Question

Explain how you find ordered-pair solutions of the equation 3x – 5y = 15, and then find two such solutions.'

Answer 1

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
_____
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.