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

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

Whakaoti mō y, x

$$x=-1$$

$$y=1$$

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. Tangohia te x mai i ngā taha e rua.

$$y-x=2$$

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.

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

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-x=2.

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

Tāpiri -4x ki te -x.

$$-5x-3=2$$

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

$$-5x=5$$

Whakawehea ngā taha e rua ki te -5.

$$x=-1$$

Whakaurua te -1 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.

$$y=-4\left(-1\right)-3$$

Whakareatia -4 ki te -1.

$$y=4-3$$

Tāpiri -3 ki te 4.

$$y=1$$

Kua oti te pūnaha te whakatau.

$$y=1,x=-1$$

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. Tangohia te x mai i ngā taha e rua.

$$y-x=2$$

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

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{1}{5}&\frac{4}{5}\\\frac{1}{5}&-\frac{1}{5}\end{matrix}\right)\left(\begin{matrix}-3\\2\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{1}{5}\left(-3\right)+\frac{4}{5}\times 2\\\frac{1}{5}\left(-3\right)-\frac{1}{5}\times 2\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa y me x.

$$y=1,x=-1$$

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. Tangohia te x mai i ngā taha e rua.

$$y-x=2$$

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.

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

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

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

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.

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

Tāpiri 4x ki te x.

$$5x=-3-2$$

Tāpiri -3 ki te -2.

$$5x=-5$$

Whakawehea ngā taha e rua ki te 5.

$$x=-1$$

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

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

Whakareatia -1 ki te -1.

$$y+1=2$$

Me tango 1 mai i ngā taha e rua o te whārite.

$$y=1$$

Kua oti te pūnaha te whakatau.

$$y=1,x=-1$$