Whakaoti i te {c}{3x-3y=-2^2}{(4-2)x-y=2(2)}

$$\left. \begin{array} { c } { 3 x - 3 y = - 2 ^ { 2 } } \\ { ( 4 - 2 ) x - y = 2 ( 2 ) } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{16}{3} = 5\frac{1}{3} \approx 5.333333333$$

$$y = \frac{20}{3} = 6\frac{2}{3} \approx 6.666666667$$

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.

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

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

$$3x-3y=-4$$

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

$$3x=3y-4$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{1}{3}\left(3y-4\right)$$

Whakareatia \frac{1}{3} ki te 3y-4.

$$x=y-\frac{4}{3}$$

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

$$2\left(y-\frac{4}{3}\right)-y=4$$

Whakareatia 2 ki te y-\frac{4}{3}.

$$2y-\frac{8}{3}-y=4$$

Tāpiri 2y ki te -y.

$$y-\frac{8}{3}=4$$

Me tāpiri \frac{8}{3} ki ngā taha e rua o te whārite.

$$y=\frac{20}{3}$$

Whakaurua te \frac{20}{3} mō y ki x=y-\frac{4}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{20-4}{3}$$

Tāpiri -\frac{4}{3} ki te \frac{20}{3} 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.

$$x=\frac{16}{3}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{16}{3},y=\frac{20}{3}$$

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.

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{16}{3}\\\frac{20}{3}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{16}{3},y=\frac{20}{3}$$

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.

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

Kia ōrite ai a 3x 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 3.

$$2\times 3x+2\left(-3\right)y=2\left(-4\right),3\times 2x+3\left(-1\right)y=3\times 4$$

Whakarūnātia.

$$6x-6y=-8,6x-3y=12$$

Me tango 6x-3y=12 mai i 6x-6y=-8 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$6x-6x-6y+3y=-8-12$$

Tāpiri 6x ki te -6x. Ka whakakore atu ngā kupu 6x me -6x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-6y+3y=-8-12$$

Tāpiri -6y ki te 3y.

$$-3y=-8-12$$

Tāpiri -8 ki te -12.

$$-3y=-20$$

Whakawehea ngā taha e rua ki te -3.

$$y=\frac{20}{3}$$

Whakaurua te \frac{20}{3} mō y ki 2x-y=4. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$2x-\frac{20}{3}=4$$

Me tāpiri \frac{20}{3} ki ngā taha e rua o te whārite.

$$2x=\frac{32}{3}$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{16}{3}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{16}{3},y=\frac{20}{3}$$