Whakaoti i te {l}{-2x+4y=14}{x-4y=-7}

$$\left. \begin{array} { l } { - 2 x + 4 y = 14 } \\ { x - 4 y = - 7 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-7$$

$$y=0$$

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.

$$-2x+4y=14,x-4y=-7$$

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

$$-2x+4y=14$$

Me tango 4y mai i ngā taha e rua o te whārite.

$$-2x=-4y+14$$

Whakawehea ngā taha e rua ki te -2.

$$x=-\frac{1}{2}\left(-4y+14\right)$$

Whakareatia -\frac{1}{2} ki te -4y+14.

$$x=2y-7$$

Whakakapia te 2y-7 mō te x ki tērā atu whārite, x-4y=-7.

$$2y-7-4y=-7$$

Tāpiri 2y ki te -4y.

$$-2y-7=-7$$

Me tāpiri 7 ki ngā taha e rua o te whārite.

$$-2y=0$$

Whakawehea ngā taha e rua ki te -2.

$$y=0$$

Whakaurua te 0 mō y ki x=2y-7. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-7$$

Kua oti te pūnaha te whakatau.

$$x=-7,y=0$$

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.

$$-2x+4y=14,x-4y=-7$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-2&4\\1&-4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}14\\-7\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}-2&4\\1&-4\end{matrix}\right))\left(\begin{matrix}-2&4\\1&-4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-2&4\\1&-4\end{matrix}\right))\left(\begin{matrix}14\\-7\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-2&4\\1&-4\end{matrix}\right))\left(\begin{matrix}14\\-7\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-2&4\\1&-4\end{matrix}\right))\left(\begin{matrix}14\\-7\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{4}{-2\left(-4\right)-4}&-\frac{4}{-2\left(-4\right)-4}\\-\frac{1}{-2\left(-4\right)-4}&-\frac{2}{-2\left(-4\right)-4}\end{matrix}\right)\left(\begin{matrix}14\\-7\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-1&-1\\-\frac{1}{4}&-\frac{1}{2}\end{matrix}\right)\left(\begin{matrix}14\\-7\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-14-\left(-7\right)\\-\frac{1}{4}\times 14-\frac{1}{2}\left(-7\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-7\\0\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-7,y=0$$

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.

$$-2x+4y=14,x-4y=-7$$

Kia ōrite ai a -2x 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 -2.

$$-2x+4y=14,-2x-2\left(-4\right)y=-2\left(-7\right)$$

Whakarūnātia.

$$-2x+4y=14,-2x+8y=14$$

Me tango -2x+8y=14 mai i -2x+4y=14 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-2x+2x+4y-8y=14-14$$

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.

$$4y-8y=14-14$$

Tāpiri 4y ki te -8y.

$$-4y=14-14$$

Tāpiri 14 ki te -14.

$$-4y=0$$

Whakawehea ngā taha e rua ki te -4.

$$y=0$$

Whakaurua te 0 mō y ki x-4y=-7. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-7$$

Kua oti te pūnaha te whakatau.

$$x=-7,y=0$$