Whakaoti i te {l}{-x+3y=6}{x-7y=14}

$$\left\{ \begin{array} { l } { - x + 3 y = 6 } \\ { x - 7 y = 14 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-21$$

$$y=-5$$

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.

$$-x+3y=6,x-7y=14$$

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

$$-x+3y=6$$

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

$$-x=-3y+6$$

Whakawehea ngā taha e rua ki te -1.

$$x=-\left(-3y+6\right)$$

Whakareatia -1 ki te -3y+6.

$$x=3y-6$$

Whakakapia te -6+3y mō te x ki tērā atu whārite, x-7y=14.

$$3y-6-7y=14$$

Tāpiri 3y ki te -7y.

$$-4y-6=14$$

Me tāpiri 6 ki ngā taha e rua o te whārite.

$$-4y=20$$

Whakawehea ngā taha e rua ki te -4.

$$y=-5$$

Whakaurua te -5 mō y ki x=3y-6. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=3\left(-5\right)-6$$

Whakareatia 3 ki te -5.

$$x=-15-6$$

Tāpiri -6 ki te -15.

$$x=-21$$

Kua oti te pūnaha te whakatau.

$$x=-21,y=-5$$

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.

$$-x+3y=6,x-7y=14$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}-1&3\\1&-7\end{matrix}\right).

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

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-1&3\\1&-7\end{matrix}\right))\left(\begin{matrix}6\\14\end{matrix}\right)$$

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

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

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.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{7}{4}&-\frac{3}{4}\\-\frac{1}{4}&-\frac{1}{4}\end{matrix}\right)\left(\begin{matrix}6\\14\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{7}{4}\times 6-\frac{3}{4}\times 14\\-\frac{1}{4}\times 6-\frac{1}{4}\times 14\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-21\\-5\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-21,y=-5$$

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.

$$-x+3y=6,x-7y=14$$

Kia ōrite ai a -x 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 -1.

$$-x+3y=6,-x-\left(-7y\right)=-14$$

Whakarūnātia.

$$-x+3y=6,-x+7y=-14$$

Me tango -x+7y=-14 mai i -x+3y=6 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-x+x+3y-7y=6+14$$

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

$$3y-7y=6+14$$

Tāpiri 3y ki te -7y.

$$-4y=6+14$$

Tāpiri 6 ki te 14.

$$-4y=20$$

Whakawehea ngā taha e rua ki te -4.

$$y=-5$$

Whakaurua te -5 mō y ki x-7y=14. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x-7\left(-5\right)=14$$

Whakareatia -7 ki te -5.

$$x+35=14$$

Me tango 35 mai i ngā taha e rua o te whārite.

$$x=-21$$

Kua oti te pūnaha te whakatau.

$$x=-21,y=-5$$