The constant on the right is divisible by the coefficients on the left, so two solutions are
(x, y) = (5, 0) or (0, -3)
The slope of the line is 3/5, so increaing y by 3 and x by 5 will result in another solution. One general parametric solution is then
x = 5n
y = 3n-3
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Integer values of n will result in integer solutions. Any value of n will yield numbers that satisfy the equation.
In general, the extended Euclidean algorithm can be used to find integer solutions. Then adding multiples of some number n to y and x in the ratio of the slope of the line will yield other solutions.