Whakaoti i te {l}{y=-2x+13}{4x+8y=20}

$$\left. \begin{array} { l } { y = - 2 x + 13 } \\ { 4 x + 8 y = 20 } \end{array} \right.$$

Whakaoti mō y, x

$$x=7$$

$$y=-1$$

Ngā Upane e Whakamahi ana i te Whakakapinga

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

$$y+2x=13$$

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.

$$y+2x=13,8y+4x=20$$

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

$$y+2x=13$$

Me tango 2x mai i ngā taha e rua o te whārite.

$$y=-2x+13$$

Whakakapia te -2x+13 mō te y ki tērā atu whārite, 8y+4x=20.

$$8\left(-2x+13\right)+4x=20$$

Whakareatia 8 ki te -2x+13.

$$-16x+104+4x=20$$

Tāpiri -16x ki te 4x.

$$-12x+104=20$$

Me tango 104 mai i ngā taha e rua o te whārite.

$$-12x=-84$$

Whakawehea ngā taha e rua ki te -12.

$$x=7$$

Whakaurua te 7 mō x ki y=-2x+13. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y=-2\times 7+13$$

Whakareatia -2 ki te 7.

$$y=-14+13$$

Tāpiri 13 ki te -14.

$$y=-1$$

Kua oti te pūnaha te whakatau.

$$y=-1,x=7$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

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

$$y+2x=13$$

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

$$y+2x=13,8y+4x=20$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&2\\8&4\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}13\\20\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&2\\8&4\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&2\\8&4\end{matrix}\right))\left(\begin{matrix}1&2\\8&4\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&2\\8&4\end{matrix}\right))\left(\begin{matrix}13\\20\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&2\\8&4\end{matrix}\right))\left(\begin{matrix}13\\20\end{matrix}\right)$$

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

$$\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&2\\8&4\end{matrix}\right))\left(\begin{matrix}13\\20\end{matrix}\right)$$

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.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{4}{4-2\times 8}&-\frac{2}{4-2\times 8}\\-\frac{8}{4-2\times 8}&\frac{1}{4-2\times 8}\end{matrix}\right)\left(\begin{matrix}13\\20\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{3}&\frac{1}{6}\\\frac{2}{3}&-\frac{1}{12}\end{matrix}\right)\left(\begin{matrix}13\\20\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{3}\times 13+\frac{1}{6}\times 20\\\frac{2}{3}\times 13-\frac{1}{12}\times 20\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-1\\7\end{matrix}\right)$$

Tangohia ngā huānga poukapa y me x.

$$y=-1,x=7$$

Ngā Upane e Whakamahi i te Tangohanga

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

$$y+2x=13$$

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.

$$y+2x=13,8y+4x=20$$

Kia ōrite ai a y me 8y, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 8 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 1.

$$8y+8\times 2x=8\times 13,8y+4x=20$$

Whakarūnātia.

$$8y+16x=104,8y+4x=20$$

Me tango 8y+4x=20 mai i 8y+16x=104 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$8y-8y+16x-4x=104-20$$

Tāpiri 8y ki te -8y. Ka whakakore atu ngā kupu 8y me -8y, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$16x-4x=104-20$$

Tāpiri 16x ki te -4x.

$$12x=104-20$$

Tāpiri 104 ki te -20.

$$12x=84$$

Whakawehea ngā taha e rua ki te 12.

$$x=7$$

Whakaurua te 7 mō x ki 8y+4x=20. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$8y+4\times 7=20$$

Whakareatia 4 ki te 7.

$$8y+28=20$$

Me tango 28 mai i ngā taha e rua o te whārite.

$$8y=-8$$

Whakawehea ngā taha e rua ki te 8.

$$y=-1$$

Kua oti te pūnaha te whakatau.

$$y=-1,x=7$$