Whakaoti i te {l}{3x+6y=1}{x+y=0}

$$\left. \begin{array} { l } { 3 x + 6 y = 1 } \\ { x + y = 0 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-\frac{1}{3}\approx -0.333333333$$

$$y=\frac{1}{3}\approx 0.333333333$$

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.

$$3x+6y=1,x+y=0$$

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

$$3x+6y=1$$

Me tango 6y mai i ngā taha e rua o te whārite.

$$3x=-6y+1$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{1}{3}\left(-6y+1\right)$$

Whakareatia \frac{1}{3} ki te -6y+1.

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

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

$$-2y+\frac{1}{3}+y=0$$

Tāpiri -2y ki te y.

$$-y+\frac{1}{3}=0$$

Me tango \frac{1}{3} mai i ngā taha e rua o te whārite.

$$-y=-\frac{1}{3}$$

Whakawehea ngā taha e rua ki te -1.

$$y=\frac{1}{3}$$

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

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

Whakareatia -2 ki te \frac{1}{3}.

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

Tāpiri \frac{1}{3} ki te -\frac{2}{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.

$$x=-\frac{1}{3}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{1}{3},y=\frac{1}{3}$$

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.

$$3x+6y=1,x+y=0$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}3&6\\1&1\end{matrix}\right).

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Tangohia ngā huānga poukapa x me y.

$$x=-\frac{1}{3},y=\frac{1}{3}$$

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.

$$3x+6y=1,x+y=0$$

Kia ōrite ai a 3x me x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 1 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 3.

$$3x+6y=1,3x+3y=0$$

Me tango 3x+3y=0 mai i 3x+6y=1 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$3x-3x+6y-3y=1$$

Tāpiri 3x ki te -3x. Ka whakakore atu ngā kupu 3x me -3x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$6y-3y=1$$

Tāpiri 6y ki te -3y.

$$3y=1$$

Whakawehea ngā taha e rua ki te 3.

$$y=\frac{1}{3}$$

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

$$x+\frac{1}{3}=0$$

Me tango \frac{1}{3} mai i ngā taha e rua o te whārite.

$$x=-\frac{1}{3}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{1}{3},y=\frac{1}{3}$$