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Whakaoti mō x, y (complex solution)
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Whakaoti mō x, y
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3x-\left(ky+y\right)=20
Whakaarohia te whārite tuatahi. Whakamahia te āhuatanga tohatoha hei whakarea te k+1 ki te y.
3x-ky-y=20
Hei kimi i te tauaro o ky+y, kimihia te tauaro o ia taurangi.
3x+\left(-k-1\right)y=20
Pahekotia ngā kīanga tau katoa e whai ana i te x,y.
kx+2x-10y=40
Whakaarohia te whārite tuarua. Whakamahia te āhuatanga tohatoha hei whakarea te k+2 ki te x.
\left(k+2\right)x-10y=40
Pahekotia ngā kīanga tau katoa e whai ana i te x,y.
3x+\left(-k-1\right)y=20,\left(k+2\right)x-10y=40
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.
3x+\left(-k-1\right)y=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.
3x=\left(k+1\right)y+20
Me tāpiri \left(k+1\right)y ki ngā taha e rua o te whārite.
x=\frac{1}{3}\left(\left(k+1\right)y+20\right)
Whakawehea ngā taha e rua ki te 3.
x=\frac{k+1}{3}y+\frac{20}{3}
Whakareatia \frac{1}{3} ki te yk+y+20.
\left(k+2\right)\left(\frac{k+1}{3}y+\frac{20}{3}\right)-10y=40
Whakakapia te \frac{yk+y+20}{3} mō te x ki tērā atu whārite, \left(k+2\right)x-10y=40.
\frac{\left(k+1\right)\left(k+2\right)}{3}y+\frac{20k+40}{3}-10y=40
Whakareatia k+2 ki te \frac{yk+y+20}{3}.
\frac{\left(k-4\right)\left(k+7\right)}{3}y+\frac{20k+40}{3}=40
Tāpiri \frac{\left(k+2\right)\left(k+1\right)y}{3} ki te -10y.
\frac{\left(k-4\right)\left(k+7\right)}{3}y=\frac{80-20k}{3}
Me tango \frac{40+20k}{3} mai i ngā taha e rua o te whārite.
y=-\frac{20}{k+7}
Whakawehea ngā taha e rua ki te \frac{\left(-4+k\right)\left(7+k\right)}{3}.
x=\frac{k+1}{3}\left(-\frac{20}{k+7}\right)+\frac{20}{3}
Whakaurua te -\frac{20}{7+k} mō y ki x=\frac{k+1}{3}y+\frac{20}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.
x=-\frac{20\left(k+1\right)}{3\left(k+7\right)}+\frac{20}{3}
Whakareatia \frac{k+1}{3} ki te -\frac{20}{7+k}.
x=\frac{40}{k+7}
Tāpiri \frac{20}{3} ki te -\frac{20\left(k+1\right)}{3\left(7+k\right)}.
x=\frac{40}{k+7},y=-\frac{20}{k+7}
Kua oti te pūnaha te whakatau.
3x-\left(ky+y\right)=20
Whakaarohia te whārite tuatahi. Whakamahia te āhuatanga tohatoha hei whakarea te k+1 ki te y.
3x-ky-y=20
Hei kimi i te tauaro o ky+y, kimihia te tauaro o ia taurangi.
3x+\left(-k-1\right)y=20
Pahekotia ngā kīanga tau katoa e whai ana i te x,y.
kx+2x-10y=40
Whakaarohia te whārite tuarua. Whakamahia te āhuatanga tohatoha hei whakarea te k+2 ki te x.
\left(k+2\right)x-10y=40
Pahekotia ngā kīanga tau katoa e whai ana i te x,y.
3x+\left(-k-1\right)y=20,\left(k+2\right)x-10y=40
Tuhia ngā whārite ki te tānga ngahuru ka whakamahi i ngā poukapa hei whakaoti i te pūnaha o ngā whārite.
\left(\begin{matrix}3&-k-1\\k+2&-10\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}20\\40\end{matrix}\right)
Tuhia ngā whārite ki te tikanga tātai poukapa.
inverse(\left(\begin{matrix}3&-k-1\\k+2&-10\end{matrix}\right))\left(\begin{matrix}3&-k-1\\k+2&-10\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-k-1\\k+2&-10\end{matrix}\right))\left(\begin{matrix}20\\40\end{matrix}\right)
Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}3&-k-1\\k+2&-10\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}3&-k-1\\k+2&-10\end{matrix}\right))\left(\begin{matrix}20\\40\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}3&-k-1\\k+2&-10\end{matrix}\right))\left(\begin{matrix}20\\40\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{10}{3\left(-10\right)-\left(-k-1\right)\left(k+2\right)}&-\frac{-k-1}{3\left(-10\right)-\left(-k-1\right)\left(k+2\right)}\\-\frac{k+2}{3\left(-10\right)-\left(-k-1\right)\left(k+2\right)}&\frac{3}{3\left(-10\right)-\left(-k-1\right)\left(k+2\right)}\end{matrix}\right)\left(\begin{matrix}20\\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{10}{\left(k-4\right)\left(k+7\right)}&\frac{k+1}{\left(k-4\right)\left(k+7\right)}\\-\frac{k+2}{\left(k-4\right)\left(k+7\right)}&\frac{3}{\left(k-4\right)\left(k+7\right)}\end{matrix}\right)\left(\begin{matrix}20\\40\end{matrix}\right)
Mahia ngā tātaitanga.
\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\left(-\frac{10}{\left(k-4\right)\left(k+7\right)}\right)\times 20+\frac{k+1}{\left(k-4\right)\left(k+7\right)}\times 40\\\left(-\frac{k+2}{\left(k-4\right)\left(k+7\right)}\right)\times 20+\frac{3}{\left(k-4\right)\left(k+7\right)}\times 40\end{matrix}\right)
Whakareatia ngā poukapa.
\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{40}{k+7}\\-\frac{20}{k+7}\end{matrix}\right)
Mahia ngā tātaitanga.
x=\frac{40}{k+7},y=-\frac{20}{k+7}
Tangohia ngā huānga poukapa x me y.
3x-\left(ky+y\right)=20
Whakaarohia te whārite tuatahi. Whakamahia te āhuatanga tohatoha hei whakarea te k+1 ki te y.
3x-ky-y=20
Hei kimi i te tauaro o ky+y, kimihia te tauaro o ia taurangi.
3x+\left(-k-1\right)y=20
Pahekotia ngā kīanga tau katoa e whai ana i te x,y.
kx+2x-10y=40
Whakaarohia te whārite tuarua. Whakamahia te āhuatanga tohatoha hei whakarea te k+2 ki te x.
\left(k+2\right)x-10y=40
Pahekotia ngā kīanga tau katoa e whai ana i te x,y.
3x+\left(-k-1\right)y=20,\left(k+2\right)x-10y=40
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.
\left(k+2\right)\times 3x+\left(k+2\right)\left(-k-1\right)y=\left(k+2\right)\times 20,3\left(k+2\right)x+3\left(-10\right)y=3\times 40
Kia ōrite ai a 3x me \left(k+2\right)x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te k+2 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 3.
\left(3k+6\right)x+\left(-\left(k+1\right)\left(k+2\right)\right)y=20k+40,\left(3k+6\right)x-30y=120
Whakarūnātia.
\left(3k+6\right)x+\left(-3k-6\right)x+\left(-\left(k+1\right)\left(k+2\right)\right)y+30y=20k+40-120
Me tango \left(3k+6\right)x-30y=120 mai i \left(3k+6\right)x+\left(-\left(k+1\right)\left(k+2\right)\right)y=20k+40 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.
\left(-\left(k+1\right)\left(k+2\right)\right)y+30y=20k+40-120
Tāpiri 3\left(2+k\right)x ki te -6x-3xk. Ka whakakore atu ngā kupu 3\left(2+k\right)x me -6x-3xk, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.
\left(4-k\right)\left(k+7\right)y=20k+40-120
Tāpiri -\left(k+2\right)\left(k+1\right)y ki te 30y.
\left(4-k\right)\left(k+7\right)y=20k-80
Tāpiri 20k+40 ki te -120.
y=-\frac{20}{k+7}
Whakawehea ngā taha e rua ki te \left(4-k\right)\left(7+k\right).
\left(k+2\right)x-10\left(-\frac{20}{k+7}\right)=40
Whakaurua te -\frac{20}{7+k} mō y ki \left(k+2\right)x-10y=40. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.
\left(k+2\right)x+\frac{200}{k+7}=40
Whakareatia -10 ki te -\frac{20}{7+k}.
\left(k+2\right)x=\frac{40\left(k+2\right)}{k+7}
Me tango \frac{200}{7+k} mai i ngā taha e rua o te whārite.
x=\frac{40}{k+7}
Whakawehea ngā taha e rua ki te k+2.
x=\frac{40}{k+7},y=-\frac{20}{k+7}
Kua oti te pūnaha te whakatau.
3x-\left(ky+y\right)=20
Whakaarohia te whārite tuatahi. Whakamahia te āhuatanga tohatoha hei whakarea te k+1 ki te y.
3x-ky-y=20
Hei kimi i te tauaro o ky+y, kimihia te tauaro o ia taurangi.
3x+\left(-k-1\right)y=20
Pahekotia ngā kīanga tau katoa e whai ana i te x,y.
kx+2x-10y=40
Whakaarohia te whārite tuarua. Whakamahia te āhuatanga tohatoha hei whakarea te k+2 ki te x.
\left(k+2\right)x-10y=40
Pahekotia ngā kīanga tau katoa e whai ana i te x,y.
3x+\left(-k-1\right)y=20,\left(k+2\right)x-10y=40
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.
3x+\left(-k-1\right)y=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.
3x=\left(k+1\right)y+20
Me tāpiri \left(k+1\right)y ki ngā taha e rua o te whārite.
x=\frac{1}{3}\left(\left(k+1\right)y+20\right)
Whakawehea ngā taha e rua ki te 3.
x=\frac{k+1}{3}y+\frac{20}{3}
Whakareatia \frac{1}{3} ki te yk+y+20.
\left(k+2\right)\left(\frac{k+1}{3}y+\frac{20}{3}\right)-10y=40
Whakakapia te \frac{yk+y+20}{3} mō te x ki tērā atu whārite, \left(k+2\right)x-10y=40.
\frac{\left(k+1\right)\left(k+2\right)}{3}y+\frac{20k+40}{3}-10y=40
Whakareatia k+2 ki te \frac{yk+y+20}{3}.
\frac{\left(k-4\right)\left(k+7\right)}{3}y+\frac{20k+40}{3}=40
Tāpiri \frac{\left(k+2\right)\left(k+1\right)y}{3} ki te -10y.
\frac{\left(k-4\right)\left(k+7\right)}{3}y=\frac{80-20k}{3}
Me tango \frac{40+20k}{3} mai i ngā taha e rua o te whārite.
y=-\frac{20}{k+7}
Whakawehea ngā taha e rua ki te \frac{\left(-4+k\right)\left(7+k\right)}{3}.
x=\frac{k+1}{3}\left(-\frac{20}{k+7}\right)+\frac{20}{3}
Whakaurua te -\frac{20}{7+k} mō y ki x=\frac{k+1}{3}y+\frac{20}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.
x=-\frac{20\left(k+1\right)}{3\left(k+7\right)}+\frac{20}{3}
Whakareatia \frac{k+1}{3} ki te -\frac{20}{7+k}.
x=\frac{40}{k+7}
Tāpiri \frac{20}{3} ki te -\frac{20\left(k+1\right)}{3\left(7+k\right)}.
x=\frac{40}{k+7},y=-\frac{20}{k+7}
Kua oti te pūnaha te whakatau.
3x-\left(ky+y\right)=20
Whakaarohia te whārite tuatahi. Whakamahia te āhuatanga tohatoha hei whakarea te k+1 ki te y.
3x-ky-y=20
Hei kimi i te tauaro o ky+y, kimihia te tauaro o ia taurangi.
3x+\left(-k-1\right)y=20
Pahekotia ngā kīanga tau katoa e whai ana i te x,y.
kx+2x-10y=40
Whakaarohia te whārite tuarua. Whakamahia te āhuatanga tohatoha hei whakarea te k+2 ki te x.
\left(k+2\right)x-10y=40
Pahekotia ngā kīanga tau katoa e whai ana i te x,y.
3x+\left(-k-1\right)y=20,\left(k+2\right)x-10y=40
Tuhia ngā whārite ki te tānga ngahuru ka whakamahi i ngā poukapa hei whakaoti i te pūnaha o ngā whārite.
\left(\begin{matrix}3&-k-1\\k+2&-10\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}20\\40\end{matrix}\right)
Tuhia ngā whārite ki te tikanga tātai poukapa.
inverse(\left(\begin{matrix}3&-k-1\\k+2&-10\end{matrix}\right))\left(\begin{matrix}3&-k-1\\k+2&-10\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-k-1\\k+2&-10\end{matrix}\right))\left(\begin{matrix}20\\40\end{matrix}\right)
Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}3&-k-1\\k+2&-10\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}3&-k-1\\k+2&-10\end{matrix}\right))\left(\begin{matrix}20\\40\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}3&-k-1\\k+2&-10\end{matrix}\right))\left(\begin{matrix}20\\40\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{10}{3\left(-10\right)-\left(-k-1\right)\left(k+2\right)}&-\frac{-k-1}{3\left(-10\right)-\left(-k-1\right)\left(k+2\right)}\\-\frac{k+2}{3\left(-10\right)-\left(-k-1\right)\left(k+2\right)}&\frac{3}{3\left(-10\right)-\left(-k-1\right)\left(k+2\right)}\end{matrix}\right)\left(\begin{matrix}20\\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{10}{\left(k-4\right)\left(k+7\right)}&\frac{k+1}{\left(k-4\right)\left(k+7\right)}\\-\frac{k+2}{\left(k-4\right)\left(k+7\right)}&\frac{3}{\left(k-4\right)\left(k+7\right)}\end{matrix}\right)\left(\begin{matrix}20\\40\end{matrix}\right)
Mahia ngā tātaitanga.
\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\left(-\frac{10}{\left(k-4\right)\left(k+7\right)}\right)\times 20+\frac{k+1}{\left(k-4\right)\left(k+7\right)}\times 40\\\left(-\frac{k+2}{\left(k-4\right)\left(k+7\right)}\right)\times 20+\frac{3}{\left(k-4\right)\left(k+7\right)}\times 40\end{matrix}\right)
Whakareatia ngā poukapa.
\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{40}{k+7}\\-\frac{20}{k+7}\end{matrix}\right)
Mahia ngā tātaitanga.
x=\frac{40}{k+7},y=-\frac{20}{k+7}
Tangohia ngā huānga poukapa x me y.
3x-\left(ky+y\right)=20
Whakaarohia te whārite tuatahi. Whakamahia te āhuatanga tohatoha hei whakarea te k+1 ki te y.
3x-ky-y=20
Hei kimi i te tauaro o ky+y, kimihia te tauaro o ia taurangi.
3x+\left(-k-1\right)y=20
Pahekotia ngā kīanga tau katoa e whai ana i te x,y.
kx+2x-10y=40
Whakaarohia te whārite tuarua. Whakamahia te āhuatanga tohatoha hei whakarea te k+2 ki te x.
\left(k+2\right)x-10y=40
Pahekotia ngā kīanga tau katoa e whai ana i te x,y.
3x+\left(-k-1\right)y=20,\left(k+2\right)x-10y=40
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.
\left(k+2\right)\times 3x+\left(k+2\right)\left(-k-1\right)y=\left(k+2\right)\times 20,3\left(k+2\right)x+3\left(-10\right)y=3\times 40
Kia ōrite ai a 3x me \left(k+2\right)x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te k+2 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 3.
\left(3k+6\right)x+\left(-\left(k+1\right)\left(k+2\right)\right)y=20k+40,\left(3k+6\right)x-30y=120
Whakarūnātia.
\left(3k+6\right)x+\left(-3k-6\right)x+\left(-\left(k+1\right)\left(k+2\right)\right)y+30y=20k+40-120
Me tango \left(3k+6\right)x-30y=120 mai i \left(3k+6\right)x+\left(-\left(k+1\right)\left(k+2\right)\right)y=20k+40 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.
\left(-\left(k+1\right)\left(k+2\right)\right)y+30y=20k+40-120
Tāpiri 3\left(2+k\right)x ki te -6x-3xk. Ka whakakore atu ngā kupu 3\left(2+k\right)x me -6x-3xk, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.
\left(4-k\right)\left(k+7\right)y=20k+40-120
Tāpiri -\left(k+2\right)\left(k+1\right)y ki te 30y.
\left(4-k\right)\left(k+7\right)y=20k-80
Tāpiri 20k+40 ki te -120.
y=-\frac{20}{k+7}
Whakawehea ngā taha e rua ki te \left(4-k\right)\left(7+k\right).
\left(k+2\right)x-10\left(-\frac{20}{k+7}\right)=40
Whakaurua te -\frac{20}{7+k} mō y ki \left(k+2\right)x-10y=40. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.
\left(k+2\right)x+\frac{200}{k+7}=40
Whakareatia -10 ki te -\frac{20}{7+k}.
\left(k+2\right)x=\frac{40\left(k+2\right)}{k+7}
Me tango \frac{200}{7+k} mai i ngā taha e rua o te whārite.
x=\frac{40}{k+7}
Whakawehea ngā taha e rua ki te k+2.
x=\frac{40}{k+7},y=-\frac{20}{k+7}
Kua oti te pūnaha te whakatau.