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

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

Whakaoti mō x, y

$$x = \frac{3}{2} = 1\frac{1}{2} = 1.5$$

$$y=\frac{1}{6}\approx 0.166666667$$

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.

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

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 tāpiri 3y ki ngā taha e rua o te whārite.

$$x=3y+1$$

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

$$3y+1+3y=2$$

Tāpiri 3y ki te 3y.

$$6y+1=2$$

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

$$6y=1$$

Whakawehea ngā taha e rua ki te 6.

$$y=\frac{1}{6}$$

Whakaurua te \frac{1}{6} 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.

$$x=3\times \frac{1}{6}+1$$

Whakareatia 3 ki te \frac{1}{6}.

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

Tāpiri 1 ki te \frac{1}{2}.

$$x=\frac{3}{2}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{3}{2},y=\frac{1}{6}$$

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.

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2}+\frac{1}{2}\times 2\\-\frac{1}{6}+\frac{1}{6}\times 2\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=\frac{3}{2},y=\frac{1}{6}$$

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.

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

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

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

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

$$-3y-3y=1-2$$

Tāpiri -3y ki te -3y.

$$-6y=1-2$$

Tāpiri 1 ki te -2.

$$-6y=-1$$

Whakawehea ngā taha e rua ki te -6.

$$y=\frac{1}{6}$$

Whakaurua te \frac{1}{6} mō y ki x+3y=2. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x+3\times \frac{1}{6}=2$$

Whakareatia 3 ki te \frac{1}{6}.

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

Me tango \frac{1}{2} mai i ngā taha e rua o te whārite.

$$x=\frac{3}{2}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{3}{2},y=\frac{1}{6}$$