Whakaoti i te {r}{2x-3y=1}{x+2y=4}

$$\left.\begin{array} { r } { 2 x - 3 y = 1 } \\ { x + 2 y = 4 } \end{array} \right\}$$

Whakaoti mō x, y

$$x=2$$

$$y=1$$

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.

$$2x-3y=1,x+2y=4$$

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

$$2x-3y=1$$

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

$$2x=3y+1$$

Whakawehea ngā taha e rua ki te 2.

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

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

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

Whakakapia te \frac{3y+1}{2} mō te x ki tērā atu whārite, x+2y=4.

$$\frac{3}{2}y+\frac{1}{2}+2y=4$$

Tāpiri \frac{3y}{2} ki te 2y.

$$\frac{7}{2}y+\frac{1}{2}=4$$

Me tango \frac{1}{2} mai i ngā taha e rua o te whārite.

$$\frac{7}{2}y=\frac{7}{2}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{7}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=1$$

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

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

Tāpiri \frac{1}{2} ki te \frac{3}{2} mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=2$$

Kua oti te pūnaha te whakatau.

$$x=2,y=1$$

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.

$$2x-3y=1,x+2y=4$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}2&-3\\1&2\end{matrix}\right).

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

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-3\\1&2\end{matrix}\right))\left(\begin{matrix}1\\4\end{matrix}\right)$$

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

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

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.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{7}&\frac{3}{7}\\-\frac{1}{7}&\frac{2}{7}\end{matrix}\right)\left(\begin{matrix}1\\4\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{7}+\frac{3}{7}\times 4\\-\frac{1}{7}+\frac{2}{7}\times 4\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=2,y=1$$

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.

$$2x-3y=1,x+2y=4$$

Kia ōrite ai a 2x me x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 1 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 2.

$$2x-3y=1,2x+2\times 2y=2\times 4$$

Whakarūnātia.

$$2x-3y=1,2x+4y=8$$

Me tango 2x+4y=8 mai i 2x-3y=1 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$2x-2x-3y-4y=1-8$$

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

$$-3y-4y=1-8$$

Tāpiri -3y ki te -4y.

$$-7y=1-8$$

Tāpiri 1 ki te -8.

$$-7y=-7$$

Whakawehea ngā taha e rua ki te -7.

$$y=1$$

Whakaurua te 1 mō y ki x+2y=4. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x+2=4$$

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

$$x=2$$

Kua oti te pūnaha te whakatau.

$$x=2,y=1$$