Whakaoti i te {l}{n+y=4}{2n+3y=12}

$$\left. \begin{array} { l } { n + y = 4 } \\ { 2 n + 3 y = 12 } \end{array} \right.$$

Whakaoti mō n, y

$$y=4$$

$$n=0$$

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.

$$n+y=4,2n+3y=12$$

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

$$n+y=4$$

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

$$n=-y+4$$

Whakakapia te -y+4 mō te n ki tērā atu whārite, 2n+3y=12.

$$2\left(-y+4\right)+3y=12$$

Whakareatia 2 ki te -y+4.

$$-2y+8+3y=12$$

Tāpiri -2y ki te 3y.

$$y+8=12$$

Me tango 8 mai i ngā taha e rua o te whārite.

$$y=4$$

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

$$n=-4+4$$

Tāpiri 4 ki te -4.

$$n=0$$

Kua oti te pūnaha te whakatau.

$$n=0,y=4$$

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.

$$n+y=4,2n+3y=12$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&1\\2&3\end{matrix}\right)\left(\begin{matrix}n\\y\end{matrix}\right)=\left(\begin{matrix}4\\12\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}1&1\\2&3\end{matrix}\right))\left(\begin{matrix}1&1\\2&3\end{matrix}\right)\left(\begin{matrix}n\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\2&3\end{matrix}\right))\left(\begin{matrix}4\\12\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}n\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\2&3\end{matrix}\right))\left(\begin{matrix}4\\12\end{matrix}\right)$$

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

$$\left(\begin{matrix}n\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\2&3\end{matrix}\right))\left(\begin{matrix}4\\12\end{matrix}\right)$$

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.

$$\left(\begin{matrix}n\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{3-2}&-\frac{1}{3-2}\\-\frac{2}{3-2}&\frac{1}{3-2}\end{matrix}\right)\left(\begin{matrix}4\\12\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}n\\y\end{matrix}\right)=\left(\begin{matrix}3&-1\\-2&1\end{matrix}\right)\left(\begin{matrix}4\\12\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}n\\y\end{matrix}\right)=\left(\begin{matrix}3\times 4-12\\-2\times 4+12\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}n\\y\end{matrix}\right)=\left(\begin{matrix}0\\4\end{matrix}\right)$$

Tangohia ngā huānga poukapa n me y.

$$n=0,y=4$$

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.

$$n+y=4,2n+3y=12$$

Kia ōrite ai a n me 2n, 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.

$$2n+2y=2\times 4,2n+3y=12$$

Whakarūnātia.

$$2n+2y=8,2n+3y=12$$

Me tango 2n+3y=12 mai i 2n+2y=8 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$2n-2n+2y-3y=8-12$$

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

$$2y-3y=8-12$$

Tāpiri 2y ki te -3y.

$$-y=8-12$$

Tāpiri 8 ki te -12.

$$-y=-4$$

Whakawehea ngā taha e rua ki te -1.

$$y=4$$

Whakaurua te 4 mō y ki 2n+3y=12. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō n hāngai tonu.

$$2n+3\times 4=12$$

Whakareatia 3 ki te 4.

$$2n+12=12$$

Me tango 12 mai i ngā taha e rua o te whārite.

$$2n=0$$

Whakawehea ngā taha e rua ki te 2.

$$n=0$$

Kua oti te pūnaha te whakatau.

$$n=0,y=4$$