# Inequality solution verification(positive solution) I have the following system of equation where a

Inequality solution verification(positive solution)
I have the following system of equation where all the free variables live in the integers that is are allowed to be integers only.
$x=l$
$y=5n-2l+25$
$z=10-2n$
For $l,n\in \mathbb{N}$
I need to compute how many positive solutions are there so I computed the following inequalities:
$x>0\to l>0$
$y>0\to n>-5+2l/5$
$z>0\to n<-5$
To get how many solution we keep a fixed l and see how many n we get and do that for each n until we get values that pop us out of the inequality for example for $l=1$ we have $n>-4.6$ hence we have $-4\le n<5$ so in this case their is $10$ solutions and we proceed like this I just want to verify my answer I got $105$ possible positive solutions is that correct?
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antantil0
Mathematically Your equations are equivalent to :
$5\left(n+5\right)>2l>0\phantom{\rule{0ex}{0ex}}5>n$
so for fixed $l$ we have $\frac{2l}{5} this means that $0 and the number of solution possible value of $n$ with $l$ fixed is exactly is $9-⌊\frac{2l}{5}⌋$ and so the number of solution is:
using calculator: we compute the sum, and we get the total number $216-110=106$