Whakaoti i te {l}{-8x-6y=30}{-6x+2y=-10}

$$\left. \begin{array} { l } { - 8 x - 6 y = 30 } \\ { - 6 x + 2 y = - 10 } \end{array} \right.$$

Whakaoti mō x, y

$$x=0$$

$$y=-5$$

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.

$$-8x-6y=30,-6x+2y=-10$$

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

$$-8x-6y=30$$

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

$$-8x=6y+30$$

Whakawehea ngā taha e rua ki te -8.

$$x=-\frac{1}{8}\left(6y+30\right)$$

Whakareatia -\frac{1}{8} ki te 30+6y.

$$x=-\frac{3}{4}y-\frac{15}{4}$$

Whakakapia te \frac{-3y-15}{4} mō te x ki tērā atu whārite, -6x+2y=-10.

$$-6\left(-\frac{3}{4}y-\frac{15}{4}\right)+2y=-10$$

Whakareatia -6 ki te \frac{-3y-15}{4}.

$$\frac{9}{2}y+\frac{45}{2}+2y=-10$$

Tāpiri \frac{9y}{2} ki te 2y.

$$\frac{13}{2}y+\frac{45}{2}=-10$$

Me tango \frac{45}{2} mai i ngā taha e rua o te whārite.

$$\frac{13}{2}y=-\frac{65}{2}$$

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.

$$y=-5$$

Whakaurua te -5 mō y ki x=-\frac{3}{4}y-\frac{15}{4}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{3}{4}\left(-5\right)-\frac{15}{4}$$

Whakareatia -\frac{3}{4} ki te -5.

$$x=\frac{15-15}{4}$$

Tāpiri -\frac{15}{4} ki te \frac{15}{4} 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.

$$x=0,y=-5$$

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.

$$-8x-6y=30,-6x+2y=-10$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-8&-6\\-6&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}30\\-10\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}-8&-6\\-6&2\end{matrix}\right))\left(\begin{matrix}-8&-6\\-6&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-8&-6\\-6&2\end{matrix}\right))\left(\begin{matrix}30\\-10\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-8&-6\\-6&2\end{matrix}\right))\left(\begin{matrix}30\\-10\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-8&-6\\-6&2\end{matrix}\right))\left(\begin{matrix}30\\-10\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{-8\times 2-\left(-6\left(-6\right)\right)}&-\frac{-6}{-8\times 2-\left(-6\left(-6\right)\right)}\\-\frac{-6}{-8\times 2-\left(-6\left(-6\right)\right)}&-\frac{8}{-8\times 2-\left(-6\left(-6\right)\right)}\end{matrix}\right)\left(\begin{matrix}30\\-10\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{26}&-\frac{3}{26}\\-\frac{3}{26}&\frac{2}{13}\end{matrix}\right)\left(\begin{matrix}30\\-10\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{26}\times 30-\frac{3}{26}\left(-10\right)\\-\frac{3}{26}\times 30+\frac{2}{13}\left(-10\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}0\\-5\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=0,y=-5$$

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.

$$-8x-6y=30,-6x+2y=-10$$

Kia ōrite ai a -8x 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 -8.

$$-6\left(-8\right)x-6\left(-6\right)y=-6\times 30,-8\left(-6\right)x-8\times 2y=-8\left(-10\right)$$

Whakarūnātia.

$$48x+36y=-180,48x-16y=80$$

Me tango 48x-16y=80 mai i 48x+36y=-180 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$48x-48x+36y+16y=-180-80$$

Tāpiri 48x ki te -48x. Ka whakakore atu ngā kupu 48x me -48x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$36y+16y=-180-80$$

Tāpiri 36y ki te 16y.

$$52y=-180-80$$

Tāpiri -180 ki te -80.

$$52y=-260$$

Whakawehea ngā taha e rua ki te 52.

$$y=-5$$

Whakaurua te -5 mō y ki -6x+2y=-10. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-6x+2\left(-5\right)=-10$$

Whakareatia 2 ki te -5.

$$-6x-10=-10$$

Me tāpiri 10 ki ngā taha e rua o te whārite.

$$-6x=0$$

Whakawehea ngā taha e rua ki te -6.

$$x=0$$

Kua oti te pūnaha te whakatau.

$$x=0,y=-5$$