6+2y.

$$12y-36+3y=2$$

Tāpiri 12y ki te 3y.

$$15y-36=2$$

Me tāpiri 36 ki ngā taha e rua o te whārite.

$$15y=38$$

Whakawehea ngā taha e rua ki te 15.

$$y=\frac{38}{15}$$

Whakaurua te \frac{38}{15} mō y ki x=2y-6. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=2\times \frac{38}{15}-6$$

Whakareatia 2 ki te \frac{38}{15}.

$$x=\frac{76}{15}-6$$

Tāpiri -6 ki te \frac{76}{15}.

$$x=-\frac{14}{15}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{14}{15},y=\frac{38}{15}$$

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.

$$x-2y=-6,6x+3y=2$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&-2\\6&3\end{matrix}\right).

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

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

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

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

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

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{3-\left(-2\times 6\right)}&-\frac{-2}{3-\left(-2\times 6\right)}\\-\frac{6}{3-\left(-2\times 6\right)}&\frac{1}{3-\left(-2\times 6\right)}\end{matrix}\right)\left(\begin{matrix}-6\\2\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{5}&\frac{2}{15}\\-\frac{2}{5}&\frac{1}{15}\end{matrix}\right)\left(\begin{matrix}-6\\2\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{5}\left(-6\right)+\frac{2}{15}\times 2\\-\frac{2}{5}\left(-6\right)+\frac{1}{15}\times 2\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{14}{15}\\\frac{38}{15}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-\frac{14}{15},y=\frac{38}{15}$$

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.

$$x-2y=-6,6x+3y=2$$

Kia ōrite ai a x me 6x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 6 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 1.

$$6x+6\left(-2\right)y=6\left(-6\right),6x+3y=2$$

Whakarūnātia.

$$6x-12y=-36,6x+3y=2$$

Me tango 6x+3y=2 mai i 6x-12y=-36 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$6x-6x-12y-3y=-36-2$$

Tāpiri 6x ki te -6x. Ka whakakore atu ngā kupu 6x me -6x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-12y-3y=-36-2$$

Tāpiri -12y ki te -3y.

$$-15y=-36-2$$

Tāpiri -36 ki te -2.

$$-15y=-38$$

Whakawehea ngā taha e rua ki te -15.

$$y=\frac{38}{15}$$

Whakaurua te \frac{38}{15} mō y ki 6x+3y=2. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$6x+3\times \frac{38}{15}=2$$

Whakareatia 3 ki te \frac{38}{15}.

$$6x+\frac{38}{5}=2$$

Me tango \frac{38}{5} mai i ngā taha e rua o te whārite.

$$6x=-\frac{28}{5}$$

Whakawehea ngā taha e rua ki te 6.

$$x=-\frac{14}{15}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{14}{15},y=\frac{38}{15}$$