\left\{ \begin{array} { l } { 2 x + y = 3 } \\ { x - y = 1 } \end{array} \right.
Whakaoti mō x, y
x = \frac{4}{3} = 1\frac{1}{3} \approx 1.333333333
y=\frac{1}{3}\approx 0.333333333
Graph
Tohaina
Kua tāruatia ki te papatopenga
2x+y=3,x-y=1
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.
2x+y=3
Kōwhiria tētahi o ngā whārite ka whakaotia mō te x mā te wehe i te x i te taha mauī o te tohu ōrite.
2x=-y+3
Me tango y mai i ngā taha e rua o te whārite.
x=\frac{1}{2}\left(-y+3\right)
Whakawehea ngā taha e rua ki te 2.
x=-\frac{1}{2}y+\frac{3}{2}
Whakareatia \frac{1}{2} ki te -y+3.
-\frac{1}{2}y+\frac{3}{2}-y=1
Whakakapia te \frac{-y+3}{2} mō te x ki tērā atu whārite, x-y=1.
-\frac{3}{2}y+\frac{3}{2}=1
Tāpiri -\frac{y}{2} ki te -y.
-\frac{3}{2}y=-\frac{1}{2}
Me tango \frac{3}{2} mai i ngā taha e rua o te whārite.
y=\frac{1}{3}
Whakawehea ngā taha e rua o te whārite ki te -\frac{3}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.
x=-\frac{1}{2}\times \frac{1}{3}+\frac{3}{2}
Whakaurua te \frac{1}{3} mō y ki x=-\frac{1}{2}y+\frac{3}{2}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.
x=-\frac{1}{6}+\frac{3}{2}
Whakareatia -\frac{1}{2} ki te \frac{1}{3} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.
x=\frac{4}{3}
Tāpiri \frac{3}{2} ki te -\frac{1}{6} mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.
x=\frac{4}{3},y=\frac{1}{3}
Kua oti te pūnaha te whakatau.
2x+y=3,x-y=1
Tuhia ngā whārite ki te tānga ngahuru ka whakamahi i ngā poukapa hei whakaoti i te pūnaha o ngā whārite.
\left(\begin{matrix}2&1\\1&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}3\\1\end{matrix}\right)
Tuhia ngā whārite ki te tikanga tātai poukapa.
inverse(\left(\begin{matrix}2&1\\1&-1\end{matrix}\right))\left(\begin{matrix}2&1\\1&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&1\\1&-1\end{matrix}\right))\left(\begin{matrix}3\\1\end{matrix}\right)
Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}2&1\\1&-1\end{matrix}\right).
\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&1\\1&-1\end{matrix}\right))\left(\begin{matrix}3\\1\end{matrix}\right)
Ko te hua o tētahi poukapa me te kōaro ko te poukapa tuakiri.
\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&1\\1&-1\end{matrix}\right))\left(\begin{matrix}3\\1\end{matrix}\right)
Whakareatia ngā poukapa kei te taha mauī o te tohu ōrite.
\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{2\left(-1\right)-1}&-\frac{1}{2\left(-1\right)-1}\\-\frac{1}{2\left(-1\right)-1}&\frac{2}{2\left(-1\right)-1}\end{matrix}\right)\left(\begin{matrix}3\\1\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}{3}&\frac{1}{3}\\\frac{1}{3}&-\frac{2}{3}\end{matrix}\right)\left(\begin{matrix}3\\1\end{matrix}\right)
Mahia ngā tātaitanga.
\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{3}\times 3+\frac{1}{3}\\\frac{1}{3}\times 3-\frac{2}{3}\end{matrix}\right)
Whakareatia ngā poukapa.
\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{3}\\\frac{1}{3}\end{matrix}\right)
Mahia ngā tātaitanga.
x=\frac{4}{3},y=\frac{1}{3}
Tangohia ngā huānga poukapa x me y.
2x+y=3,x-y=1
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.
2x+y=3,2x+2\left(-1\right)y=2
Kia ōrite ai a 2x me x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 1 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 2.
2x+y=3,2x-2y=2
Whakarūnātia.
2x-2x+y+2y=3-2
Me tango 2x-2y=2 mai i 2x+y=3 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.
y+2y=3-2
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.
3y=3-2
Tāpiri y ki te 2y.
3y=1
Tāpiri 3 ki te -2.
y=\frac{1}{3}
Whakawehea ngā taha e rua ki te 3.
x-\frac{1}{3}=1
Whakaurua te \frac{1}{3} mō y ki x-y=1. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.
x=\frac{4}{3}
Me tāpiri \frac{1}{3} ki ngā taha e rua o te whārite.
x=\frac{4}{3},y=\frac{1}{3}
Kua oti te pūnaha te whakatau.
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