Whakaoti i te {l}{3x+4y=0}{6x+2y=0}

$$\left. \begin{array} { l } { 3 x + 4 y = 0 } \\ { 6 x + 2 y = 0 } \end{array} \right.$$

Whakaoti mō x, y

$$x=0$$

$$y=0$$

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.

$$3x+4y=0,6x+2y=0$$

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

$$3x+4y=0$$

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

$$3x=-4y$$

Whakawehea ngā taha e rua ki te 3.

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

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

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

Whakakapia te -\frac{4y}{3} mō te x ki tērā atu whārite, 6x+2y=0.

$$6\left(-\frac{4}{3}\right)y+2y=0$$

Whakareatia 6 ki te -\frac{4y}{3}.

$$-8y+2y=0$$

Tāpiri -8y ki te 2y.

$$-6y=0$$

Whakawehea ngā taha e rua ki te -6.

$$y=0$$

Whakaurua te 0 mō y ki x=-\frac{4}{3}y. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=0$$

Kua oti te pūnaha te whakatau.

$$x=0,y=0$$

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.

$$3x+4y=0,6x+2y=0$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&4\\6&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}0\\0\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}3&4\\6&2\end{matrix}\right).

$$inverse(\left(\begin{matrix}3&4\\6&2\end{matrix}\right))\left(\begin{matrix}3&4\\6&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&4\\6&2\end{matrix}\right))\left(\begin{matrix}0\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&4\\6&2\end{matrix}\right))\left(\begin{matrix}0\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&4\\6&2\end{matrix}\right))\left(\begin{matrix}0\\0\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{3\times 2-4\times 6}&-\frac{4}{3\times 2-4\times 6}\\-\frac{6}{3\times 2-4\times 6}&\frac{3}{3\times 2-4\times 6}\end{matrix}\right)\left(\begin{matrix}0\\0\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{9}&\frac{2}{9}\\\frac{1}{3}&-\frac{1}{6}\end{matrix}\right)\left(\begin{matrix}0\\0\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}0\\0\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=0,y=0$$

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.

$$3x+4y=0,6x+2y=0$$

Kia ōrite ai a 3x me 6x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 6 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 3.

$$6\times 3x+6\times 4y=0,3\times 6x+3\times 2y=0$$

Whakarūnātia.

$$18x+24y=0,18x+6y=0$$

Me tango 18x+6y=0 mai i 18x+24y=0 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$18x-18x+24y-6y=0$$

Tāpiri 18x ki te -18x. Ka whakakore atu ngā kupu 18x me -18x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$24y-6y=0$$

Tāpiri 24y ki te -6y.

$$18y=0$$

Whakawehea ngā taha e rua ki te 18.

$$y=0$$

Whakaurua te 0 mō y ki 6x+2y=0. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$6x=0$$

Whakawehea ngā taha e rua ki te 6.

$$x=0$$

Kua oti te pūnaha te whakatau.

$$x=0,y=0$$