Whakaoti i te {l}{3x+2y=-8}{-x-2y=12}

$$\left. \begin{array} { l } { 3 x + 2 y = - 8 } \\ { - x - 2 y = 12 } \end{array} \right.$$

Whakaoti mō x, y

$$x=2$$

$$y=-7$$

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.

$$3x+2y=-8,-x-2y=12$$

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

$$3x+2y=-8$$

Me tango 2y mai i ngā taha e rua o te whārite.

$$3x=-2y-8$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{1}{3}\left(-2y-8\right)$$

Whakareatia \frac{1}{3} ki te -2y-8.

$$x=-\frac{2}{3}y-\frac{8}{3}$$

Whakakapia te \frac{-2y-8}{3} mō te x ki tērā atu whārite, -x-2y=12.

$$-\left(-\frac{2}{3}y-\frac{8}{3}\right)-2y=12$$

Whakareatia -1 ki te \frac{-2y-8}{3}.

$$\frac{2}{3}y+\frac{8}{3}-2y=12$$

Tāpiri \frac{2y}{3} ki te -2y.

$$-\frac{4}{3}y+\frac{8}{3}=12$$

Me tango \frac{8}{3} mai i ngā taha e rua o te whārite.

$$-\frac{4}{3}y=\frac{28}{3}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{4}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-7$$

Whakaurua te -7 mō y ki x=-\frac{2}{3}y-\frac{8}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{2}{3}\left(-7\right)-\frac{8}{3}$$

Whakareatia -\frac{2}{3} ki te -7.

$$x=\frac{14-8}{3}$$

Tāpiri -\frac{8}{3} ki te \frac{14}{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.

$$x=2$$

Kua oti te pūnaha te whakatau.

$$x=2,y=-7$$

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.

$$3x+2y=-8,-x-2y=12$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&2\\-1&-2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-8\\12\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}3&2\\-1&-2\end{matrix}\right))\left(\begin{matrix}3&2\\-1&-2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&2\\-1&-2\end{matrix}\right))\left(\begin{matrix}-8\\12\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&2\\-1&-2\end{matrix}\right))\left(\begin{matrix}-8\\12\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&2\\-1&-2\end{matrix}\right))\left(\begin{matrix}-8\\12\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{3\left(-2\right)-2\left(-1\right)}&-\frac{2}{3\left(-2\right)-2\left(-1\right)}\\-\frac{-1}{3\left(-2\right)-2\left(-1\right)}&\frac{3}{3\left(-2\right)-2\left(-1\right)}\end{matrix}\right)\left(\begin{matrix}-8\\12\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2}&\frac{1}{2}\\-\frac{1}{4}&-\frac{3}{4}\end{matrix}\right)\left(\begin{matrix}-8\\12\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2}\left(-8\right)+\frac{1}{2}\times 12\\-\frac{1}{4}\left(-8\right)-\frac{3}{4}\times 12\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}2\\-7\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=2,y=-7$$

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.

$$3x+2y=-8,-x-2y=12$$

Kia ōrite ai a 3x me -x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te -1 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 3.

$$-3x-2y=-\left(-8\right),3\left(-1\right)x+3\left(-2\right)y=3\times 12$$

Whakarūnātia.

$$-3x-2y=8,-3x-6y=36$$

Me tango -3x-6y=36 mai i -3x-2y=8 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-3x+3x-2y+6y=8-36$$

Tāpiri -3x ki te 3x. Ka whakakore atu ngā kupu -3x me 3x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-2y+6y=8-36$$

Tāpiri -2y ki te 6y.

$$4y=8-36$$

Tāpiri 8 ki te -36.

$$4y=-28$$

Whakawehea ngā taha e rua ki te 4.

$$y=-7$$

Whakaurua te -7 mō y ki -x-2y=12. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-x-2\left(-7\right)=12$$

Whakareatia -2 ki te -7.

$$-x+14=12$$

Me tango 14 mai i ngā taha e rua o te whārite.

$$-x=-2$$

Whakawehea ngā taha e rua ki te -1.

$$x=2$$

Kua oti te pūnaha te whakatau.

$$x=2,y=-7$$