Whakaoti i te {l}{y=3x-2}{5x+4y=9}

$$\left\{ \begin{array} { l } { y = 3 x - 2 } \\ { 5 x + 4 y = 9 } \end{array} \right.$$

Whakaoti mō y, x

$$x=1$$

$$y=1$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Tangohia te 3x mai i ngā taha e rua.

$$y-3x=-2$$

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.

$$y-3x=-2,4y+5x=9$$

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

$$y-3x=-2$$

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

$$y=3x-2$$

Whakakapia te 3x-2 mō te y ki tērā atu whārite, 4y+5x=9.

$$4\left(3x-2\right)+5x=9$$

Whakareatia 4 ki te 3x-2.

$$12x-8+5x=9$$

Tāpiri 12x ki te 5x.

$$17x-8=9$$

Me tāpiri 8 ki ngā taha e rua o te whārite.

$$17x=17$$

Whakawehea ngā taha e rua ki te 17.

$$x=1$$

Whakaurua te 1 mō x ki y=3x-2. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y=3-2$$

Tāpiri -2 ki te 3.

$$y=1$$

Kua oti te pūnaha te whakatau.

$$y=1,x=1$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Tangohia te 3x mai i ngā taha e rua.

$$y-3x=-2$$

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

$$y-3x=-2,4y+5x=9$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&-3\\4&5\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-2\\9\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&-3\\4&5\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&-3\\4&5\end{matrix}\right))\left(\begin{matrix}1&-3\\4&5\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-3\\4&5\end{matrix}\right))\left(\begin{matrix}-2\\9\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-3\\4&5\end{matrix}\right))\left(\begin{matrix}-2\\9\end{matrix}\right)$$

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

$$\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-3\\4&5\end{matrix}\right))\left(\begin{matrix}-2\\9\end{matrix}\right)$$

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.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{5}{5-\left(-3\times 4\right)}&-\frac{-3}{5-\left(-3\times 4\right)}\\-\frac{4}{5-\left(-3\times 4\right)}&\frac{1}{5-\left(-3\times 4\right)}\end{matrix}\right)\left(\begin{matrix}-2\\9\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{5}{17}&\frac{3}{17}\\-\frac{4}{17}&\frac{1}{17}\end{matrix}\right)\left(\begin{matrix}-2\\9\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{5}{17}\left(-2\right)+\frac{3}{17}\times 9\\-\frac{4}{17}\left(-2\right)+\frac{1}{17}\times 9\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa y me x.

$$y=1,x=1$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Tangohia te 3x mai i ngā taha e rua.

$$y-3x=-2$$

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.

$$y-3x=-2,4y+5x=9$$

Kia ōrite ai a y me 4y, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 4 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 1.

$$4y+4\left(-3\right)x=4\left(-2\right),4y+5x=9$$

Whakarūnātia.

$$4y-12x=-8,4y+5x=9$$

Me tango 4y+5x=9 mai i 4y-12x=-8 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$4y-4y-12x-5x=-8-9$$

Tāpiri 4y ki te -4y. Ka whakakore atu ngā kupu 4y me -4y, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-12x-5x=-8-9$$

Tāpiri -12x ki te -5x.

$$-17x=-8-9$$

Tāpiri -8 ki te -9.

$$-17x=-17$$

Whakawehea ngā taha e rua ki te -17.

$$x=1$$

Whakaurua te 1 mō x ki 4y+5x=9. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$4y+5=9$$

Me tango 5 mai i ngā taha e rua o te whārite.

$$4y=4$$

Whakawehea ngā taha e rua ki te 4.

$$y=1$$

Kua oti te pūnaha te whakatau.

$$y=1,x=1$$