Whakaoti i te {c}{x+y=0}{4y+x=1}

$$\left. \begin{array} { c } { x + y = 0 } \\ { 4 y + x = 1 } \end{array} \right.$$

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.

$$x+y=0,x+4y=1$$

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

$$x+y=0$$

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

$$x=-y$$

Whakakapia te -y mō te x ki tērā atu whārite, x+4y=1.

$$-y+4y=1$$

Tāpiri -y ki te 4y.

$$3y=1$$

Whakawehea ngā taha e rua ki te 3.

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

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

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

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.

$$x+y=0,x+4y=1$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&1\\1&4\end{matrix}\right).

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

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

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

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

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

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{4-1}&-\frac{1}{4-1}\\-\frac{1}{4-1}&\frac{1}{4-1}\end{matrix}\right)\left(\begin{matrix}0\\1\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{3}&-\frac{1}{3}\\-\frac{1}{3}&\frac{1}{3}\end{matrix}\right)\left(\begin{matrix}0\\1\end{matrix}\right)$$

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.

$$x+y=0,x+4y=1$$

Me tango x+4y=1 mai i x+y=0 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$x-x+y-4y=-1$$

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.

$$y-4y=-1$$

Tāpiri y ki te -4y.

$$-3y=-1$$

Whakawehea ngā taha e rua ki te -3.

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

Whakaurua te \frac{1}{3} mō y ki x+4y=1. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x+4\times \frac{1}{3}=1$$

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

$$x+\frac{4}{3}=1$$

Me tango \frac{4}{3} mai i ngā taha e rua o te whārite.

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

Kua oti te pūnaha te whakatau.

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