Whakaoti i te {c}{2x+2y=4}{-2x+3y=-9}

$$\left. \begin{array} { c } { 2 x + 2 y = 4 } \\ { - 2 x + 3 y = - 9 } \end{array} \right.$$

Whakaoti mō x, y

$$x=3$$

$$y=-1$$

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.

$$2x+2y=4,-2x+3y=-9$$

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

$$2x+2y=4$$

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

$$2x=-2y+4$$

Whakawehea ngā taha e rua ki te 2.

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

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

$$x=-y+2$$

Whakakapia te -y+2 mō te x ki tērā atu whārite, -2x+3y=-9.

$$-2\left(-y+2\right)+3y=-9$$

Whakareatia -2 ki te -y+2.

$$2y-4+3y=-9$$

Tāpiri 2y ki te 3y.

$$5y-4=-9$$

Me tāpiri 4 ki ngā taha e rua o te whārite.

$$5y=-5$$

Whakawehea ngā taha e rua ki te 5.

$$y=-1$$

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

$$x=-\left(-1\right)+2$$

Whakareatia -1 ki te -1.

$$x=1+2$$

Tāpiri 2 ki te 1.

$$x=3$$

Kua oti te pūnaha te whakatau.

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

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.

$$2x+2y=4,-2x+3y=-9$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&2\\-2&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}4\\-9\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}2&2\\-2&3\end{matrix}\right))\left(\begin{matrix}2&2\\-2&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&2\\-2&3\end{matrix}\right))\left(\begin{matrix}4\\-9\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&2\\-2&3\end{matrix}\right))\left(\begin{matrix}4\\-9\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&2\\-2&3\end{matrix}\right))\left(\begin{matrix}4\\-9\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{2\times 3-2\left(-2\right)}&-\frac{2}{2\times 3-2\left(-2\right)}\\-\frac{-2}{2\times 3-2\left(-2\right)}&\frac{2}{2\times 3-2\left(-2\right)}\end{matrix}\right)\left(\begin{matrix}4\\-9\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{10}&-\frac{1}{5}\\\frac{1}{5}&\frac{1}{5}\end{matrix}\right)\left(\begin{matrix}4\\-9\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{10}\times 4-\frac{1}{5}\left(-9\right)\\\frac{1}{5}\times 4+\frac{1}{5}\left(-9\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

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

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.

$$2x+2y=4,-2x+3y=-9$$

Kia ōrite ai a 2x 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 2.

$$-2\times 2x-2\times 2y=-2\times 4,2\left(-2\right)x+2\times 3y=2\left(-9\right)$$

Whakarūnātia.

$$-4x-4y=-8,-4x+6y=-18$$

Me tango -4x+6y=-18 mai i -4x-4y=-8 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-4x+4x-4y-6y=-8+18$$

Tāpiri -4x ki te 4x. Ka whakakore atu ngā kupu -4x me 4x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-4y-6y=-8+18$$

Tāpiri -4y ki te -6y.

$$-10y=-8+18$$

Tāpiri -8 ki te 18.

$$-10y=10$$

Whakawehea ngā taha e rua ki te -10.

$$y=-1$$

Whakaurua te -1 mō y ki -2x+3y=-9. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-2x+3\left(-1\right)=-9$$

Whakareatia 3 ki te -1.

$$-2x-3=-9$$

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

$$-2x=-6$$

Whakawehea ngā taha e rua ki te -2.

$$x=3$$

Kua oti te pūnaha te whakatau.

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