Whakaoti i te {l}{y=-4x-3}{y=-2x+1}

$$\left\{ \begin{array} { l } { y = - 4 x - 3 } \\ { y = - 2 x + 1 } \end{array} \right.$$

Whakaoti mō y, x

$$x=-2$$

$$y=5$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Me tāpiri te 4x ki ngā taha e rua.

$$y+4x=-3$$

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

$$y+2x=1$$

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.

$$y+4x=-3,y+2x=1$$

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

$$y+4x=-3$$

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

$$y=-4x-3$$

Whakakapia te -4x-3 mō te y ki tērā atu whārite, y+2x=1.

$$-4x-3+2x=1$$

Tāpiri -4x ki te 2x.

$$-2x-3=1$$

Me tāpiri 3 ki ngā taha e rua o te whārite.

$$-2x=4$$

Whakawehea ngā taha e rua ki te -2.

$$x=-2$$

Whakaurua te -2 mō x ki y=-4x-3. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y=-4\left(-2\right)-3$$

Whakareatia -4 ki te -2.

$$y=8-3$$

Tāpiri -3 ki te 8.

$$y=5$$

Kua oti te pūnaha te whakatau.

$$y=5,x=-2$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Me tāpiri te 4x ki ngā taha e rua.

$$y+4x=-3$$

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

$$y+2x=1$$

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

$$y+4x=-3,y+2x=1$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}5\\-2\end{matrix}\right)$$

Tangohia ngā huānga poukapa y me x.

$$y=5,x=-2$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Me tāpiri te 4x ki ngā taha e rua.

$$y+4x=-3$$

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

$$y+2x=1$$

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.

$$y+4x=-3,y+2x=1$$

Me tango y+2x=1 mai i y+4x=-3 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$y-y+4x-2x=-3-1$$

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

$$4x-2x=-3-1$$

Tāpiri 4x ki te -2x.

$$2x=-3-1$$

Tāpiri -3 ki te -1.

$$2x=-4$$

Whakawehea ngā taha e rua ki te 2.

$$x=-2$$

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.

$$y+2\left(-2\right)=1$$

Whakareatia 2 ki te -2.

$$y-4=1$$

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

$$y=5$$

Kua oti te pūnaha te whakatau.

$$y=5,x=-2$$