Whakaoti i te {c}{x=6+y}{4y=-2x}

$$\left. \begin{array} { c } { x = 6 + y } \\ { 4 y = - 2 x } \end{array} \right.$$

Whakaoti mō x, y

$$x=4$$

$$y=-2$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Tangohia te y mai i ngā taha e rua.

$$x-y=6$$

Whakaarohia te whārite tuarua. Me tāpiri te 2x ki ngā taha e rua.

$$4y+2x=0$$

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.

$$x-y=6,2x+4y=0$$

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

$$x-y=6$$

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

$$x=y+6$$

Whakakapia te y+6 mō te x ki tērā atu whārite, 2x+4y=0.

$$2\left(y+6\right)+4y=0$$

Whakareatia 2 ki te y+6.

$$2y+12+4y=0$$

Tāpiri 2y ki te 4y.

$$6y+12=0$$

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

$$6y=-12$$

Whakawehea ngā taha e rua ki te 6.

$$y=-2$$

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

$$x=-2+6$$

Tāpiri 6 ki te -2.

$$x=4$$

Kua oti te pūnaha te whakatau.

$$x=4,y=-2$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Tangohia te y mai i ngā taha e rua.

$$x-y=6$$

Whakaarohia te whārite tuarua. Me tāpiri te 2x ki ngā taha e rua.

$$4y+2x=0$$

Tuhia ngā whārite ki te tānga ngahuru ka whakamahi i ngā poukapa hei whakaoti i te pūnaha o ngā whārite.

$$x-y=6,2x+4y=0$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&-1\\2&4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}6\\0\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&-1\\2&4\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&-1\\2&4\end{matrix}\right))\left(\begin{matrix}1&-1\\2&4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-1\\2&4\end{matrix}\right))\left(\begin{matrix}6\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-1\\2&4\end{matrix}\right))\left(\begin{matrix}6\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-1\\2&4\end{matrix}\right))\left(\begin{matrix}6\\0\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{4-\left(-2\right)}&-\frac{-1}{4-\left(-2\right)}\\-\frac{2}{4-\left(-2\right)}&\frac{1}{4-\left(-2\right)}\end{matrix}\right)\left(\begin{matrix}6\\0\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}4\\-2\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=4,y=-2$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Tangohia te y mai i ngā taha e rua.

$$x-y=6$$

Whakaarohia te whārite tuarua. Me tāpiri te 2x ki ngā taha e rua.

$$4y+2x=0$$

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.

$$x-y=6,2x+4y=0$$

Kia ōrite ai a x me 2x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 2 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 1.

$$2x+2\left(-1\right)y=2\times 6,2x+4y=0$$

Whakarūnātia.

$$2x-2y=12,2x+4y=0$$

Me tango 2x+4y=0 mai i 2x-2y=12 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$2x-2x-2y-4y=12$$

Tāpiri 2x ki te -2x. Ka whakakore atu ngā kupu 2x me -2x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-2y-4y=12$$

Tāpiri -2y ki te -4y.

$$-6y=12$$

Whakawehea ngā taha e rua ki te -6.

$$y=-2$$

Whakaurua te -2 mō y ki 2x+4y=0. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$2x+4\left(-2\right)=0$$

Whakareatia 4 ki te -2.

$$2x-8=0$$

Me tāpiri 8 ki ngā taha e rua o te whārite.

$$2x=8$$

Whakawehea ngā taha e rua ki te 2.

$$x=4$$

Kua oti te pūnaha te whakatau.

$$x=4,y=-2$$