Whakaoti i te {c}{-x+2y=0}{2x-3y+1=0}

$$\left\{ \begin{array} { c } { - x + 2 y = 0 } \\ { 2 x - 3 y + 1 = 0 } \end{array} \right.$$

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.

$$-x+2y=0,2x-3y+1=0$$

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

$$-x+2y=0$$

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

$$-x=-2y$$

Whakawehea ngā taha e rua ki te -1.

$$x=-\left(-2\right)y$$

Whakareatia -1 ki te -2y.

$$x=2y$$

Whakakapia te 2y mō te x ki tērā atu whārite, 2x-3y+1=0.

$$2\times 2y-3y+1=0$$

Whakareatia 2 ki te 2y.

$$4y-3y+1=0$$

Tāpiri 4y ki te -3y.

$$y+1=0$$

Me tango 1 mai i ngā taha e rua o te whārite.

$$y=-1$$

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

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

Whakareatia 2 ki te -1.

$$x=-2$$

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.

$$-x+2y=0,2x-3y+1=0$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

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.

$$-x+2y=0,2x-3y+1=0$$

Kia ōrite ai a -x 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 -1.

$$2\left(-1\right)x+2\times 2y=0,-2x-\left(-3y\right)-1=0$$

Whakarūnātia.

$$-2x+4y=0,-2x+3y-1=0$$

Me tango -2x+3y-1=0 mai i -2x+4y=0 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-2x+2x+4y-3y+1=0$$

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.

$$4y-3y+1=0$$

Tāpiri 4y ki te -3y.

$$y+1=0$$

Me tango 1 mai i ngā taha e rua o te whārite.

$$y=-1$$

Whakaurua te -1 mō y ki 2x-3y+1=0. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$2x-3\left(-1\right)+1=0$$

Whakareatia -3 ki te -1.

$$2x+3+1=0$$

Tāpiri 3 ki te 1.

$$2x+4=0$$

Me tango 4 mai i ngā taha e rua o te whārite.

$$2x=-4$$

Whakawehea ngā taha e rua ki te 2.

$$x=-2$$

Kua oti te pūnaha te whakatau.

$$x=-2,y=-1$$