Whakaoti i te {l}{x+y=7}{2x+3y=18}

$$\left. \begin{array} { l } { x + y = 7 } \\ { 2 x + 3 y = 18 } \end{array} \right.$$

Whakaoti mō x, y

$$x=3$$

$$y=4$$

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.

$$x+y=7,2x+3y=18$$

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

$$x+y=7$$

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

$$x=-y+7$$

Whakakapia te -y+7 mō te x ki tērā atu whārite, 2x+3y=18.

$$2\left(-y+7\right)+3y=18$$

Whakareatia 2 ki te -y+7.

$$-2y+14+3y=18$$

Tāpiri -2y ki te 3y.

$$y+14=18$$

Me tango 14 mai i ngā taha e rua o te whārite.

$$y=4$$

Whakaurua te 4 mō y ki x=-y+7. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-4+7$$

Tāpiri 7 ki te -4.

$$x=3$$

Kua oti te pūnaha te whakatau.

$$x=3,y=4$$

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.

$$x+y=7,2x+3y=18$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&1\\2&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}7\\18\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}1&1\\2&3\end{matrix}\right))\left(\begin{matrix}1&1\\2&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\2&3\end{matrix}\right))\left(\begin{matrix}7\\18\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\2&3\end{matrix}\right))\left(\begin{matrix}7\\18\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\2&3\end{matrix}\right))\left(\begin{matrix}7\\18\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{3-2}&-\frac{1}{3-2}\\-\frac{2}{3-2}&\frac{1}{3-2}\end{matrix}\right)\left(\begin{matrix}7\\18\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}3&-1\\-2&1\end{matrix}\right)\left(\begin{matrix}7\\18\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}3\times 7-18\\-2\times 7+18\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}3\\4\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=3,y=4$$

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.

$$x+y=7,2x+3y=18$$

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.

$$2x+2y=2\times 7,2x+3y=18$$

Whakarūnātia.

$$2x+2y=14,2x+3y=18$$

Me tango 2x+3y=18 mai i 2x+2y=14 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$2x-2x+2y-3y=14-18$$

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.

$$2y-3y=14-18$$

Tāpiri 2y ki te -3y.

$$-y=14-18$$

Tāpiri 14 ki te -18.

$$-y=-4$$

Whakawehea ngā taha e rua ki te -1.

$$y=4$$

Whakaurua te 4 mō y ki 2x+3y=18. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$2x+3\times 4=18$$

Whakareatia 3 ki te 4.

$$2x+12=18$$

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

$$2x=6$$

Whakawehea ngā taha e rua ki te 2.

$$x=3$$

Kua oti te pūnaha te whakatau.

$$x=3,y=4$$