Datrys ar gyfer x, y (complex solution)
\left\{\begin{matrix}x=-\frac{c\left(1-b\right)}{a\left(b-a\right)}\text{, }y=-\frac{c\left(a-1\right)}{b\left(b-a\right)}\text{, }&b\neq 0\text{ and }a\neq b\text{ and }a\neq 0\\x=-\frac{by-c}{a}\text{, }y\in \mathrm{C}\text{, }&\left(c=0\text{ and }b=0\text{ and }a\neq 0\right)\text{ or }\left(c=0\text{ and }a=b\text{ and }b\neq 0\right)\text{ or }\left(a=1\text{ and }b=1\right)\text{ or }\left(a=1\text{ and }b=0\right)\\x\in \mathrm{C}\text{, }y=0\text{, }&c=0\text{ and }a=0\\x\in \mathrm{C}\text{, }y=c\text{, }&b=1\text{ and }a=0\\x\in \mathrm{C}\text{, }y\in \mathrm{C}\text{, }&c=0\text{ and }b=0\text{ and }a=0\end{matrix}\right.
Datrys ar gyfer x, y
\left\{\begin{matrix}x=-\frac{c\left(1-b\right)}{a\left(b-a\right)}\text{, }y=-\frac{c\left(a-1\right)}{b\left(b-a\right)}\text{, }&b\neq 0\text{ and }a\neq b\text{ and }a\neq 0\\x=-\frac{by-c}{a}\text{, }y\in \mathrm{R}\text{, }&\left(c=0\text{ and }b=0\text{ and }a\neq 0\right)\text{ or }\left(c=0\text{ and }a=b\text{ and }b\neq 0\right)\text{ or }\left(a=1\text{ and }b=1\right)\text{ or }\left(a=1\text{ and }b=0\right)\\x\in \mathrm{R}\text{, }y=0\text{, }&c=0\text{ and }a=0\text{ and }b\neq 1\text{ and }b\neq 0\\x\in \mathrm{R}\text{, }y=c\text{, }&b=1\text{ and }a=0\\x\in \mathrm{R}\text{, }y\in \mathrm{R}\text{, }&c=0\text{ and }b=0\text{ and }a=0\end{matrix}\right.
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ax+by=c,a^{2}x+b^{2}y=c
I ddatrys pâr o hafaliadau gan ddefnyddio amnewid, yn gyntaf datryswch un o'r hafaliadau ar gyfer un o'r newidynnau. Yna amnewidiwch y canlyniad am y newidyn hwnnw yn yr hafaliad arall.
ax+by=c
Dewiswch un o'r hafaliadau a’i ddatrys ar gyfer x drwy ynysu x ar ochr chwith yr arwydd hafal.
ax=\left(-b\right)y+c
Tynnu by o ddwy ochr yr hafaliad.
x=\frac{1}{a}\left(\left(-b\right)y+c\right)
Rhannu’r ddwy ochr â a.
x=\left(-\frac{b}{a}\right)y+\frac{c}{a}
Lluoswch \frac{1}{a} â -by+c.
a^{2}\left(\left(-\frac{b}{a}\right)y+\frac{c}{a}\right)+b^{2}y=c
Amnewid \frac{-by+c}{a} am x yn yr hafaliad arall, a^{2}x+b^{2}y=c.
\left(-ab\right)y+ac+b^{2}y=c
Lluoswch a^{2} â \frac{-by+c}{a}.
b\left(b-a\right)y+ac=c
Adio -bay at b^{2}y.
b\left(b-a\right)y=c-ac
Tynnu ca o ddwy ochr yr hafaliad.
y=\frac{c\left(1-a\right)}{b\left(b-a\right)}
Rhannu’r ddwy ochr â b\left(b-a\right).
x=\left(-\frac{b}{a}\right)\times \frac{c\left(1-a\right)}{b\left(b-a\right)}+\frac{c}{a}
Cyfnewidiwch \frac{c\left(1-a\right)}{b\left(b-a\right)} am y yn x=\left(-\frac{b}{a}\right)y+\frac{c}{a}. Am fod yr hafaliad canlynol yn cynnwys dim ond un newidyn, gallwch ddatrys ar gyfer x yn uniongyrchol.
x=-\frac{c\left(1-a\right)}{a\left(b-a\right)}+\frac{c}{a}
Lluoswch -\frac{b}{a} â \frac{c\left(1-a\right)}{b\left(b-a\right)}.
x=\frac{c\left(b-1\right)}{a\left(b-a\right)}
Adio \frac{c}{a} at -\frac{\left(1-a\right)c}{\left(b-a\right)a}.
x=\frac{c\left(b-1\right)}{a\left(b-a\right)},y=\frac{c\left(1-a\right)}{b\left(b-a\right)}
Mae’r system wedi’i datrys nawr.
ax+by=c,a^{2}x+b^{2}y=c
Rhowch yr hafaliadau yn y ffurf safonol ac yna defnyddio’r matricsau i ddatrys y system o hafaliadau.
\left(\begin{matrix}a&b\\a^{2}&b^{2}\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}c\\c\end{matrix}\right)
Ysgrifennu’r hafaliadau ar ffurf matrics.
inverse(\left(\begin{matrix}a&b\\a^{2}&b^{2}\end{matrix}\right))\left(\begin{matrix}a&b\\a^{2}&b^{2}\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}a&b\\a^{2}&b^{2}\end{matrix}\right))\left(\begin{matrix}c\\c\end{matrix}\right)
Lluoswch chwith yr hafaliad gan y matrics gwrthdro o \left(\begin{matrix}a&b\\a^{2}&b^{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}a&b\\a^{2}&b^{2}\end{matrix}\right))\left(\begin{matrix}c\\c\end{matrix}\right)
Cynnyrch matrics a'i wrthdro ydy'r matrics hunaniaeth.
\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}a&b\\a^{2}&b^{2}\end{matrix}\right))\left(\begin{matrix}c\\c\end{matrix}\right)
Lluoswch y matricsau ar ochr chwith yr arwydd hafal.
\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{b^{2}}{ab^{2}-ba^{2}}&-\frac{b}{ab^{2}-ba^{2}}\\-\frac{a^{2}}{ab^{2}-ba^{2}}&\frac{a}{ab^{2}-ba^{2}}\end{matrix}\right)\left(\begin{matrix}c\\c\end{matrix}\right)
Ar gyfer y matrics 2\times 2 \left(\begin{matrix}a&b\\c&d\end{matrix}\right), y matrics gwrthdro yw \left(\begin{matrix}\frac{d}{ad-bc}&\frac{-b}{ad-bc}\\\frac{-c}{ad-bc}&\frac{a}{ad-bc}\end{matrix}\right), felly gellir ailysgrifennu hafaliad y matrics fel problem lluosi matrics.
\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{b}{a\left(b-a\right)}&-\frac{1}{a\left(b-a\right)}\\-\frac{a}{b\left(b-a\right)}&\frac{1}{b\left(b-a\right)}\end{matrix}\right)\left(\begin{matrix}c\\c\end{matrix}\right)
Gwneud y symiau.
\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{b}{a\left(b-a\right)}c+\left(-\frac{1}{a\left(b-a\right)}\right)c\\\left(-\frac{a}{b\left(b-a\right)}\right)c+\frac{1}{b\left(b-a\right)}c\end{matrix}\right)
Lluosi’r matricsau.
\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{c\left(b-1\right)}{a\left(b-a\right)}\\\frac{c\left(1-a\right)}{b\left(b-a\right)}\end{matrix}\right)
Gwneud y symiau.
x=\frac{c\left(b-1\right)}{a\left(b-a\right)},y=\frac{c\left(1-a\right)}{b\left(b-a\right)}
Echdynnu yr elfennau matrics x a y.
ax+by=c,a^{2}x+b^{2}y=c
Er mwyn datrys drwy ddileu, mae’n rhaid i gyfernodau un o'r newidynnau fod yr un peth yn y ddau hafaliad fel bod y newidyn yn cael ei ddiddymu pan fydd un hafaliad yn cael ei dynnu o’r llall.
a^{2}ax+a^{2}by=a^{2}c,aa^{2}x+ab^{2}y=ac
I wneud ax a a^{2}x yn gyfartal, lluoswch yr holl dermau ar bob ochr yr hafaliad cyntaf â a^{2} a holl dermau naill ochr yr ail â a.
a^{3}x+ba^{2}y=ca^{2},a^{3}x+ab^{2}y=ac
Symleiddio.
a^{3}x+\left(-a^{3}\right)x+ba^{2}y+\left(-ab^{2}\right)y=ca^{2}-ac
Tynnwch a^{3}x+ab^{2}y=ac o a^{3}x+ba^{2}y=ca^{2} trwy dynnu termau sydd yr un fath ar bob ochr yr arwydd hafal.
ba^{2}y+\left(-ab^{2}\right)y=ca^{2}-ac
Adio a^{3}x at -a^{3}x. Mae'r termau a^{3}x a -a^{3}x yn diddymu ei gilydd, gan adael hafaliad gyda dim ond un newidyn y gellir ei datrys.
ab\left(a-b\right)y=ca^{2}-ac
Adio a^{2}by at -ab^{2}y.
ab\left(a-b\right)y=ac\left(a-1\right)
Adio a^{2}c at -ac.
y=\frac{c\left(a-1\right)}{b\left(a-b\right)}
Rhannu’r ddwy ochr â ab\left(a-b\right).
a^{2}x+b^{2}\times \frac{c\left(a-1\right)}{b\left(a-b\right)}=c
Cyfnewidiwch \frac{\left(-1+a\right)c}{b\left(a-b\right)} am y yn a^{2}x+b^{2}y=c. Am fod yr hafaliad canlynol yn cynnwys dim ond un newidyn, gallwch ddatrys ar gyfer x yn uniongyrchol.
a^{2}x+\frac{bc\left(a-1\right)}{a-b}=c
Lluoswch b^{2} â \frac{\left(-1+a\right)c}{b\left(a-b\right)}.
a^{2}x=\frac{ac\left(1-b\right)}{a-b}
Tynnu \frac{b\left(-1+a\right)c}{a-b} o ddwy ochr yr hafaliad.
x=\frac{c\left(1-b\right)}{a\left(a-b\right)}
Rhannu’r ddwy ochr â a^{2}.
x=\frac{c\left(1-b\right)}{a\left(a-b\right)},y=\frac{c\left(a-1\right)}{b\left(a-b\right)}
Mae’r system wedi’i datrys nawr.
ax+by=c,a^{2}x+b^{2}y=c
I ddatrys pâr o hafaliadau gan ddefnyddio amnewid, yn gyntaf datryswch un o'r hafaliadau ar gyfer un o'r newidynnau. Yna amnewidiwch y canlyniad am y newidyn hwnnw yn yr hafaliad arall.
ax+by=c
Dewiswch un o'r hafaliadau a’i ddatrys ar gyfer x drwy ynysu x ar ochr chwith yr arwydd hafal.
ax=\left(-b\right)y+c
Tynnu by o ddwy ochr yr hafaliad.
x=\frac{1}{a}\left(\left(-b\right)y+c\right)
Rhannu’r ddwy ochr â a.
x=\left(-\frac{b}{a}\right)y+\frac{c}{a}
Lluoswch \frac{1}{a} â -by+c.
a^{2}\left(\left(-\frac{b}{a}\right)y+\frac{c}{a}\right)+b^{2}y=c
Amnewid \frac{-by+c}{a} am x yn yr hafaliad arall, a^{2}x+b^{2}y=c.
\left(-ab\right)y+ac+b^{2}y=c
Lluoswch a^{2} â \frac{-by+c}{a}.
b\left(b-a\right)y+ac=c
Adio -bay at b^{2}y.
b\left(b-a\right)y=c-ac
Tynnu ca o ddwy ochr yr hafaliad.
y=\frac{c\left(1-a\right)}{b\left(b-a\right)}
Rhannu’r ddwy ochr â b\left(-a+b\right).
x=\left(-\frac{b}{a}\right)\times \frac{c\left(1-a\right)}{b\left(b-a\right)}+\frac{c}{a}
Cyfnewidiwch \frac{c\left(1-a\right)}{b\left(-a+b\right)} am y yn x=\left(-\frac{b}{a}\right)y+\frac{c}{a}. Am fod yr hafaliad canlynol yn cynnwys dim ond un newidyn, gallwch ddatrys ar gyfer x yn uniongyrchol.
x=-\frac{c\left(1-a\right)}{a\left(b-a\right)}+\frac{c}{a}
Lluoswch -\frac{b}{a} â \frac{c\left(1-a\right)}{b\left(-a+b\right)}.
x=\frac{c\left(b-1\right)}{a\left(b-a\right)}
Adio \frac{c}{a} at -\frac{\left(1-a\right)c}{\left(-a+b\right)a}.
x=\frac{c\left(b-1\right)}{a\left(b-a\right)},y=\frac{c\left(1-a\right)}{b\left(b-a\right)}
Mae’r system wedi’i datrys nawr.
ax+by=c,a^{2}x+b^{2}y=c
Rhowch yr hafaliadau yn y ffurf safonol ac yna defnyddio’r matricsau i ddatrys y system o hafaliadau.
\left(\begin{matrix}a&b\\a^{2}&b^{2}\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}c\\c\end{matrix}\right)
Ysgrifennu’r hafaliadau ar ffurf matrics.
inverse(\left(\begin{matrix}a&b\\a^{2}&b^{2}\end{matrix}\right))\left(\begin{matrix}a&b\\a^{2}&b^{2}\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}a&b\\a^{2}&b^{2}\end{matrix}\right))\left(\begin{matrix}c\\c\end{matrix}\right)
Lluoswch chwith yr hafaliad gan y matrics gwrthdro o \left(\begin{matrix}a&b\\a^{2}&b^{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}a&b\\a^{2}&b^{2}\end{matrix}\right))\left(\begin{matrix}c\\c\end{matrix}\right)
Cynnyrch matrics a'i wrthdro ydy'r matrics hunaniaeth.
\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}a&b\\a^{2}&b^{2}\end{matrix}\right))\left(\begin{matrix}c\\c\end{matrix}\right)
Lluoswch y matricsau ar ochr chwith yr arwydd hafal.
\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{b^{2}}{ab^{2}-ba^{2}}&-\frac{b}{ab^{2}-ba^{2}}\\-\frac{a^{2}}{ab^{2}-ba^{2}}&\frac{a}{ab^{2}-ba^{2}}\end{matrix}\right)\left(\begin{matrix}c\\c\end{matrix}\right)
Ar gyfer y matrics 2\times 2 \left(\begin{matrix}a&b\\c&d\end{matrix}\right), y matrics gwrthdro yw \left(\begin{matrix}\frac{d}{ad-bc}&\frac{-b}{ad-bc}\\\frac{-c}{ad-bc}&\frac{a}{ad-bc}\end{matrix}\right), felly gellir ailysgrifennu hafaliad y matrics fel problem lluosi matrics.
\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{b}{a\left(b-a\right)}&-\frac{1}{a\left(b-a\right)}\\-\frac{a}{b\left(b-a\right)}&\frac{1}{b\left(b-a\right)}\end{matrix}\right)\left(\begin{matrix}c\\c\end{matrix}\right)
Gwneud y symiau.
\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{b}{a\left(b-a\right)}c+\left(-\frac{1}{a\left(b-a\right)}\right)c\\\left(-\frac{a}{b\left(b-a\right)}\right)c+\frac{1}{b\left(b-a\right)}c\end{matrix}\right)
Lluosi’r matricsau.
\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{c\left(b-1\right)}{a\left(b-a\right)}\\\frac{c\left(1-a\right)}{b\left(b-a\right)}\end{matrix}\right)
Gwneud y symiau.
x=\frac{c\left(b-1\right)}{a\left(b-a\right)},y=\frac{c\left(1-a\right)}{b\left(b-a\right)}
Echdynnu yr elfennau matrics x a y.
ax+by=c,a^{2}x+b^{2}y=c
Er mwyn datrys drwy ddileu, mae’n rhaid i gyfernodau un o'r newidynnau fod yr un peth yn y ddau hafaliad fel bod y newidyn yn cael ei ddiddymu pan fydd un hafaliad yn cael ei dynnu o’r llall.
a^{2}ax+a^{2}by=a^{2}c,aa^{2}x+ab^{2}y=ac
I wneud ax a a^{2}x yn gyfartal, lluoswch yr holl dermau ar bob ochr yr hafaliad cyntaf â a^{2} a holl dermau naill ochr yr ail â a.
a^{3}x+ba^{2}y=ca^{2},a^{3}x+ab^{2}y=ac
Symleiddio.
a^{3}x+\left(-a^{3}\right)x+ba^{2}y+\left(-ab^{2}\right)y=ca^{2}-ac
Tynnwch a^{3}x+ab^{2}y=ac o a^{3}x+ba^{2}y=ca^{2} trwy dynnu termau sydd yr un fath ar bob ochr yr arwydd hafal.
ba^{2}y+\left(-ab^{2}\right)y=ca^{2}-ac
Adio a^{3}x at -a^{3}x. Mae'r termau a^{3}x a -a^{3}x yn diddymu ei gilydd, gan adael hafaliad gyda dim ond un newidyn y gellir ei datrys.
ab\left(a-b\right)y=ca^{2}-ac
Adio a^{2}by at -ab^{2}y.
ab\left(a-b\right)y=ac\left(a-1\right)
Adio a^{2}c at -ac.
y=\frac{c\left(a-1\right)}{b\left(a-b\right)}
Rhannu’r ddwy ochr â ab\left(a-b\right).
a^{2}x+b^{2}\times \frac{c\left(a-1\right)}{b\left(a-b\right)}=c
Cyfnewidiwch \frac{\left(-1+a\right)c}{b\left(a-b\right)} am y yn a^{2}x+b^{2}y=c. Am fod yr hafaliad canlynol yn cynnwys dim ond un newidyn, gallwch ddatrys ar gyfer x yn uniongyrchol.
a^{2}x+\frac{bc\left(a-1\right)}{a-b}=c
Lluoswch b^{2} â \frac{\left(-1+a\right)c}{b\left(a-b\right)}.
a^{2}x=\frac{ac\left(1-b\right)}{a-b}
Tynnu \frac{b\left(-1+a\right)c}{a-b} o ddwy ochr yr hafaliad.
x=\frac{c\left(1-b\right)}{a\left(a-b\right)}
Rhannu’r ddwy ochr â a^{2}.
x=\frac{c\left(1-b\right)}{a\left(a-b\right)},y=\frac{c\left(a-1\right)}{b\left(a-b\right)}
Mae’r system wedi’i datrys nawr.
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