Solve x-m=2-x-n/m-3

$$x - m = 2 - \frac { x - n } { m } - 3$$

Solve for n

$$n=m\left(x-m+1\right)+x$$

$$m\neq 0$$

Solution Steps

Multiply both sides of the equation by m.

$$mx-mm=m\times 2-\left(x-n\right)+m\left(-3\right)$$

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

$$mx-m^{2}=m\times 2-\left(x-n\right)+m\left(-3\right)$$

To find the opposite of x-n, find the opposite of each term.

$$mx-m^{2}=m\times 2-x+n+m\left(-3\right)$$

Combine m\times 2 and m\left(-3\right) to get -m.

$$mx-m^{2}=-m-x+n$$

Swap sides so that all variable terms are on the left hand side.

$$-m-x+n=mx-m^{2}$$

Add m to both sides.

$$-x+n=mx-m^{2}+m$$

Add x to both sides.

$$n=mx-m^{2}+m+x$$

Solve for m (complex solution)

$$\left\{\begin{matrix}m=\frac{\sqrt{x^{2}+6x-4n+1}+x+1}{2}\text{, }&\left(n\neq -1\text{ and }arg(n+1)<\pi \right)\text{ or }x\neq n\\m=\frac{-\sqrt{x^{2}+6x-4n+1}+x+1}{2}\text{, }&\left(n\neq -1\text{ and }arg(n+1)\geq \pi \right)\text{ or }x\neq n\end{matrix}\right.$$

Solve for m

$$\left\{\begin{matrix}m=\frac{\sqrt{x^{2}+6x-4n+1}+x+1}{2}\text{, }&\left(x\neq n\text{ and }x\geq 2\sqrt{n+2}-3\text{ and }n\leq \frac{x^{2}}{4}+\frac{3x}{2}+\frac{1}{4}\text{ and }n\geq -2\right)\text{ or }\left(x\neq n\text{ and }x\leq -2\sqrt{n+2}-3\text{ and }n\leq \frac{x^{2}}{4}+\frac{3x}{2}+\frac{1}{4}\text{ and }n\geq -2\right)\text{ or }\left(n>-1\text{ and }x\geq 2\sqrt{n+2}-3\text{ and }n\leq \frac{x^{2}}{4}+\frac{3x}{2}+\frac{1}{4}\right)\text{ or }\left(n>-1\text{ and }x\leq -2\sqrt{n+2}-3\text{ and }n\leq \frac{x^{2}}{4}+\frac{3x}{2}+\frac{1}{4}\right)\text{ or }\left(x\neq n\text{ and }n\leq -2\text{ and }n\leq \frac{x^{2}}{4}+\frac{3x}{2}+\frac{1}{4}\right)\\m=\frac{-\sqrt{x^{2}+6x-4n+1}+x+1}{2}\text{, }&\left(x\neq n\text{ or }n<-1\right)\text{ and }\left(n\leq -2\text{ or }x\leq -2\sqrt{n+2}-3\text{ or }x\geq 2\sqrt{n+2}-3\right)\text{ and }n\leq \frac{x^{2}}{4}+\frac{3x}{2}+\frac{1}{4}\text{ and }\left(x\geq 2\sqrt{n+2}-3\text{ or }x\neq n\text{ or }n<-1\right)\text{ and }\left(x\geq 2\sqrt{n+2}-3\text{ or }x\leq -2\sqrt{n+2}-3\text{ or }n\leq -2\text{ or }x\neq n\right)\end{matrix}\right.$$