Whakaoti i te {l}{x+3y=1}{2x+3y=1}

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

Whakaoti mō x, y

$$x=0$$

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

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.

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

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

$$x+3y=1$$

Me tango 3y mai i ngā taha e rua o te whārite.

$$x=-3y+1$$

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

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

Whakareatia 2 ki te -3y+1.

$$-6y+2+3y=1$$

Tāpiri -6y ki te 3y.

$$-3y+2=1$$

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

$$-3y=-1$$

Whakawehea ngā taha e rua ki te -3.

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

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

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

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

$$x=-1+1$$

Tāpiri 1 ki te -1.

$$x=0$$

Kua oti te pūnaha te whakatau.

$$x=0,y=\frac{1}{3}$$

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.

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=0,y=\frac{1}{3}$$

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.

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

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

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

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

$$x-2x=1-1$$

Tāpiri x ki te -2x.

$$-x=1-1$$

Tāpiri 1 ki te -1.

$$-x=0$$

Whakawehea ngā taha e rua ki te -1.

$$x=0$$

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

$$3y=1$$

Whakawehea ngā taha e rua ki te 3.

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

Kua oti te pūnaha te whakatau.

$$x=0,y=\frac{1}{3}$$