Whakaoti i te {l}{x+y=14}{2x+4y=38}

$$\left\{ \begin{array} { l } { x + y = 14 } \\ { 2 x + 4 y = 38 } \end{array} \right.$$

Whakaoti mō x, y

$$x=9$$

$$y=5$$

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.

$$x+y=14,2x+4y=38$$

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

$$x+y=14$$

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

$$x=-y+14$$

Whakakapia te -y+14 mō te x ki tērā atu whārite, 2x+4y=38.

$$2\left(-y+14\right)+4y=38$$

Whakareatia 2 ki te -y+14.

$$-2y+28+4y=38$$

Tāpiri -2y ki te 4y.

$$2y+28=38$$

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

$$2y=10$$

Whakawehea ngā taha e rua ki te 2.

$$y=5$$

Whakaurua te 5 mō y ki x=-y+14. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-5+14$$

Tāpiri 14 ki te -5.

$$x=9$$

Kua oti te pūnaha te whakatau.

$$x=9,y=5$$

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.

$$x+y=14,2x+4y=38$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&1\\2&4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}14\\38\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}1&1\\2&4\end{matrix}\right))\left(\begin{matrix}1&1\\2&4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\2&4\end{matrix}\right))\left(\begin{matrix}14\\38\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\2&4\end{matrix}\right))\left(\begin{matrix}14\\38\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\2&4\end{matrix}\right))\left(\begin{matrix}14\\38\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{4-2}&-\frac{1}{4-2}\\-\frac{2}{4-2}&\frac{1}{4-2}\end{matrix}\right)\left(\begin{matrix}14\\38\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}2&-\frac{1}{2}\\-1&\frac{1}{2}\end{matrix}\right)\left(\begin{matrix}14\\38\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}2\times 14-\frac{1}{2}\times 38\\-14+\frac{1}{2}\times 38\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}9\\5\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=9,y=5$$

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.

$$x+y=14,2x+4y=38$$

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.

$$2x+2y=2\times 14,2x+4y=38$$

Whakarūnātia.

$$2x+2y=28,2x+4y=38$$

Me tango 2x+4y=38 mai i 2x+2y=28 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$2x-2x+2y-4y=28-38$$

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.

$$2y-4y=28-38$$

Tāpiri 2y ki te -4y.

$$-2y=28-38$$

Tāpiri 28 ki te -38.

$$-2y=-10$$

Whakawehea ngā taha e rua ki te -2.

$$y=5$$

Whakaurua te 5 mō y ki 2x+4y=38. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$2x+4\times 5=38$$

Whakareatia 4 ki te 5.

$$2x+20=38$$

Me tango 20 mai i ngā taha e rua o te whārite.

$$2x=18$$

Whakawehea ngā taha e rua ki te 2.

$$x=9$$

Kua oti te pūnaha te whakatau.

$$x=9,y=5$$