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

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

Whakaoti mō x, y

$$x=1$$

$$y=0$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuarua. Me tāpiri te 4x ki ngā taha e rua.

$$y+4x=4$$

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.

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

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

$$x-y=1$$

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

$$x=y+1$$

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

$$4\left(y+1\right)+y=4$$

Whakareatia 4 ki te y+1.

$$4y+4+y=4$$

Tāpiri 4y ki te y.

$$5y+4=4$$

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

$$5y=0$$

Whakawehea ngā taha e rua ki te 5.

$$y=0$$

Whakaurua te 0 mō y ki x=y+1. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=1$$

Kua oti te pūnaha te whakatau.

$$x=1,y=0$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuarua. Me tāpiri te 4x ki ngā taha e rua.

$$y+4x=4$$

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

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{5}&\frac{1}{5}\\-\frac{4}{5}&\frac{1}{5}\end{matrix}\right)\left(\begin{matrix}1\\4\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{5}+\frac{1}{5}\times 4\\-\frac{4}{5}+\frac{1}{5}\times 4\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}1\\0\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=1,y=0$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuarua. Me tāpiri te 4x ki ngā taha e rua.

$$y+4x=4$$

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.

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

Kia ōrite ai a x me 4x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 4 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 1.

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

Whakarūnātia.

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

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

$$4x-4x-4y-y=4-4$$

Tāpiri 4x ki te -4x. Ka whakakore atu ngā kupu 4x me -4x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-4y-y=4-4$$

Tāpiri -4y ki te -y.

$$-5y=4-4$$

Tāpiri 4 ki te -4.

$$-5y=0$$

Whakawehea ngā taha e rua ki te -5.

$$y=0$$

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

$$4x=4$$

Whakawehea ngā taha e rua ki te 4.

$$x=1$$

Kua oti te pūnaha te whakatau.

$$x=1,y=0$$