Whakaoti i te {c}{x+2y=50}{2x+y=40}

$$\left. \begin{array} { c } { x + 2 y = 50 } \\ { 2 x + y = 40 } \end{array} \right.$$

Whakaoti mō x, y

$$x=10$$

$$y=20$$

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.

$$x+2y=50,2x+y=40$$

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

$$x+2y=50$$

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

$$x=-2y+50$$

Whakakapia te -2y+50 mō te x ki tērā atu whārite, 2x+y=40.

$$2\left(-2y+50\right)+y=40$$

Whakareatia 2 ki te -2y+50.

$$-4y+100+y=40$$

Tāpiri -4y ki te y.

$$-3y+100=40$$

Me tango 100 mai i ngā taha e rua o te whārite.

$$-3y=-60$$

Whakawehea ngā taha e rua ki te -3.

$$y=20$$

Whakaurua te 20 mō y ki x=-2y+50. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-2\times 20+50$$

Whakareatia -2 ki te 20.

$$x=-40+50$$

Tāpiri 50 ki te -40.

$$x=10$$

Kua oti te pūnaha te whakatau.

$$x=10,y=20$$

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.

$$x+2y=50,2x+y=40$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&2\\2&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}50\\40\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}1&2\\2&1\end{matrix}\right))\left(\begin{matrix}1&2\\2&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&2\\2&1\end{matrix}\right))\left(\begin{matrix}50\\40\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&2\\2&1\end{matrix}\right))\left(\begin{matrix}50\\40\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&2\\2&1\end{matrix}\right))\left(\begin{matrix}50\\40\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{1-2\times 2}&-\frac{2}{1-2\times 2}\\-\frac{2}{1-2\times 2}&\frac{1}{1-2\times 2}\end{matrix}\right)\left(\begin{matrix}50\\40\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{3}&\frac{2}{3}\\\frac{2}{3}&-\frac{1}{3}\end{matrix}\right)\left(\begin{matrix}50\\40\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{3}\times 50+\frac{2}{3}\times 40\\\frac{2}{3}\times 50-\frac{1}{3}\times 40\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}10\\20\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=10,y=20$$

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.

$$x+2y=50,2x+y=40$$

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.

$$2x+2\times 2y=2\times 50,2x+y=40$$

Whakarūnātia.

$$2x+4y=100,2x+y=40$$

Me tango 2x+y=40 mai i 2x+4y=100 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$2x-2x+4y-y=100-40$$

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.

$$4y-y=100-40$$

Tāpiri 4y ki te -y.

$$3y=100-40$$

Tāpiri 100 ki te -40.

$$3y=60$$

Whakawehea ngā taha e rua ki te 3.

$$y=20$$

Whakaurua te 20 mō y ki 2x+y=40. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$2x+20=40$$

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

$$2x=20$$

Whakawehea ngā taha e rua ki te 2.

$$x=10$$

Kua oti te pūnaha te whakatau.

$$x=10,y=20$$