Whakaoti i te {c}{x-4y=-13}{6x+4y=6}

$$\left\{ \begin{array} { c } { x - 4 y = - 13 } \\ { 6 x + 4 y = 6 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-1$$

$$y=3$$

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.

$$x-4y=-13,6x+4y=6$$

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

$$x-4y=-13$$

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

$$x=4y-13$$

Whakakapia te 4y-13 mō te x ki tērā atu whārite, 6x+4y=6.

$$6\left(4y-13\right)+4y=6$$

Whakareatia 6 ki te 4y-13.

$$24y-78+4y=6$$

Tāpiri 24y ki te 4y.

$$28y-78=6$$

Me tāpiri 78 ki ngā taha e rua o te whārite.

$$28y=84$$

Whakawehea ngā taha e rua ki te 28.

$$y=3$$

Whakaurua te 3 mō y ki x=4y-13. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=4\times 3-13$$

Whakareatia 4 ki te 3.

$$x=12-13$$

Tāpiri -13 ki te 12.

$$x=-1$$

Kua oti te pūnaha te whakatau.

$$x=-1,y=3$$

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.

$$x-4y=-13,6x+4y=6$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&-4\\6&4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-13\\6\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&-4\\6&4\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&-4\\6&4\end{matrix}\right))\left(\begin{matrix}1&-4\\6&4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-4\\6&4\end{matrix}\right))\left(\begin{matrix}-13\\6\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-4\\6&4\end{matrix}\right))\left(\begin{matrix}-13\\6\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-4\\6&4\end{matrix}\right))\left(\begin{matrix}-13\\6\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{4-\left(-4\times 6\right)}&-\frac{-4}{4-\left(-4\times 6\right)}\\-\frac{6}{4-\left(-4\times 6\right)}&\frac{1}{4-\left(-4\times 6\right)}\end{matrix}\right)\left(\begin{matrix}-13\\6\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{7}&\frac{1}{7}\\-\frac{3}{14}&\frac{1}{28}\end{matrix}\right)\left(\begin{matrix}-13\\6\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{7}\left(-13\right)+\frac{1}{7}\times 6\\-\frac{3}{14}\left(-13\right)+\frac{1}{28}\times 6\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=-1,y=3$$

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.

$$x-4y=-13,6x+4y=6$$

Kia ōrite ai a x 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 1.

$$6x+6\left(-4\right)y=6\left(-13\right),6x+4y=6$$

Whakarūnātia.

$$6x-24y=-78,6x+4y=6$$

Me tango 6x+4y=6 mai i 6x-24y=-78 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$6x-6x-24y-4y=-78-6$$

Tāpiri 6x ki te -6x. Ka whakakore atu ngā kupu 6x me -6x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-24y-4y=-78-6$$

Tāpiri -24y ki te -4y.

$$-28y=-78-6$$

Tāpiri -78 ki te -6.

$$-28y=-84$$

Whakawehea ngā taha e rua ki te -28.

$$y=3$$

Whakaurua te 3 mō y ki 6x+4y=6. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$6x+4\times 3=6$$

Whakareatia 4 ki te 3.

$$6x+12=6$$

Me tango 12 mai i 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.

$$x=-1,y=3$$