Solve :- $\frac{x - n}{m} - \frac{x - m}{n} = \frac{m}{n}$ for all $mn \neq 0$

Question

Solve :- $\frac{x - n}{m} - \frac{x - m}{n} = \frac{m}{n}$ for all $mn \neq 0$

What I Tried :- I just simplified the expression step by step and got :-

$$=> \frac{n(x - n) - m(x - m)}{mn} = \frac{m}{n}$$

$$=> n(x - n) - m(x - m) = m^2$$

$$=> nx - n^2 - mx + m^2 = m^2$$

$$=> nx - n^2 - mx = 0$$

From here I don't know how to proceed in a good way. Can anyone help?

Answer 1

Now $$(n-m)x=n^2.$$ If $n=m$, so $n=0$, which is impossible.

But for $n\neq m$ we obtain: $$x=\frac{n^2}{n-m}.$$ Id est, we got the following answer for $mn\neq0$.

If $m=n$, so our equation has no roots.

if $n\neq m$ we obtain: $\left\{\dfrac{n^2}{n-m}\right\}$