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

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

Whakaoti mō x, y

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

$$y=\frac{1}{4}=0.25$$

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.

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

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

$$2x+4y=2$$

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

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

Whakawehea ngā taha e rua ki te 2.

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

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

$$x=-2y+1$$

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

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

Whakareatia 6 ki te -2y+1.

$$-12y+6+4y=4$$

Tāpiri -12y ki te 4y.

$$-8y+6=4$$

Me tango 6 mai i ngā taha e rua o te whārite.

$$-8y=-2$$

Whakawehea ngā taha e rua ki te -8.

$$y=\frac{1}{4}$$

Whakaurua te \frac{1}{4} 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.

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

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

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

Tāpiri 1 ki te -\frac{1}{2}.

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

Kua oti te pūnaha te whakatau.

$$x=\frac{1}{2},y=\frac{1}{4}$$

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.

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{4}&\frac{1}{4}\\\frac{3}{8}&-\frac{1}{8}\end{matrix}\right)\left(\begin{matrix}2\\4\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{4}\times 2+\frac{1}{4}\times 4\\\frac{3}{8}\times 2-\frac{1}{8}\times 4\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=\frac{1}{2},y=\frac{1}{4}$$

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.

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

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

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

Tāpiri 4y ki te -4y. Ka whakakore atu ngā kupu 4y me -4y, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$2x-6x=2-4$$

Tāpiri 2x ki te -6x.

$$-4x=2-4$$

Tāpiri 2 ki te -4.

$$-4x=-2$$

Whakawehea ngā taha e rua ki te -4.

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

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

$$6\times \frac{1}{2}+4y=4$$

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

$$3+4y=4$$

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

$$4y=1$$

Whakawehea ngā taha e rua ki te 4.

$$y=\frac{1}{4}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{1}{2},y=\frac{1}{4}$$