Whakaoti i te {l}{2x+4y=1}{2x-6y=-4}

$$\left\{ \begin{array} { l } { 2 x + 4 y = 1 } \\ { 2 x - 6 y = - 4 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-\frac{1}{2}=-0.5$$

$$y=\frac{1}{2}=0.5$$

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.

$$2x+4y=1,2x-6y=-4$$

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

$$2x+4y=1$$

Me tango 4y mai i ngā taha e rua o te whārite.

$$2x=-4y+1$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{1}{2}\left(-4y+1\right)$$

Whakareatia \frac{1}{2} ki te -4y+1.

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

Whakakapia te -2y+\frac{1}{2} mō te x ki tērā atu whārite, 2x-6y=-4.

$$2\left(-2y+\frac{1}{2}\right)-6y=-4$$

Whakareatia 2 ki te -2y+\frac{1}{2}.

$$-4y+1-6y=-4$$

Tāpiri -4y ki te -6y.

$$-10y+1=-4$$

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

$$-10y=-5$$

Whakawehea ngā taha e rua ki te -10.

$$y=\frac{1}{2}$$

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

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

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

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

Tāpiri \frac{1}{2} ki te -1.

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

Kua oti te pūnaha te whakatau.

$$x=-\frac{1}{2},y=\frac{1}{2}$$

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.

$$2x+4y=1,2x-6y=-4$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&4\\2&-6\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}1\\-4\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}2&4\\2&-6\end{matrix}\right).

$$inverse(\left(\begin{matrix}2&4\\2&-6\end{matrix}\right))\left(\begin{matrix}2&4\\2&-6\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&4\\2&-6\end{matrix}\right))\left(\begin{matrix}1\\-4\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&4\\2&-6\end{matrix}\right))\left(\begin{matrix}1\\-4\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&4\\2&-6\end{matrix}\right))\left(\begin{matrix}1\\-4\end{matrix}\right)$$

Mō te poukapa 2\times 2 \left(\begin{matrix}a&b\\c&d\end{matrix}\right), ko te poukapa kōaro ko \left(\begin{matrix}\frac{d}{ad-bc}&\frac{-b}{ad-bc}\\\frac{-c}{ad-bc}&\frac{a}{ad-bc}\end{matrix}\right), kia tuhia anō ai te whārite poukapa hei rapanga whakarea poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{6}{2\left(-6\right)-4\times 2}&-\frac{4}{2\left(-6\right)-4\times 2}\\-\frac{2}{2\left(-6\right)-4\times 2}&\frac{2}{2\left(-6\right)-4\times 2}\end{matrix}\right)\left(\begin{matrix}1\\-4\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{10}&\frac{1}{5}\\\frac{1}{10}&-\frac{1}{10}\end{matrix}\right)\left(\begin{matrix}1\\-4\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{10}+\frac{1}{5}\left(-4\right)\\\frac{1}{10}-\frac{1}{10}\left(-4\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=-\frac{1}{2},y=\frac{1}{2}$$

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.

$$2x+4y=1,2x-6y=-4$$

Me tango 2x-6y=-4 mai i 2x+4y=1 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$2x-2x+4y+6y=1+4$$

Tāpiri 2x ki te -2x. Ka whakakore atu ngā kupu 2x me -2x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$4y+6y=1+4$$

Tāpiri 4y ki te 6y.

$$10y=1+4$$

Tāpiri 1 ki te 4.

$$10y=5$$

Whakawehea ngā taha e rua ki te 10.

$$y=\frac{1}{2}$$

Whakaurua te \frac{1}{2} mō y ki 2x-6y=-4. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$2x-6\times \frac{1}{2}=-4$$

Whakareatia -6 ki te \frac{1}{2}.

$$2x-3=-4$$

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

$$2x=-1$$

Whakawehea ngā taha e rua ki te 2.

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

Kua oti te pūnaha te whakatau.

$$x=-\frac{1}{2},y=\frac{1}{2}$$