Whakaoti i te {l}{3y=6x-3}{2x=7-y}

$$\left. \begin{array} { l } { 3 y = 6 x - 3 } \\ { 2 x = 7 - y } \end{array} \right.$$

Whakaoti mō y, x

$$x=2$$

$$y=3$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Tangohia te 6x mai i ngā taha e rua.

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

Whakaarohia te whārite tuarua. Me tāpiri te y ki ngā taha e rua.

$$2x+y=7$$

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.

$$3y-6x=-3,y+2x=7$$

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

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

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

$$3y=6x-3$$

Whakawehea ngā taha e rua ki te 3.

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

Whakareatia \frac{1}{3} ki te 6x-3.

$$y=2x-1$$

Whakakapia te 2x-1 mō te y ki tērā atu whārite, y+2x=7.

$$2x-1+2x=7$$

Tāpiri 2x ki te 2x.

$$4x-1=7$$

Me tāpiri 1 ki ngā taha e rua o te whārite.

$$4x=8$$

Whakawehea ngā taha e rua ki te 4.

$$x=2$$

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

$$y=2\times 2-1$$

Whakareatia 2 ki te 2.

$$y=4-1$$

Tāpiri -1 ki te 4.

$$y=3$$

Kua oti te pūnaha te whakatau.

$$y=3,x=2$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Tangohia te 6x mai i ngā taha e rua.

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

Whakaarohia te whārite tuarua. Me tāpiri te y ki ngā taha e rua.

$$2x+y=7$$

Tuhia ngā whārite ki te tānga ngahuru ka whakamahi i ngā poukapa hei whakaoti i te pūnaha o ngā whārite.

$$3y-6x=-3,y+2x=7$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&-6\\1&2\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-3\\7\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}3&-6\\1&2\end{matrix}\right).

$$inverse(\left(\begin{matrix}3&-6\\1&2\end{matrix}\right))\left(\begin{matrix}3&-6\\1&2\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}3&-6\\1&2\end{matrix}\right))\left(\begin{matrix}-3\\7\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}3&-6\\1&2\end{matrix}\right))\left(\begin{matrix}-3\\7\end{matrix}\right)$$

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

$$\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}3&-6\\1&2\end{matrix}\right))\left(\begin{matrix}-3\\7\end{matrix}\right)$$

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.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{2}{3\times 2-\left(-6\right)}&-\frac{-6}{3\times 2-\left(-6\right)}\\-\frac{1}{3\times 2-\left(-6\right)}&\frac{3}{3\times 2-\left(-6\right)}\end{matrix}\right)\left(\begin{matrix}-3\\7\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{1}{6}&\frac{1}{2}\\-\frac{1}{12}&\frac{1}{4}\end{matrix}\right)\left(\begin{matrix}-3\\7\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{1}{6}\left(-3\right)+\frac{1}{2}\times 7\\-\frac{1}{12}\left(-3\right)+\frac{1}{4}\times 7\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa y me x.

$$y=3,x=2$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Tangohia te 6x mai i ngā taha e rua.

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

Whakaarohia te whārite tuarua. Me tāpiri te y ki ngā taha e rua.

$$2x+y=7$$

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.

$$3y-6x=-3,y+2x=7$$

Kia ōrite ai a 3y me y, 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 3.

$$3y-6x=-3,3y+3\times 2x=3\times 7$$

Whakarūnātia.

$$3y-6x=-3,3y+6x=21$$

Me tango 3y+6x=21 mai i 3y-6x=-3 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$3y-3y-6x-6x=-3-21$$

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

$$-6x-6x=-3-21$$

Tāpiri -6x ki te -6x.

$$-12x=-3-21$$

Tāpiri -3 ki te -21.

$$-12x=-24$$

Whakawehea ngā taha e rua ki te -12.

$$x=2$$

Whakaurua te 2 mō x ki y+2x=7. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y+2\times 2=7$$

Whakareatia 2 ki te 2.

$$y+4=7$$

Me tango 4 mai i ngā taha e rua o te whārite.

$$y=3$$

Kua oti te pūnaha te whakatau.

$$y=3,x=2$$