Solve for a (complex solution)
\left\{\begin{matrix}a=-\frac{b\left(2-x\right)}{x+1}\text{, }&x\neq -1\\a\in \mathrm{C}\text{, }&b=0\text{ and }x=-1\end{matrix}\right.
Solve for b (complex solution)
\left\{\begin{matrix}b=\frac{a\left(x+1\right)}{x-2}\text{, }&x\neq 2\\b\in \mathrm{C}\text{, }&a=0\text{ and }x=2\end{matrix}\right.
Solve for a
\left\{\begin{matrix}a=-\frac{b\left(2-x\right)}{x+1}\text{, }&x\neq -1\\a\in \mathrm{R}\text{, }&b=0\text{ and }x=-1\end{matrix}\right.
Solve for b
\left\{\begin{matrix}b=\frac{a\left(x+1\right)}{x-2}\text{, }&x\neq 2\\b\in \mathrm{R}\text{, }&a=0\text{ and }x=2\end{matrix}\right.
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ax+a=bx-2b
Add a to both sides.
\left(x+1\right)a=bx-2b
Combine all terms containing a.
\frac{\left(x+1\right)a}{x+1}=\frac{b\left(x-2\right)}{x+1}
Divide both sides by x+1.
a=\frac{b\left(x-2\right)}{x+1}
Dividing by x+1 undoes the multiplication by x+1.
bx-2b-a=ax
Swap sides so that all variable terms are on the left hand side.
bx-2b=ax+a
Add a to both sides.
\left(x-2\right)b=ax+a
Combine all terms containing b.
\frac{\left(x-2\right)b}{x-2}=\frac{ax+a}{x-2}
Divide both sides by x-2.
b=\frac{ax+a}{x-2}
Dividing by x-2 undoes the multiplication by x-2.
b=\frac{a\left(x+1\right)}{x-2}
Divide ax+a by x-2.
ax+a=bx-2b
Add a to both sides.
\left(x+1\right)a=bx-2b
Combine all terms containing a.
\frac{\left(x+1\right)a}{x+1}=\frac{b\left(x-2\right)}{x+1}
Divide both sides by x+1.
a=\frac{b\left(x-2\right)}{x+1}
Dividing by x+1 undoes the multiplication by x+1.
bx-2b-a=ax
Swap sides so that all variable terms are on the left hand side.
bx-2b=ax+a
Add a to both sides.
\left(x-2\right)b=ax+a
Combine all terms containing b.
\frac{\left(x-2\right)b}{x-2}=\frac{ax+a}{x-2}
Divide both sides by x-2.
b=\frac{ax+a}{x-2}
Dividing by x-2 undoes the multiplication by x-2.
b=\frac{a\left(x+1\right)}{x-2}
Divide ax+a by x-2.
Examples
Quadratic equation
{ x } ^ { 2 } - 4 x - 5 = 0
Trigonometry
4 \sin \theta \cos \theta = 2 \sin \theta
Linear equation
y = 3x + 4
Arithmetic
699 * 533
Matrix
\left[ \begin{array} { l l } { 2 } & { 3 } \\ { 5 } & { 4 } \end{array} \right] \left[ \begin{array} { l l l } { 2 } & { 0 } & { 3 } \\ { -1 } & { 1 } & { 5 } \end{array} \right]
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}