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

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

Whakaoti mō x, y

$$x=1$$

$$y=4$$

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.

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

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

$$2x+y=6$$

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

$$2x=-y+6$$

Whakawehea ngā taha e rua ki te 2.

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

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

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

Whakakapia te -\frac{y}{2}+3 mō te x ki tērā atu whārite, 6x-y=2.

$$6\left(-\frac{1}{2}y+3\right)-y=2$$

Whakareatia 6 ki te -\frac{y}{2}+3.

$$-3y+18-y=2$$

Tāpiri -3y ki te -y.

$$-4y+18=2$$

Me tango 18 mai i ngā taha e rua o te whārite.

$$-4y=-16$$

Whakawehea ngā taha e rua ki te -4.

$$y=4$$

Whakaurua te 4 mō y ki x=-\frac{1}{2}y+3. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{1}{2}\times 4+3$$

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

$$x=-2+3$$

Tāpiri 3 ki te -2.

$$x=1$$

Kua oti te pūnaha te whakatau.

$$x=1,y=4$$

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.

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}2&1\\6&-1\end{matrix}\right).

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{8}&\frac{1}{8}\\\frac{3}{4}&-\frac{1}{4}\end{matrix}\right)\left(\begin{matrix}6\\2\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{8}\times 6+\frac{1}{8}\times 2\\\frac{3}{4}\times 6-\frac{1}{4}\times 2\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=1,y=4$$

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.

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

Kia ōrite ai a 2x me 6x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 6 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 2.

$$6\times 2x+6y=6\times 6,2\times 6x+2\left(-1\right)y=2\times 2$$

Whakarūnātia.

$$12x+6y=36,12x-2y=4$$

Me tango 12x-2y=4 mai i 12x+6y=36 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$12x-12x+6y+2y=36-4$$

Tāpiri 12x ki te -12x. Ka whakakore atu ngā kupu 12x me -12x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$6y+2y=36-4$$

Tāpiri 6y ki te 2y.

$$8y=36-4$$

Tāpiri 36 ki te -4.

$$8y=32$$

Whakawehea ngā taha e rua ki te 8.

$$y=4$$

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

$$6x-4=2$$

Me tāpiri 4 ki ngā taha e rua o te whārite.

$$6x=6$$

Whakawehea ngā taha e rua ki te 6.

$$x=1$$

Kua oti te pūnaha te whakatau.

$$x=1,y=4$$