Solve for a (complex solution)
\left\{\begin{matrix}a=-\frac{-bx+by-A}{x-y}\text{, }&x\neq y\\a\in \mathrm{C}\text{, }&A=0\text{ and }x=y\end{matrix}\right.
Solve for a
\left\{\begin{matrix}a=-\frac{-bx+by-A}{x-y}\text{, }&x\neq y\\a\in \mathrm{R}\text{, }&A=0\text{ and }x=y\end{matrix}\right.
Solve for A
A=\left(a-b\right)\left(x-y\right)
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ax+by-ay-bx=A
Swap sides so that all variable terms are on the left hand side.
ax-ay-bx=A-by
Subtract by from both sides.
ax-ay=A-by+bx
Add bx to both sides.
\left(x-y\right)a=A-by+bx
Combine all terms containing a.
\left(x-y\right)a=bx-by+A
The equation is in standard form.
\frac{\left(x-y\right)a}{x-y}=\frac{bx-by+A}{x-y}
Divide both sides by x-y.
a=\frac{bx-by+A}{x-y}
Dividing by x-y undoes the multiplication by x-y.
ax+by-ay-bx=A
Swap sides so that all variable terms are on the left hand side.
ax-ay-bx=A-by
Subtract by from both sides.
ax-ay=A-by+bx
Add bx to both sides.
\left(x-y\right)a=A-by+bx
Combine all terms containing a.
\left(x-y\right)a=bx-by+A
The equation is in standard form.
\frac{\left(x-y\right)a}{x-y}=\frac{bx-by+A}{x-y}
Divide both sides by x-y.
a=\frac{bx-by+A}{x-y}
Dividing by x-y undoes the multiplication by x-y.
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