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

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

Whakaoti mō y, x

$$x=1$$

$$y=-4$$

Ngā Upane e Whakamahi ana i te Whakakapinga

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

$$y+6x=2$$

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

$$y+x=-3$$

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.

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

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

$$y+6x=2$$

Me tango 6x mai i ngā taha e rua o te whārite.

$$y=-6x+2$$

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

$$-6x+2+x=-3$$

Tāpiri -6x ki te x.

$$-5x+2=-3$$

Me tango 2 mai i 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=-6x+2. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y=-6+2$$

Tāpiri 2 ki te -6.

$$y=-4$$

Kua oti te pūnaha te whakatau.

$$y=-4,x=1$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

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

$$y+6x=2$$

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

$$y+x=-3$$

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

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{5}&\frac{6}{5}\\\frac{1}{5}&-\frac{1}{5}\end{matrix}\right)\left(\begin{matrix}2\\-3\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{5}\times 2+\frac{6}{5}\left(-3\right)\\\frac{1}{5}\times 2-\frac{1}{5}\left(-3\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa y me x.

$$y=-4,x=1$$

Ngā Upane e Whakamahi i te Tangohanga

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

$$y+6x=2$$

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

$$y+x=-3$$

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.

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

Me tango y+x=-3 mai i y+6x=2 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

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

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.

$$6x-x=2+3$$

Tāpiri 6x ki te -x.

$$5x=2+3$$

Tāpiri 2 ki te 3.

$$5x=5$$

Whakawehea ngā taha e rua ki te 5.

$$x=1$$

Whakaurua te 1 mō x ki y+x=-3. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y+1=-3$$

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

$$y=-4$$

Kua oti te pūnaha te whakatau.

$$y=-4,x=1$$