Is set of three linear equations with three unknown solvable?

Question

I have the following set of linear equations with the unknowns $h, n, i$ which I would like to express as a function of my known quantities, $e, f, g$:

$$ e = h - n\\ f = h - i\\ g = i -n $$

with the constraint $0 \leq h, n, i \leq 1$.

Unfortunately, any type of simplification I tried lead to $0 = 0$. Is this system uniquely solvable or not? If uniquely solvable, I would appreciate any hints.

Answer 1

As it is, your equations are not linearly independent, since you have $(2)+(3)=(1)$. This means that you can pick arbitrarily one of the variables $h$, $i$ or $n$, and only then will the other two follow by your system of equations.

Notice that you can derive an additional constraint on your known parameters, namely that $f+g=e$.

Answer 2

Sometimes it is solvable - and has infinitely many solutions - but sometimes it is not. It depends on the values of $e,f$ and $g$.
In particular, if there is a solution then $h-i+i-n+n-h=f+g-e=0$.
In that case, $h$ can be anything, say $=x$, then $n=x-e$ and $i=x-f$