Whakaoti i te {l}{y=-x-3}{y=-8x+4}

$$\left. \begin{array} { l } { y = - x - 3 } \\ { y = - 8 x + 4 } \end{array} \right.$$

Whakaoti mō y, x

$$x=1$$

$$y=-4$$

Ngā Upane e Whakamahi ana i te Whakakapinga

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

$$y+x=-3$$

Whakaarohia te whārite tuarua. Me tāpiri te 8x ki ngā taha e rua.

$$y+8x=4$$

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.

$$y+x=-3,y+8x=4$$

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

$$y+x=-3$$

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

$$y=-x-3$$

Whakakapia te -x-3 mō te y ki tērā atu whārite, y+8x=4.

$$-x-3+8x=4$$

Tāpiri -x ki te 8x.

$$7x-3=4$$

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

$$7x=7$$

Whakawehea ngā taha e rua ki te 7.

$$x=1$$

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.

$$y=-1-3$$

Tāpiri -3 ki te -1.

$$y=-4$$

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 x ki ngā taha e rua.

$$y+x=-3$$

Whakaarohia te whārite tuarua. Me tāpiri te 8x ki ngā taha e rua.

$$y+8x=4$$

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

$$y+x=-3,y+8x=4$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&1\\1&8\end{matrix}\right).

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{8}{7}&-\frac{1}{7}\\-\frac{1}{7}&\frac{1}{7}\end{matrix}\right)\left(\begin{matrix}-3\\4\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{8}{7}\left(-3\right)-\frac{1}{7}\times 4\\-\frac{1}{7}\left(-3\right)+\frac{1}{7}\times 4\end{matrix}\right)$$

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 x ki ngā taha e rua.

$$y+x=-3$$

Whakaarohia te whārite tuarua. Me tāpiri te 8x ki ngā taha e rua.

$$y+8x=4$$

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.

$$y+x=-3,y+8x=4$$

Me tango y+8x=4 mai i y+x=-3 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$y-y+x-8x=-3-4$$

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.

$$x-8x=-3-4$$

Tāpiri x ki te -8x.

$$-7x=-3-4$$

Tāpiri -3 ki te -4.

$$-7x=-7$$

Whakawehea ngā taha e rua ki te -7.

$$x=1$$

Whakaurua te 1 mō x ki y+8x=4. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y+8=4$$

Me tango 8 mai i ngā taha e rua o te whārite.

$$y=-4$$

Kua oti te pūnaha te whakatau.

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