Whakaoti i te 2x-y=2,4x-y=2

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

Whakaoti mō x, y

$$x=0$$

$$y=-2$$

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.

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

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

$$2x-y=2$$

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

$$2x=y+2$$

Whakawehea ngā taha e rua ki te 2.

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

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

$$x=\frac{1}{2}y+1$$

Whakakapia te \frac{y}{2}+1 mō te x ki tērā atu whārite, 4x-y=2.

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

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

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

Tāpiri 2y ki te -y.

$$y+4=2$$

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

$$y=-2$$

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

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

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

$$x=-1+1$$

Tāpiri 1 ki te -1.

$$x=0$$

Kua oti te pūnaha te whakatau.

$$x=0,y=-2$$

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.

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=0,y=-2$$

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.

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

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

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

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.

$$2x-4x=2-2$$

Tāpiri 2x ki te -4x.

$$-2x=2-2$$

Tāpiri 2 ki te -2.

$$-2x=0$$

Whakawehea ngā taha e rua ki te -2.

$$x=0$$

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

$$-y=2$$

Whakawehea ngā taha e rua ki te -1.

$$y=-2$$

Kua oti te pūnaha te whakatau.

$$x=0,y=-2$$