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

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

Whakaoti mō x, y

$$x=2$$

$$y=1$$

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.

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

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

$$4x-2y=6$$

Me tāpiri 2y ki ngā taha e rua o te whārite.

$$4x=2y+6$$

Whakawehea ngā taha e rua ki te 4.

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

Whakareatia \frac{1}{4} ki te 6+2y.

$$x=\frac{1}{2}y+\frac{3}{2}$$

Whakakapia te \frac{3+y}{2} mō te x ki tērā atu whārite, x+2y=4.

$$\frac{1}{2}y+\frac{3}{2}+2y=4$$

Tāpiri \frac{y}{2} ki te 2y.

$$\frac{5}{2}y+\frac{3}{2}=4$$

Me tango \frac{3}{2} mai i ngā taha e rua o te whārite.

$$\frac{5}{2}y=\frac{5}{2}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{5}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=1$$

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

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

Tāpiri \frac{3}{2} ki te \frac{1}{2} mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=2$$

Kua oti te pūnaha te whakatau.

$$x=2,y=1$$

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.

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}4&-2\\1&2\end{matrix}\right).

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{5}&\frac{1}{5}\\-\frac{1}{10}&\frac{2}{5}\end{matrix}\right)\left(\begin{matrix}6\\4\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{5}\times 6+\frac{1}{5}\times 4\\-\frac{1}{10}\times 6+\frac{2}{5}\times 4\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=2,y=1$$

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.

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

Kia ōrite ai a 4x me x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 1 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 4.

$$4x-2y=6,4x+4\times 2y=4\times 4$$

Whakarūnātia.

$$4x-2y=6,4x+8y=16$$

Me tango 4x+8y=16 mai i 4x-2y=6 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$4x-4x-2y-8y=6-16$$

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

$$-2y-8y=6-16$$

Tāpiri -2y ki te -8y.

$$-10y=6-16$$

Tāpiri 6 ki te -16.

$$-10y=-10$$

Whakawehea ngā taha e rua ki te -10.

$$y=1$$

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

$$x+2=4$$

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

$$x=2$$

Kua oti te pūnaha te whakatau.

$$x=2,y=1$$