c d g - e g = \partial b
Solve for b (complex solution)
\left\{\begin{matrix}b=\frac{g\left(cd-e\right)}{∂}\text{, }&∂\neq 0\\b\in \mathrm{C}\text{, }&\left(c=\frac{e}{d}\text{ and }d\neq 0\text{ and }∂=0\right)\text{ or }\left(g=0\text{ and }∂=0\right)\end{matrix}\right.
Solve for c (complex solution)
\left\{\begin{matrix}c=\frac{b∂+eg}{dg}\text{, }&g\neq 0\text{ and }d\neq 0\\c\in \mathrm{C}\text{, }&\left(g=0\text{ and }∂=0\right)\text{ or }\left(g=0\text{ and }b=0\right)\text{ or }\left(g=-\frac{b∂}{e}\text{ and }d=0\text{ and }b\neq 0\text{ and }∂\neq 0\right)\end{matrix}\right.
Solve for b
\left\{\begin{matrix}b=\frac{g\left(cd-e\right)}{∂}\text{, }&∂\neq 0\\b\in \mathrm{R}\text{, }&\left(c=\frac{e}{d}\text{ and }d\neq 0\text{ and }∂=0\right)\text{ or }\left(g=0\text{ and }∂=0\right)\end{matrix}\right.
Solve for c
\left\{\begin{matrix}c=\frac{b∂+eg}{dg}\text{, }&g\neq 0\text{ and }d\neq 0\\c\in \mathrm{R}\text{, }&\left(g=0\text{ and }∂=0\right)\text{ or }\left(g=0\text{ and }b=0\right)\text{ or }\left(g=-\frac{b∂}{e}\text{ and }d=0\text{ and }b\neq 0\text{ and }∂\neq 0\right)\end{matrix}\right.
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∂b=cdg-eg
Swap sides so that all variable terms are on the left hand side.
\frac{∂b}{∂}=\frac{g\left(cd-e\right)}{∂}
Divide both sides by ∂.
b=\frac{g\left(cd-e\right)}{∂}
Dividing by ∂ undoes the multiplication by ∂.
cdg=∂b+eg
Add eg to both sides.
dgc=b∂+eg
The equation is in standard form.
\frac{dgc}{dg}=\frac{b∂+eg}{dg}
Divide both sides by dg.
c=\frac{b∂+eg}{dg}
Dividing by dg undoes the multiplication by dg.
∂b=cdg-eg
Swap sides so that all variable terms are on the left hand side.
\frac{∂b}{∂}=\frac{g\left(cd-e\right)}{∂}
Divide both sides by ∂.
b=\frac{g\left(cd-e\right)}{∂}
Dividing by ∂ undoes the multiplication by ∂.
cdg=∂b+eg
Add eg to both sides.
dgc=b∂+eg
The equation is in standard form.
\frac{dgc}{dg}=\frac{b∂+eg}{dg}
Divide both sides by dg.
c=\frac{b∂+eg}{dg}
Dividing by dg undoes the multiplication by dg.
Examples
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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
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