Whakaoti i te {c}{x-3y=6}{-8x-y=6}

$$\left. \begin{array} { c } { x - 3 y = 6 } \\ { - 8 x - y = 6 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-\frac{12}{25}=-0.48$$

$$y = -\frac{54}{25} = -2\frac{4}{25} = -2.16$$

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.

$$x-3y=6,-8x-y=6$$

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

$$x-3y=6$$

Me tāpiri 3y ki ngā taha e rua o te whārite.

$$x=3y+6$$

Whakakapia te 6+3y mō te x ki tērā atu whārite, -8x-y=6.

$$-8\left(3y+6\right)-y=6$$

Whakareatia -8 ki te 6+3y.

$$-24y-48-y=6$$

Tāpiri -24y ki te -y.

$$-25y-48=6$$

Me tāpiri 48 ki ngā taha e rua o te whārite.

$$-25y=54$$

Whakawehea ngā taha e rua ki te -25.

$$y=-\frac{54}{25}$$

Whakaurua te -\frac{54}{25} mō y ki x=3y+6. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=3\left(-\frac{54}{25}\right)+6$$

Whakareatia 3 ki te -\frac{54}{25}.

$$x=-\frac{162}{25}+6$$

Tāpiri 6 ki te -\frac{162}{25}.

$$x=-\frac{12}{25}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{12}{25},y=-\frac{54}{25}$$

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.

$$x-3y=6,-8x-y=6$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&-3\\-8&-1\end{matrix}\right).

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

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

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

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

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

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{-1-\left(-3\left(-8\right)\right)}&-\frac{-3}{-1-\left(-3\left(-8\right)\right)}\\-\frac{-8}{-1-\left(-3\left(-8\right)\right)}&\frac{1}{-1-\left(-3\left(-8\right)\right)}\end{matrix}\right)\left(\begin{matrix}6\\6\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{25}&-\frac{3}{25}\\-\frac{8}{25}&-\frac{1}{25}\end{matrix}\right)\left(\begin{matrix}6\\6\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{25}\times 6-\frac{3}{25}\times 6\\-\frac{8}{25}\times 6-\frac{1}{25}\times 6\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{12}{25}\\-\frac{54}{25}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-\frac{12}{25},y=-\frac{54}{25}$$

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.

$$x-3y=6,-8x-y=6$$

Kia ōrite ai a x 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 1.

$$-8x-8\left(-3\right)y=-8\times 6,-8x-y=6$$

Whakarūnātia.

$$-8x+24y=-48,-8x-y=6$$

Me tango -8x-y=6 mai i -8x+24y=-48 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-8x+8x+24y+y=-48-6$$

Tāpiri -8x ki te 8x. Ka whakakore atu ngā kupu -8x me 8x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$24y+y=-48-6$$

Tāpiri 24y ki te y.

$$25y=-48-6$$

Tāpiri -48 ki te -6.

$$25y=-54$$

Whakawehea ngā taha e rua ki te 25.

$$y=-\frac{54}{25}$$

Whakaurua te -\frac{54}{25} mō y ki -8x-y=6. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-8x-\left(-\frac{54}{25}\right)=6$$

Me tango \frac{54}{25} mai i ngā taha e rua o te whārite.

$$-8x=\frac{96}{25}$$

Whakawehea ngā taha e rua ki te -8.

$$x=-\frac{12}{25}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{12}{25},y=-\frac{54}{25}$$