Whakaoti i te {r}{4x-2y=6}{-2x+2y=8}

$$\left\{ \begin{array} { r } { 4 x - 2 y = 6 } \\ { - 2 x + 2 y = 8 } \end{array} \right.$$

Whakaoti mō x, y

$$x=7$$

$$y=11$$

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.

$$4x-2y=6,-2x+2y=8$$

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

$$4x-2y=6$$

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

$$4x=2y+6$$

Whakawehea ngā taha e rua ki te 4.

$$x=\frac{1}{4}\left(2y+6\right)$$

Whakareatia \frac{1}{4} ki te 6+2y.

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

Whakakapia te \frac{3+y}{2} mō te x ki tērā atu whārite, -2x+2y=8.

$$-2\left(\frac{1}{2}y+\frac{3}{2}\right)+2y=8$$

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

$$-y-3+2y=8$$

Tāpiri -y ki te 2y.

$$y-3=8$$

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

$$y=11$$

Whakaurua te 11 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.

$$x=\frac{1}{2}\times 11+\frac{3}{2}$$

Whakareatia \frac{1}{2} ki te 11.

$$x=\frac{11+3}{2}$$

Tāpiri \frac{3}{2} ki te \frac{11}{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.

$$x=7$$

Kua oti te pūnaha te whakatau.

$$x=7,y=11$$

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.

$$4x-2y=6,-2x+2y=8$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}4&-2\\-2&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}6\\8\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}4&-2\\-2&2\end{matrix}\right))\left(\begin{matrix}4&-2\\-2&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&-2\\-2&2\end{matrix}\right))\left(\begin{matrix}6\\8\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&-2\\-2&2\end{matrix}\right))\left(\begin{matrix}6\\8\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&-2\\-2&2\end{matrix}\right))\left(\begin{matrix}6\\8\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{4\times 2-\left(-2\left(-2\right)\right)}&-\frac{-2}{4\times 2-\left(-2\left(-2\right)\right)}\\-\frac{-2}{4\times 2-\left(-2\left(-2\right)\right)}&\frac{4}{4\times 2-\left(-2\left(-2\right)\right)}\end{matrix}\right)\left(\begin{matrix}6\\8\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2}&\frac{1}{2}\\\frac{1}{2}&1\end{matrix}\right)\left(\begin{matrix}6\\8\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2}\times 6+\frac{1}{2}\times 8\\\frac{1}{2}\times 6+8\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}7\\11\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=7,y=11$$

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.

$$4x-2y=6,-2x+2y=8$$

Kia ōrite ai a 4x me -2x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te -2 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 4.

$$-2\times 4x-2\left(-2\right)y=-2\times 6,4\left(-2\right)x+4\times 2y=4\times 8$$

Whakarūnātia.

$$-8x+4y=-12,-8x+8y=32$$

Me tango -8x+8y=32 mai i -8x+4y=-12 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-8x+8x+4y-8y=-12-32$$

Tāpiri -8x ki te 8x. Ka whakakore atu ngā kupu -8x me 8x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$4y-8y=-12-32$$

Tāpiri 4y ki te -8y.

$$-4y=-12-32$$

Tāpiri -12 ki te -32.

$$-4y=-44$$

Whakawehea ngā taha e rua ki te -4.

$$y=11$$

Whakaurua te 11 mō y ki -2x+2y=8. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-2x+2\times 11=8$$

Whakareatia 2 ki te 11.

$$-2x+22=8$$

Me tango 22 mai i ngā taha e rua o te whārite.

$$-2x=-14$$

Whakawehea ngā taha e rua ki te -2.

$$x=7$$

Kua oti te pūnaha te whakatau.

$$x=7,y=11$$