Whakaoti i te {l}{y=-2x}{4x+6y=24}

$$\left. \begin{array} { l } { y = - 2 x } \\ { 4 x + 6 y = 24 } \end{array} \right.$$

Whakaoti mō y, x

$$x=-3$$

$$y=6$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Me tāpiri te 2x ki ngā taha e rua.

$$y+2x=0$$

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.

$$y+2x=0,6y+4x=24$$

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

$$y+2x=0$$

Me tango 2x mai i ngā taha e rua o te whārite.

$$y=-2x$$

Whakakapia te -2x mō te y ki tērā atu whārite, 6y+4x=24.

$$6\left(-2\right)x+4x=24$$

Whakareatia 6 ki te -2x.

$$-12x+4x=24$$

Tāpiri -12x ki te 4x.

$$-8x=24$$

Whakawehea ngā taha e rua ki te -8.

$$x=-3$$

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

$$y=-2\left(-3\right)$$

Whakareatia -2 ki te -3.

$$y=6$$

Kua oti te pūnaha te whakatau.

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

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Me tāpiri te 2x ki ngā taha e rua.

$$y+2x=0$$

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

$$y+2x=0,6y+4x=24$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&2\\6&4\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}0\\24\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&2\\6&4\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&2\\6&4\end{matrix}\right))\left(\begin{matrix}1&2\\6&4\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&2\\6&4\end{matrix}\right))\left(\begin{matrix}0\\24\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&2\\6&4\end{matrix}\right))\left(\begin{matrix}0\\24\end{matrix}\right)$$

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

$$\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&2\\6&4\end{matrix}\right))\left(\begin{matrix}0\\24\end{matrix}\right)$$

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.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{4}{4-2\times 6}&-\frac{2}{4-2\times 6}\\-\frac{6}{4-2\times 6}&\frac{1}{4-2\times 6}\end{matrix}\right)\left(\begin{matrix}0\\24\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{2}&\frac{1}{4}\\\frac{3}{4}&-\frac{1}{8}\end{matrix}\right)\left(\begin{matrix}0\\24\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{1}{4}\times 24\\-\frac{1}{8}\times 24\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa y me x.

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

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Me tāpiri te 2x ki ngā taha e rua.

$$y+2x=0$$

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.

$$y+2x=0,6y+4x=24$$

Kia ōrite ai a y me 6y, 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.

$$6y+6\times 2x=0,6y+4x=24$$

Whakarūnātia.

$$6y+12x=0,6y+4x=24$$

Me tango 6y+4x=24 mai i 6y+12x=0 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$6y-6y+12x-4x=-24$$

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

$$12x-4x=-24$$

Tāpiri 12x ki te -4x.

$$8x=-24$$

Whakawehea ngā taha e rua ki te 8.

$$x=-3$$

Whakaurua te -3 mō x ki 6y+4x=24. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$6y+4\left(-3\right)=24$$

Whakareatia 4 ki te -3.

$$6y-12=24$$

Me tāpiri 12 ki ngā taha e rua o te whārite.

$$6y=36$$

Whakawehea ngā taha e rua ki te 6.

$$y=6$$

Kua oti te pūnaha te whakatau.

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