Whakaoti i te {l}{x+4y=-1}{2x-4y=4}

$$\left\{ \begin{array} { l } { x + 4 y = - 1 } \\ { 2 x - 4 y = 4 } \end{array} \right.$$

Whakaoti mō x, y

$$x=1$$

$$y=-\frac{1}{2}=-0.5$$

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.

$$x+4y=-1,2x-4y=4$$

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

$$x+4y=-1$$

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

$$x=-4y-1$$

Whakakapia te -4y-1 mō te x ki tērā atu whārite, 2x-4y=4.

$$2\left(-4y-1\right)-4y=4$$

Whakareatia 2 ki te -4y-1.

$$-8y-2-4y=4$$

Tāpiri -8y ki te -4y.

$$-12y-2=4$$

Me tāpiri 2 ki ngā taha e rua o te whārite.

$$-12y=6$$

Whakawehea ngā taha e rua ki te -12.

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

Whakaurua te -\frac{1}{2} mō y ki x=-4y-1. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

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

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

$$x=2-1$$

Tāpiri -1 ki te 2.

$$x=1$$

Kua oti te pūnaha te whakatau.

$$x=1,y=-\frac{1}{2}$$

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.

$$x+4y=-1,2x-4y=4$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{3}&\frac{1}{3}\\\frac{1}{6}&-\frac{1}{12}\end{matrix}\right)\left(\begin{matrix}-1\\4\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{3}\left(-1\right)+\frac{1}{3}\times 4\\\frac{1}{6}\left(-1\right)-\frac{1}{12}\times 4\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=1,y=-\frac{1}{2}$$

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.

$$x+4y=-1,2x-4y=4$$

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.

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

Whakarūnātia.

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

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

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

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.

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

Tāpiri 8y ki te 4y.

$$12y=-2-4$$

Tāpiri -2 ki te -4.

$$12y=-6$$

Whakawehea ngā taha e rua ki te 12.

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

Whakaurua te -\frac{1}{2} mō y ki 2x-4y=4. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

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

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

$$2x+2=4$$

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

$$2x=2$$

Whakawehea ngā taha e rua ki te 2.

$$x=1$$

Kua oti te pūnaha te whakatau.

$$x=1,y=-\frac{1}{2}$$