Whakaoti i te {l}{-6x+y=-2}{-3x-6y=12}

$$\left. \begin{array} { l } { - 6 x + y = - 2 } \\ { - 3 x - 6 y = 12 } \end{array} \right.$$

Whakaoti mō x, y

$$x=0$$

$$y=-2$$

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.

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

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

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

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

$$-6x=-y-2$$

Whakawehea ngā taha e rua ki te -6.

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

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

$$x=\frac{1}{6}y+\frac{1}{3}$$

Whakakapia te \frac{y}{6}+\frac{1}{3} mō te x ki tērā atu whārite, -3x-6y=12.

$$-3\left(\frac{1}{6}y+\frac{1}{3}\right)-6y=12$$

Whakareatia -3 ki te \frac{y}{6}+\frac{1}{3}.

$$-\frac{1}{2}y-1-6y=12$$

Tāpiri -\frac{y}{2} ki te -6y.

$$-\frac{13}{2}y-1=12$$

Me tāpiri 1 ki ngā taha e rua o te whārite.

$$-\frac{13}{2}y=13$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{13}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-2$$

Whakaurua te -2 mō y ki x=\frac{1}{6}y+\frac{1}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{1}{6}\left(-2\right)+\frac{1}{3}$$

Whakareatia \frac{1}{6} ki te -2.

$$x=\frac{-1+1}{3}$$

Tāpiri \frac{1}{3} ki te -\frac{1}{3} mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=0$$

Kua oti te pūnaha te whakatau.

$$x=0,y=-2$$

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.

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-6&1\\-3&-6\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-2\\12\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}-6&1\\-3&-6\end{matrix}\right))\left(\begin{matrix}-6&1\\-3&-6\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-6&1\\-3&-6\end{matrix}\right))\left(\begin{matrix}-2\\12\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-6&1\\-3&-6\end{matrix}\right))\left(\begin{matrix}-2\\12\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-6&1\\-3&-6\end{matrix}\right))\left(\begin{matrix}-2\\12\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{6}{-6\left(-6\right)-\left(-3\right)}&-\frac{1}{-6\left(-6\right)-\left(-3\right)}\\-\frac{-3}{-6\left(-6\right)-\left(-3\right)}&-\frac{6}{-6\left(-6\right)-\left(-3\right)}\end{matrix}\right)\left(\begin{matrix}-2\\12\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{13}&-\frac{1}{39}\\\frac{1}{13}&-\frac{2}{13}\end{matrix}\right)\left(\begin{matrix}-2\\12\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{13}\left(-2\right)-\frac{1}{39}\times 12\\\frac{1}{13}\left(-2\right)-\frac{2}{13}\times 12\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=0,y=-2$$

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.

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

Kia ōrite ai a -6x me -3x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te -3 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te -6.

$$-3\left(-6\right)x-3y=-3\left(-2\right),-6\left(-3\right)x-6\left(-6\right)y=-6\times 12$$

Whakarūnātia.

$$18x-3y=6,18x+36y=-72$$

Me tango 18x+36y=-72 mai i 18x-3y=6 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$18x-18x-3y-36y=6+72$$

Tāpiri 18x ki te -18x. Ka whakakore atu ngā kupu 18x me -18x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-3y-36y=6+72$$

Tāpiri -3y ki te -36y.

$$-39y=6+72$$

Tāpiri 6 ki te 72.

$$-39y=78$$

Whakawehea ngā taha e rua ki te -39.

$$y=-2$$

Whakaurua te -2 mō y ki -3x-6y=12. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-3x-6\left(-2\right)=12$$

Whakareatia -6 ki te -2.

$$-3x+12=12$$

Me tango 12 mai i ngā taha e rua o te whārite.

$$-3x=0$$

Whakawehea ngā taha e rua ki te -3.

$$x=0$$

Kua oti te pūnaha te whakatau.

$$x=0,y=-2$$