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

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

Whakaoti mō x, y

$$x=-\frac{1}{5}=-0.2$$

$$y=\frac{3}{5}=0.6$$

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.

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

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

$$x+2y=1$$

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

$$x=-2y+1$$

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

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

Whakareatia 3 ki te -2y+1.

$$-6y+3+y=0$$

Tāpiri -6y ki te y.

$$-5y+3=0$$

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

$$-5y=-3$$

Whakawehea ngā taha e rua ki te -5.

$$y=\frac{3}{5}$$

Whakaurua te \frac{3}{5} mō y ki x=-2y+1. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-2\times \frac{3}{5}+1$$

Whakareatia -2 ki te \frac{3}{5}.

$$x=-\frac{6}{5}+1$$

Tāpiri 1 ki te -\frac{6}{5}.

$$x=-\frac{1}{5}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{1}{5},y=\frac{3}{5}$$

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.

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{5}&\frac{2}{5}\\\frac{3}{5}&-\frac{1}{5}\end{matrix}\right)\left(\begin{matrix}1\\0\end{matrix}\right)$$

Whakareatia ngā poukapa.

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

Tangohia ngā huānga poukapa x me y.

$$x=-\frac{1}{5},y=\frac{3}{5}$$

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.

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

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

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

Whakarūnātia.

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

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

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

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.

$$6y-y=3$$

Tāpiri 6y ki te -y.

$$5y=3$$

Whakawehea ngā taha e rua ki te 5.

$$y=\frac{3}{5}$$

Whakaurua te \frac{3}{5} mō y ki 3x+y=0. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$3x+\frac{3}{5}=0$$

Me tango \frac{3}{5} mai i ngā taha e rua o te whārite.

$$3x=-\frac{3}{5}$$

Whakawehea ngā taha e rua ki te 3.

$$x=-\frac{1}{5}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{1}{5},y=\frac{3}{5}$$