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

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

Whakaoti mō x, y

$$x=a+1$$

$$y=1-a$$

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.

$$x-y=2a,2x+3y=5-a$$

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

$$x-y=2a$$

Me tāpiri y ki ngā taha e rua o te whārite.

$$x=y+2a$$

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

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

Whakareatia 2 ki te y+2a.

$$2y+4a+3y=5-a$$

Tāpiri 2y ki te 3y.

$$5y+4a=5-a$$

Me tango 4a mai i ngā taha e rua o te whārite.

$$5y=5-5a$$

Whakawehea ngā taha e rua ki te 5.

$$y=1-a$$

Whakaurua te 1-a mō y ki x=y+2a. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=1-a+2a$$

Tāpiri 2a ki te 1-a.

$$x=a+1$$

Kua oti te pūnaha te whakatau.

$$x=a+1,y=1-a$$

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.

$$x-y=2a,2x+3y=5-a$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}a+1\\1-a\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=a+1,y=1-a$$

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.

$$x-y=2a,2x+3y=5-a$$

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.

$$2x+2\left(-1\right)y=2\times 2a,2x+3y=5-a$$

Whakarūnātia.

$$2x-2y=4a,2x+3y=5-a$$

Me tango 2x+3y=5-a mai i 2x-2y=4a mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$2x-2x-2y-3y=4a+a-5$$

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.

$$-2y-3y=4a+a-5$$

Tāpiri -2y ki te -3y.

$$-5y=4a+a-5$$

Tāpiri 4a ki te -5+a.

$$-5y=5a-5$$

Whakawehea ngā taha e rua ki te -5.

$$y=1-a$$

Whakaurua te 1-a mō y ki 2x+3y=5-a. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$2x+3\left(1-a\right)=5-a$$

Whakareatia 3 ki te 1-a.

$$2x+3-3a=5-a$$

Me tango 3-3a mai i ngā taha e rua o te whārite.

$$2x=2a+2$$

Whakawehea ngā taha e rua ki te 2.

$$x=a+1$$

Kua oti te pūnaha te whakatau.

$$x=a+1,y=1-a$$