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5x-4y=19,3x+2y=71
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.
5x-4y=19
Kōwhiria tētahi o ngā whārite ka whakaotia mō te x mā te wehe i te x i te taha mauī o te tohu ōrite.
5x=4y+19
Me tāpiri 4y ki ngā taha e rua o te whārite.
x=\frac{1}{5}\left(4y+19\right)
Whakawehea ngā taha e rua ki te 5.
x=\frac{4}{5}y+\frac{19}{5}
Whakareatia \frac{1}{5} ki te 4y+19.
3\left(\frac{4}{5}y+\frac{19}{5}\right)+2y=71
Whakakapia te \frac{4y+19}{5} mō te x ki tērā atu whārite, 3x+2y=71.
\frac{12}{5}y+\frac{57}{5}+2y=71
Whakareatia 3 ki te \frac{4y+19}{5}.
\frac{22}{5}y+\frac{57}{5}=71
Tāpiri \frac{12y}{5} ki te 2y.
\frac{22}{5}y=\frac{298}{5}
Me tango \frac{57}{5} mai i ngā taha e rua o te whārite.
y=\frac{149}{11}
Whakawehea ngā taha e rua o te whārite ki te \frac{22}{5}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.
x=\frac{4}{5}\times \frac{149}{11}+\frac{19}{5}
Whakaurua te \frac{149}{11} mō y ki x=\frac{4}{5}y+\frac{19}{5}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.
x=\frac{596}{55}+\frac{19}{5}
Whakareatia \frac{4}{5} ki te \frac{149}{11} 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{161}{11}
Tāpiri \frac{19}{5} ki te \frac{596}{55} 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{161}{11},y=\frac{149}{11}
Kua oti te pūnaha te whakatau.
5x-4y=19,3x+2y=71
Tuhia ngā whārite ki te tānga ngahuru ka whakamahi i ngā poukapa hei whakaoti i te pūnaha o ngā whārite.
\left(\begin{matrix}5&-4\\3&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}19\\71\end{matrix}\right)
Tuhia ngā whārite ki te tikanga tātai poukapa.
inverse(\left(\begin{matrix}5&-4\\3&2\end{matrix}\right))\left(\begin{matrix}5&-4\\3&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-4\\3&2\end{matrix}\right))\left(\begin{matrix}19\\71\end{matrix}\right)
Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}5&-4\\3&2\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}5&-4\\3&2\end{matrix}\right))\left(\begin{matrix}19\\71\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}5&-4\\3&2\end{matrix}\right))\left(\begin{matrix}19\\71\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{2}{5\times 2-\left(-4\times 3\right)}&-\frac{-4}{5\times 2-\left(-4\times 3\right)}\\-\frac{3}{5\times 2-\left(-4\times 3\right)}&\frac{5}{5\times 2-\left(-4\times 3\right)}\end{matrix}\right)\left(\begin{matrix}19\\71\end{matrix}\right)
Mō te poukapa 2\times 2 \left(\begin{matrix}a&b\\c&d\end{matrix}\right), ko te poukapa kōaro ko \left(\begin{matrix}\frac{d}{ad-bc}&\frac{-b}{ad-bc}\\\frac{-c}{ad-bc}&\frac{a}{ad-bc}\end{matrix}\right), kia tuhia anō ai te whārite poukapa hei rapanga whakarea poukapa.
\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{11}&\frac{2}{11}\\-\frac{3}{22}&\frac{5}{22}\end{matrix}\right)\left(\begin{matrix}19\\71\end{matrix}\right)
Mahia ngā tātaitanga.
\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{11}\times 19+\frac{2}{11}\times 71\\-\frac{3}{22}\times 19+\frac{5}{22}\times 71\end{matrix}\right)
Whakareatia ngā poukapa.
\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{161}{11}\\\frac{149}{11}\end{matrix}\right)
Mahia ngā tātaitanga.
x=\frac{161}{11},y=\frac{149}{11}
Tangohia ngā huānga poukapa x me y.
5x-4y=19,3x+2y=71
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.
3\times 5x+3\left(-4\right)y=3\times 19,5\times 3x+5\times 2y=5\times 71
Kia ōrite ai a 5x me 3x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 3 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 5.
15x-12y=57,15x+10y=355
Whakarūnātia.
15x-15x-12y-10y=57-355
Me tango 15x+10y=355 mai i 15x-12y=57 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.
-12y-10y=57-355
Tāpiri 15x ki te -15x. Ka whakakore atu ngā kupu 15x me -15x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.
-22y=57-355
Tāpiri -12y ki te -10y.
-22y=-298
Tāpiri 57 ki te -355.
y=\frac{149}{11}
Whakawehea ngā taha e rua ki te -22.
3x+2\times \frac{149}{11}=71
Whakaurua te \frac{149}{11} mō y ki 3x+2y=71. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.
3x+\frac{298}{11}=71
Whakareatia 2 ki te \frac{149}{11}.
3x=\frac{483}{11}
Me tango \frac{298}{11} mai i ngā taha e rua o te whārite.
x=\frac{161}{11}
Whakawehea ngā taha e rua ki te 3.
x=\frac{161}{11},y=\frac{149}{11}
Kua oti te pūnaha te whakatau.