Whakaoti i te {l}{x+6y=19}{2x+2y=18}

$$\left. \begin{array} { l } { x + 6 y = 19 } \\ { 2 x + 2 y = 18 } \end{array} \right.$$

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.

$$x+6y=19,2x+2y=18$$

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

$$x+6y=19$$

Me tango 6y mai i ngā taha e rua o te whārite.

$$x=-6y+19$$

Whakakapia te -6y+19 mō te x ki tērā atu whārite, 2x+2y=18.

$$2\left(-6y+19\right)+2y=18$$

Whakareatia 2 ki te -6y+19.

$$-12y+38+2y=18$$

Tāpiri -12y ki te 2y.

$$-10y+38=18$$

Me tango 38 mai i ngā taha e rua o te whārite.

$$-10y=-20$$

Whakawehea ngā taha e rua ki te -10.

$$y=2$$

Whakaurua te 2 mō y ki x=-6y+19. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-6\times 2+19$$

Whakareatia -6 ki te 2.

$$x=-12+19$$

Tāpiri 19 ki te -12.

$$x=7$$

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.

$$x+6y=19,2x+2y=18$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&6\\2&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}19\\18\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&6\\2&2\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&6\\2&2\end{matrix}\right))\left(\begin{matrix}1&6\\2&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&6\\2&2\end{matrix}\right))\left(\begin{matrix}19\\18\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&6\\2&2\end{matrix}\right))\left(\begin{matrix}19\\18\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&6\\2&2\end{matrix}\right))\left(\begin{matrix}19\\18\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{2-6\times 2}&-\frac{6}{2-6\times 2}\\-\frac{2}{2-6\times 2}&\frac{1}{2-6\times 2}\end{matrix}\right)\left(\begin{matrix}19\\18\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{5}&\frac{3}{5}\\\frac{1}{5}&-\frac{1}{10}\end{matrix}\right)\left(\begin{matrix}19\\18\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{5}\times 19+\frac{3}{5}\times 18\\\frac{1}{5}\times 19-\frac{1}{10}\times 18\end{matrix}\right)$$

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.

$$x+6y=19,2x+2y=18$$

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.

$$2x+2\times 6y=2\times 19,2x+2y=18$$

Whakarūnātia.

$$2x+12y=38,2x+2y=18$$

Me tango 2x+2y=18 mai i 2x+12y=38 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$2x-2x+12y-2y=38-18$$

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.

$$12y-2y=38-18$$

Tāpiri 12y ki te -2y.

$$10y=38-18$$

Tāpiri 38 ki te -18.

$$10y=20$$

Whakawehea ngā taha e rua ki te 10.

$$y=2$$

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

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

Whakareatia 2 ki te 2.

$$2x+4=18$$

Me tango 4 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.

$$x=7,y=2$$