Whakaoti i te {l}{-x+y=-6}{3x-2y=10}

$$\left. \begin{array} { l } { - x + y = - 6 } \\ { 3 x - 2 y = 10 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-2$$

$$y=-8$$

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.

$$-x+y=-6,3x-2y=10$$

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

$$-x+y=-6$$

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

$$-x=-y-6$$

Whakawehea ngā taha e rua ki te -1.

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

Whakareatia -1 ki te -y-6.

$$x=y+6$$

Whakakapia te y+6 mō te x ki tērā atu whārite, 3x-2y=10.

$$3\left(y+6\right)-2y=10$$

Whakareatia 3 ki te y+6.

$$3y+18-2y=10$$

Tāpiri 3y ki te -2y.

$$y+18=10$$

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

$$y=-8$$

Whakaurua te -8 mō y ki x=y+6. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-8+6$$

Tāpiri 6 ki te -8.

$$x=-2$$

Kua oti te pūnaha te whakatau.

$$x=-2,y=-8$$

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.

$$-x+y=-6,3x-2y=10$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-1&1\\3&-2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-6\\10\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}-1&1\\3&-2\end{matrix}\right).

$$inverse(\left(\begin{matrix}-1&1\\3&-2\end{matrix}\right))\left(\begin{matrix}-1&1\\3&-2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-1&1\\3&-2\end{matrix}\right))\left(\begin{matrix}-6\\10\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-1&1\\3&-2\end{matrix}\right))\left(\begin{matrix}-6\\10\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-1&1\\3&-2\end{matrix}\right))\left(\begin{matrix}-6\\10\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{-\left(-2\right)-3}&-\frac{1}{-\left(-2\right)-3}\\-\frac{3}{-\left(-2\right)-3}&-\frac{1}{-\left(-2\right)-3}\end{matrix}\right)\left(\begin{matrix}-6\\10\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}2&1\\3&1\end{matrix}\right)\left(\begin{matrix}-6\\10\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}2\left(-6\right)+10\\3\left(-6\right)+10\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-2\\-8\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-2,y=-8$$

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.

$$-x+y=-6,3x-2y=10$$

Kia ōrite ai a -x me 3x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 3 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te -1.

$$3\left(-1\right)x+3y=3\left(-6\right),-3x-\left(-2y\right)=-10$$

Whakarūnātia.

$$-3x+3y=-18,-3x+2y=-10$$

Me tango -3x+2y=-10 mai i -3x+3y=-18 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-3x+3x+3y-2y=-18+10$$

Tāpiri -3x ki te 3x. Ka whakakore atu ngā kupu -3x me 3x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$3y-2y=-18+10$$

Tāpiri 3y ki te -2y.

$$y=-18+10$$

Tāpiri -18 ki te 10.

$$y=-8$$

Whakaurua te -8 mō y ki 3x-2y=10. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$3x-2\left(-8\right)=10$$

Whakareatia -2 ki te -8.

$$3x+16=10$$

Me tango 16 mai i ngā taha e rua o te whārite.

$$3x=-6$$

Whakawehea ngā taha e rua ki te 3.

$$x=-2$$

Kua oti te pūnaha te whakatau.

$$x=-2,y=-8$$