Solve a/b-b/c=k

$$\frac { a } { b } - \frac { b } { c } = k$$

Solve for a

$$a=\frac{b\left(b+ck\right)}{c}$$

$$b\neq 0\text{ and }c\neq 0$$

Steps for Solving Linear Equation

Multiply both sides of the equation by bc, the least common multiple of b,c.

$$ca-bb=kbc$$

Multiply b and b to get b^{2}.

$$ca-b^{2}=kbc$$

Add b^{2} to both sides.

$$ca=kbc+b^{2}$$

The equation is in standard form.

$$ca=b^{2}+bck$$

Divide both sides by c.

$$\frac{ca}{c}=\frac{b\left(b+ck\right)}{c}$$

Dividing by c undoes the multiplication by c.

$$a=\frac{b\left(b+ck\right)}{c}$$

Solve for b (complex solution)

$$\left\{\begin{matrix}b=\frac{\sqrt{4ac+\left(ck\right)^{2}}-ck}{2}\text{, }&\left(a\neq 0\text{ and }c\neq 0\right)\text{ or }\left(k\neq 0\text{ and }arg(ck)\geq \pi \text{ and }c\neq 0\right)\\b=\frac{-\sqrt{4ac+\left(ck\right)^{2}}-ck}{2}\text{, }&\left(a\neq 0\text{ and }c\neq 0\right)\text{ or }\left(arg(ck)<\pi \text{ and }k\neq 0\text{ and }c\neq 0\right)\end{matrix}\right.$$