Whakaoti i te y=2x-1text{and}2x+y=3

$$y = 2 x - 1 \text { and } 2 x + y = 3$$

Whakaoti mō y, x

$$x=1$$

$$y=1$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Tangohia te 2x mai i ngā taha e rua.

$$y-2x=-1$$

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.

$$y-2x=-1,y+2x=3$$

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

$$y-2x=-1$$

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

$$y=2x-1$$

Whakakapia te 2x-1 mō te y ki tērā atu whārite, y+2x=3.

$$2x-1+2x=3$$

Tāpiri 2x ki te 2x.

$$4x-1=3$$

Me tāpiri 1 ki ngā taha e rua o te whārite.

$$4x=4$$

Whakawehea ngā taha e rua ki te 4.

$$x=1$$

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

$$y=2-1$$

Tāpiri -1 ki te 2.

$$y=1$$

Kua oti te pūnaha te whakatau.

$$y=1,x=1$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Tangohia te 2x mai i ngā taha e rua.

$$y-2x=-1$$

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

$$y-2x=-1,y+2x=3$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

Mō te poukapa 2\times 2 \left(\begin{matrix}a&b\\c&d\end{matrix}\right), ko te poukapa kōaro ko \left(\begin{matrix}\frac{d}{ad-bc}&\frac{-b}{ad-bc}\\\frac{-c}{ad-bc}&\frac{a}{ad-bc}\end{matrix}\right), kia tuhia anō ai te whārite poukapa hei rapanga whakarea poukapa.

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

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa y me x.

$$y=1,x=1$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Tangohia te 2x mai i ngā taha e rua.

$$y-2x=-1$$

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.

$$y-2x=-1,y+2x=3$$

Me tango y+2x=3 mai i y-2x=-1 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$y-y-2x-2x=-1-3$$

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

$$-2x-2x=-1-3$$

Tāpiri -2x ki te -2x.

$$-4x=-1-3$$

Tāpiri -1 ki te -3.

$$-4x=-4$$

Whakawehea ngā taha e rua ki te -4.

$$x=1$$

Whakaurua te 1 mō x ki y+2x=3. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y+2=3$$

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

$$y=1$$

Kua oti te pūnaha te whakatau.

$$y=1,x=1$$