Whakaoti i te {l}{x+y=5}{2x+3y=20}

$$\left. \begin{array} { l } { x + y = 5 } \\ { 2 x + 3 y = 20 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-5$$

$$y=10$$

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.

$$x+y=5,2x+3y=20$$

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

$$x+y=5$$

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

$$x=-y+5$$

Whakakapia te -y+5 mō te x ki tērā atu whārite, 2x+3y=20.

$$2\left(-y+5\right)+3y=20$$

Whakareatia 2 ki te -y+5.

$$-2y+10+3y=20$$

Tāpiri -2y ki te 3y.

$$y+10=20$$

Me tango 10 mai i ngā taha e rua o te whārite.

$$y=10$$

Whakaurua te 10 mō y ki x=-y+5. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-10+5$$

Tāpiri 5 ki te -10.

$$x=-5$$

Kua oti te pūnaha te whakatau.

$$x=-5,y=10$$

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.

$$x+y=5,2x+3y=20$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&1\\2&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}5\\20\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}1&1\\2&3\end{matrix}\right))\left(\begin{matrix}1&1\\2&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\2&3\end{matrix}\right))\left(\begin{matrix}5\\20\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\2&3\end{matrix}\right))\left(\begin{matrix}5\\20\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\2&3\end{matrix}\right))\left(\begin{matrix}5\\20\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{3-2}&-\frac{1}{3-2}\\-\frac{2}{3-2}&\frac{1}{3-2}\end{matrix}\right)\left(\begin{matrix}5\\20\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}3&-1\\-2&1\end{matrix}\right)\left(\begin{matrix}5\\20\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}3\times 5-20\\-2\times 5+20\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-5\\10\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-5,y=10$$

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.

$$x+y=5,2x+3y=20$$

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.

$$2x+2y=2\times 5,2x+3y=20$$

Whakarūnātia.

$$2x+2y=10,2x+3y=20$$

Me tango 2x+3y=20 mai i 2x+2y=10 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$2x-2x+2y-3y=10-20$$

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.

$$2y-3y=10-20$$

Tāpiri 2y ki te -3y.

$$-y=10-20$$

Tāpiri 10 ki te -20.

$$-y=-10$$

Whakawehea ngā taha e rua ki te -1.

$$y=10$$

Whakaurua te 10 mō y ki 2x+3y=20. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$2x+3\times 10=20$$

Whakareatia 3 ki te 10.

$$2x+30=20$$

Me tango 30 mai i ngā taha e rua o te whārite.

$$2x=-10$$

Whakawehea ngā taha e rua ki te 2.

$$x=-5$$

Kua oti te pūnaha te whakatau.

$$x=-5,y=10$$