Whakaoti i te {l}{y=-4x-17}{-2x+6y=2}

$$\left. \begin{array} { l } { y = - 4 x - 17 } \\ { - 2 x + 6 y = 2 } \end{array} \right.$$

Whakaoti mō y, x

$$x=-4$$

$$y=-1$$

Ngā Upane e Whakamahi ana i te Whakakapinga

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

$$y+4x=-17$$

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.

$$y+4x=-17,6y-2x=2$$

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

$$y+4x=-17$$

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

$$y=-4x-17$$

Whakakapia te -4x-17 mō te y ki tērā atu whārite, 6y-2x=2.

$$6\left(-4x-17\right)-2x=2$$

Whakareatia 6 ki te -4x-17.

$$-24x-102-2x=2$$

Tāpiri -24x ki te -2x.

$$-26x-102=2$$

Me tāpiri 102 ki ngā taha e rua o te whārite.

$$-26x=104$$

Whakawehea ngā taha e rua ki te -26.

$$x=-4$$

Whakaurua te -4 mō x ki y=-4x-17. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y=-4\left(-4\right)-17$$

Whakareatia -4 ki te -4.

$$y=16-17$$

Tāpiri -17 ki te 16.

$$y=-1$$

Kua oti te pūnaha te whakatau.

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

Ngā Upane e Whakamahi ana i Ngā Poukapa

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

$$y+4x=-17$$

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

$$y+4x=-17,6y-2x=2$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&4\\6&-2\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-17\\2\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}1&4\\6&-2\end{matrix}\right))\left(\begin{matrix}1&4\\6&-2\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&4\\6&-2\end{matrix}\right))\left(\begin{matrix}-17\\2\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&4\\6&-2\end{matrix}\right))\left(\begin{matrix}-17\\2\end{matrix}\right)$$

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

$$\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&4\\6&-2\end{matrix}\right))\left(\begin{matrix}-17\\2\end{matrix}\right)$$

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.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{-2-4\times 6}&-\frac{4}{-2-4\times 6}\\-\frac{6}{-2-4\times 6}&\frac{1}{-2-4\times 6}\end{matrix}\right)\left(\begin{matrix}-17\\2\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{1}{13}&\frac{2}{13}\\\frac{3}{13}&-\frac{1}{26}\end{matrix}\right)\left(\begin{matrix}-17\\2\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{1}{13}\left(-17\right)+\frac{2}{13}\times 2\\\frac{3}{13}\left(-17\right)-\frac{1}{26}\times 2\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa y me x.

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

Ngā Upane e Whakamahi i te Tangohanga

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

$$y+4x=-17$$

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.

$$y+4x=-17,6y-2x=2$$

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.

$$6y+6\times 4x=6\left(-17\right),6y-2x=2$$

Whakarūnātia.

$$6y+24x=-102,6y-2x=2$$

Me tango 6y-2x=2 mai i 6y+24x=-102 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$6y-6y+24x+2x=-102-2$$

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.

$$24x+2x=-102-2$$

Tāpiri 24x ki te 2x.

$$26x=-102-2$$

Tāpiri -102 ki te -2.

$$26x=-104$$

Whakawehea ngā taha e rua ki te 26.

$$x=-4$$

Whakaurua te -4 mō x ki 6y-2x=2. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$6y-2\left(-4\right)=2$$

Whakareatia -2 ki te -4.

$$6y+8=2$$

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

$$6y=-6$$

Whakawehea ngā taha e rua ki te 6.

$$y=-1$$

Kua oti te pūnaha te whakatau.

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