Whakaoti i te {l}{x-4y=-1}{2x+y=16}

$$\left\{ \begin{array} { l } { x - 4 y = - 1 } \\ { 2 x + y = 16 } \end{array} \right.$$

Whakaoti mō x, y

$$x=7$$

$$y=2$$

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.

$$x-4y=-1,2x+y=16$$

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

$$x-4y=-1$$

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

$$x=4y-1$$

Whakakapia te 4y-1 mō te x ki tērā atu whārite, 2x+y=16.

$$2\left(4y-1\right)+y=16$$

Whakareatia 2 ki te 4y-1.

$$8y-2+y=16$$

Tāpiri 8y ki te y.

$$9y-2=16$$

Me tāpiri 2 ki ngā taha e rua o te whārite.

$$9y=18$$

Whakawehea ngā taha e rua ki te 9.

$$y=2$$

Whakaurua te 2 mō y ki x=4y-1. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=4\times 2-1$$

Whakareatia 4 ki te 2.

$$x=8-1$$

Tāpiri -1 ki te 8.

$$x=7$$

Kua oti te pūnaha te whakatau.

$$x=7,y=2$$

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.

$$x-4y=-1,2x+y=16$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&-4\\2&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-1\\16\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}1&-4\\2&1\end{matrix}\right))\left(\begin{matrix}1&-4\\2&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-4\\2&1\end{matrix}\right))\left(\begin{matrix}-1\\16\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-4\\2&1\end{matrix}\right))\left(\begin{matrix}-1\\16\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-4\\2&1\end{matrix}\right))\left(\begin{matrix}-1\\16\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{1-\left(-4\times 2\right)}&-\frac{-4}{1-\left(-4\times 2\right)}\\-\frac{2}{1-\left(-4\times 2\right)}&\frac{1}{1-\left(-4\times 2\right)}\end{matrix}\right)\left(\begin{matrix}-1\\16\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{9}&\frac{4}{9}\\-\frac{2}{9}&\frac{1}{9}\end{matrix}\right)\left(\begin{matrix}-1\\16\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{9}\left(-1\right)+\frac{4}{9}\times 16\\-\frac{2}{9}\left(-1\right)+\frac{1}{9}\times 16\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}7\\2\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=7,y=2$$

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.

$$x-4y=-1,2x+y=16$$

Kia ōrite ai a x me 2x, 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.

$$2x+2\left(-4\right)y=2\left(-1\right),2x+y=16$$

Whakarūnātia.

$$2x-8y=-2,2x+y=16$$

Me tango 2x+y=16 mai i 2x-8y=-2 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$2x-2x-8y-y=-2-16$$

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

$$-8y-y=-2-16$$

Tāpiri -8y ki te -y.

$$-9y=-2-16$$

Tāpiri -2 ki te -16.

$$-9y=-18$$

Whakawehea ngā taha e rua ki te -9.

$$y=2$$

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

$$2x+2=16$$

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

$$2x=14$$

Whakawehea ngā taha e rua ki te 2.

$$x=7$$

Kua oti te pūnaha te whakatau.

$$x=7,y=2$$