$a x+b y =c$
1- We find gcd of a and b: gcd(a, b)
2- the condition for existence of solution is that : $gcd(a, b) | c$
3- we find the solutions of homogeneous equation $(ax+by=0)$; in case a and b are co-primes then $gcd(a, b)=1$ and we have:
$a x+b y = 0$:
$x= b$ and $y=-a$
So general solutions of homogeneous equation is:
$x = b . t$
$y = -a . t$
Where t can be any arbitrary value in Z. If we sum these solutions with a couple of certain solutions $x_0$,and $y_0$ then we find general solutions of equation $a x+b y =c$ ; we have:
$x = b.t + x_0$
$y = -a.t + y_0$
Hence this equation has infinitely many solutions in Z in general and in N in particular.The values of $x_0$ and $y_0$ depends on the value of c, therefore for n different c there can be n different sets of positive solutions.
Note that if a and b are co primes then their common divisor is $1$.
Geometric interpretation: Equation $ax +by=c$ in fact represent a line which is the locus of infinitely many points, among them many have coordinates in Z in general and in N in particular. With certain values of a and b , which define the slope of the lines c can have various values; each value gives a line parallel to others so that if we have n different c we will have n different equation or line with the same slope, hence we have exactly n solutions(or n lines) for the equation in each there exist many points having coordinates in N.So with n different c we have n different set of point having positive coordinates. c can be any arbitrary integer. in other words multiplying both sides of equation by a common divisor like $d$ we get a new equation( a new line).In this way all equations satisfy the condition that $gcd(ad, bd)=d|cd=c_1$ where $c_1$ is the new c.