Whakaoti i te {l}{2x+y=3}{y=2x+1}

$$\left. \begin{array} { l } { 2 x + y = 3 } \\ { y = 2 x + 1 } \end{array} \right.$$

Whakaoti mō x, y

$$x=\frac{1}{2}=0.5$$

$$y=2$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuarua. 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.

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

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

$$2x+y=3$$

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

$$2x=-y+3$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{1}{2}\left(-y+3\right)$$

Whakareatia \frac{1}{2} ki te -y+3.

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

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

$$-2\left(-\frac{1}{2}y+\frac{3}{2}\right)+y=1$$

Whakareatia -2 ki te \frac{-y+3}{2}.

$$y-3+y=1$$

Tāpiri y ki te y.

$$2y-3=1$$

Me tāpiri 3 ki ngā taha e rua o te whārite.

$$2y=4$$

Whakawehea ngā taha e rua ki te 2.

$$y=2$$

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

$$x=-\frac{1}{2}\times 2+\frac{3}{2}$$

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

$$x=-1+\frac{3}{2}$$

Tāpiri \frac{3}{2} ki te -1.

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

Kua oti te pūnaha te whakatau.

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

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuarua. 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.

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

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

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuarua. 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.

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

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

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

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.

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

Tāpiri 2x ki te 2x.

$$4x=3-1$$

Tāpiri 3 ki te -1.

$$4x=2$$

Whakawehea ngā taha e rua ki te 4.

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

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

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

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

$$-1+y=1$$

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

$$y=2$$

Kua oti te pūnaha te whakatau.

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