Whakaoti i te {l}{y=9-2x}{3x+2y=16}

$$\left\{ \begin{array} { l } { y = 9 - 2 x } \\ { 3 x + 2 y = 16 } \end{array} \right.$$

Whakaoti mō y, x

$$x=2$$

$$y=5$$

Ngā Upane e Whakamahi ana i te Whakakapinga

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

$$y+2x=9$$

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.

$$y+2x=9,2y+3x=16$$

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

$$y+2x=9$$

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

$$y=-2x+9$$

Whakakapia te -2x+9 mō te y ki tērā atu whārite, 2y+3x=16.

$$2\left(-2x+9\right)+3x=16$$

Whakareatia 2 ki te -2x+9.

$$-4x+18+3x=16$$

Tāpiri -4x ki te 3x.

$$-x+18=16$$

Me tango 18 mai i ngā taha e rua o te whārite.

$$-x=-2$$

Whakawehea ngā taha e rua ki te -1.

$$x=2$$

Whakaurua te 2 mō x ki y=-2x+9. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y=-2\times 2+9$$

Whakareatia -2 ki te 2.

$$y=-4+9$$

Tāpiri 9 ki te -4.

$$y=5$$

Kua oti te pūnaha te whakatau.

$$y=5,x=2$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

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

$$y+2x=9$$

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

$$y+2x=9,2y+3x=16$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&2\\2&3\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}9\\16\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&2\\2&3\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&2\\2&3\end{matrix}\right))\left(\begin{matrix}1&2\\2&3\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&2\\2&3\end{matrix}\right))\left(\begin{matrix}9\\16\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&2\\2&3\end{matrix}\right))\left(\begin{matrix}9\\16\end{matrix}\right)$$

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

$$\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&2\\2&3\end{matrix}\right))\left(\begin{matrix}9\\16\end{matrix}\right)$$

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.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{3}{3-2\times 2}&-\frac{2}{3-2\times 2}\\-\frac{2}{3-2\times 2}&\frac{1}{3-2\times 2}\end{matrix}\right)\left(\begin{matrix}9\\16\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-3&2\\2&-1\end{matrix}\right)\left(\begin{matrix}9\\16\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-3\times 9+2\times 16\\2\times 9-16\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}5\\2\end{matrix}\right)$$

Tangohia ngā huānga poukapa y me x.

$$y=5,x=2$$

Ngā Upane e Whakamahi i te Tangohanga

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

$$y+2x=9$$

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.

$$y+2x=9,2y+3x=16$$

Kia ōrite ai a y me 2y, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 2 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 1.

$$2y+2\times 2x=2\times 9,2y+3x=16$$

Whakarūnātia.

$$2y+4x=18,2y+3x=16$$

Me tango 2y+3x=16 mai i 2y+4x=18 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$2y-2y+4x-3x=18-16$$

Tāpiri 2y ki te -2y. Ka whakakore atu ngā kupu 2y me -2y, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$4x-3x=18-16$$

Tāpiri 4x ki te -3x.

$$x=18-16$$

Tāpiri 18 ki te -16.

$$x=2$$

Whakaurua te 2 mō x ki 2y+3x=16. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$2y+3\times 2=16$$

Whakareatia 3 ki te 2.

$$2y+6=16$$

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

$$2y=10$$

Whakawehea ngā taha e rua ki te 2.

$$y=5$$

Kua oti te pūnaha te whakatau.

$$y=5,x=2$$