Solve for a (complex solution)
\left\{\begin{matrix}a=\frac{cx+cy-by}{x}\text{, }&x\neq 0\\a\in \mathrm{C}\text{, }&\left(y=0\text{ or }b=c\right)\text{ and }x=0\end{matrix}\right.
Solve for b (complex solution)
\left\{\begin{matrix}b=\frac{cx-ax+cy}{y}\text{, }&y\neq 0\\b\in \mathrm{C}\text{, }&y=0\text{ and }\left(a=c\text{ or }x=0\right)\end{matrix}\right.
Solve for a
\left\{\begin{matrix}a=\frac{cx+cy-by}{x}\text{, }&x\neq 0\\a\in \mathrm{R}\text{, }&\left(y=0\text{ or }b=c\right)\text{ and }x=0\end{matrix}\right.
Solve for b
\left\{\begin{matrix}b=\frac{cx-ax+cy}{y}\text{, }&y\neq 0\\b\in \mathrm{R}\text{, }&y=0\text{ and }\left(a=c\text{ or }x=0\right)\end{matrix}\right.
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ax+by=cx+cy
Use the distributive property to multiply c by x+y.
ax=cx+cy-by
Subtract by from both sides.
xa=cx+cy-by
The equation is in standard form.
\frac{xa}{x}=\frac{cx+cy-by}{x}
Divide both sides by x.
a=\frac{cx+cy-by}{x}
Dividing by x undoes the multiplication by x.
ax+by=cx+cy
Use the distributive property to multiply c by x+y.
by=cx+cy-ax
Subtract ax from both sides.
yb=cx-ax+cy
The equation is in standard form.
\frac{yb}{y}=\frac{cx-ax+cy}{y}
Divide both sides by y.
b=\frac{cx-ax+cy}{y}
Dividing by y undoes the multiplication by y.
ax+by=cx+cy
Use the distributive property to multiply c by x+y.
ax=cx+cy-by
Subtract by from both sides.
xa=cx+cy-by
The equation is in standard form.
\frac{xa}{x}=\frac{cx+cy-by}{x}
Divide both sides by x.
a=\frac{cx+cy-by}{x}
Dividing by x undoes the multiplication by x.
ax+by=cx+cy
Use the distributive property to multiply c by x+y.
by=cx+cy-ax
Subtract ax from both sides.
yb=cx-ax+cy
The equation is in standard form.
\frac{yb}{y}=\frac{cx-ax+cy}{y}
Divide both sides by y.
b=\frac{cx-ax+cy}{y}
Dividing by y undoes the multiplication by y.
Examples
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{ x } ^ { 2 } - 4 x - 5 = 0
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Linear equation
y = 3x + 4
Arithmetic
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Matrix
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Simultaneous equation
\left. \begin{cases} { 8x+2y = 46 } \\ { 7x+3y = 47 } \end{cases} \right.
Differentiation
\frac { d } { d x } \frac { ( 3 x ^ { 2 } - 2 ) } { ( x - 5 ) }
Integration
\int _ { 0 } ^ { 1 } x e ^ { - x ^ { 2 } } d x
Limits
\lim _{x \rightarrow-3} \frac{x^{2}-9}{x^{2}+2 x-3}