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

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

Whakaoti mō x, y

$$x=0$$

$$y = \frac{3}{2} = 1\frac{1}{2} = 1.5$$

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.

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

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

$$x+2y=3$$

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

$$x=-2y+3$$

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

$$2\left(-2y+3\right)+2y=3$$

Whakareatia 2 ki te -2y+3.

$$-4y+6+2y=3$$

Tāpiri -4y ki te 2y.

$$-2y+6=3$$

Me tango 6 mai i ngā taha e rua o te whārite.

$$-2y=-3$$

Whakawehea ngā taha e rua ki te -2.

$$y=\frac{3}{2}$$

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

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

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

$$x=-3+3$$

Tāpiri 3 ki te -3.

$$x=0$$

Kua oti te pūnaha te whakatau.

$$x=0,y=\frac{3}{2}$$

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.

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}0\\\frac{3}{2}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=0,y=\frac{3}{2}$$

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.

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

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

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

Tāpiri 2y ki te -2y. Ka whakakore atu ngā kupu 2y me -2y, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

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

Tāpiri x ki te -2x.

$$-x=3-3$$

Tāpiri 3 ki te -3.

$$-x=0$$

Whakawehea ngā taha e rua ki te -1.

$$x=0$$

Whakaurua te 0 mō x ki 2x+2y=3. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$2y=3$$

Whakawehea ngā taha e rua ki te 2.

$$y=\frac{3}{2}$$

Kua oti te pūnaha te whakatau.

$$x=0,y=\frac{3}{2}$$