Whakaoti i te {l}{x-4y=27}{3x+y=-23}

$$\left. \begin{array} { l } { x - 4 y = 27 } \\ { 3 x + y = - 23 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-5$$

$$y=-8$$

Ngā Upane e Whakamahi ana i te Whakakapinga

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.

$$x-4y=27,3x+y=-23$$

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

$$x-4y=27$$

Me tāpiri 4y ki ngā taha e rua o te whārite.

$$x=4y+27$$

Whakakapia te 4y+27 mō te x ki tērā atu whārite, 3x+y=-23.

$$3\left(4y+27\right)+y=-23$$

Whakareatia 3 ki te 4y+27.

$$12y+81+y=-23$$

Tāpiri 12y ki te y.

$$13y+81=-23$$

Me tango 81 mai i ngā taha e rua o te whārite.

$$13y=-104$$

Whakawehea ngā taha e rua ki te 13.

$$y=-8$$

Whakaurua te -8 mō y ki x=4y+27. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=4\left(-8\right)+27$$

Whakareatia 4 ki te -8.

$$x=-32+27$$

Tāpiri 27 ki te -32.

$$x=-5$$

Kua oti te pūnaha te whakatau.

$$x=-5,y=-8$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

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

$$x-4y=27,3x+y=-23$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&-4\\3&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}27\\-23\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}1&-4\\3&1\end{matrix}\right))\left(\begin{matrix}1&-4\\3&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-4\\3&1\end{matrix}\right))\left(\begin{matrix}27\\-23\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-4\\3&1\end{matrix}\right))\left(\begin{matrix}27\\-23\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-4\\3&1\end{matrix}\right))\left(\begin{matrix}27\\-23\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{1-\left(-4\times 3\right)}&-\frac{-4}{1-\left(-4\times 3\right)}\\-\frac{3}{1-\left(-4\times 3\right)}&\frac{1}{1-\left(-4\times 3\right)}\end{matrix}\right)\left(\begin{matrix}27\\-23\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{13}&\frac{4}{13}\\-\frac{3}{13}&\frac{1}{13}\end{matrix}\right)\left(\begin{matrix}27\\-23\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{13}\times 27+\frac{4}{13}\left(-23\right)\\-\frac{3}{13}\times 27+\frac{1}{13}\left(-23\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-5\\-8\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-5,y=-8$$

Ngā Upane e Whakamahi i te Tangohanga

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.

$$x-4y=27,3x+y=-23$$

Kia ōrite ai a x me 3x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 3 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 1.

$$3x+3\left(-4\right)y=3\times 27,3x+y=-23$$

Whakarūnātia.

$$3x-12y=81,3x+y=-23$$

Me tango 3x+y=-23 mai i 3x-12y=81 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$3x-3x-12y-y=81+23$$

Tāpiri 3x ki te -3x. Ka whakakore atu ngā kupu 3x me -3x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-12y-y=81+23$$

Tāpiri -12y ki te -y.

$$-13y=81+23$$

Tāpiri 81 ki te 23.

$$-13y=104$$

Whakawehea ngā taha e rua ki te -13.

$$y=-8$$

Whakaurua te -8 mō y ki 3x+y=-23. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$3x-8=-23$$

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

$$3x=-15$$

Whakawehea ngā taha e rua ki te 3.

$$x=-5$$

Kua oti te pūnaha te whakatau.

$$x=-5,y=-8$$