Multiply through the equation by a number that will make all of the coefficients integers. For example, if the equation is X/4 - Y/5 = 1, multiply through by 20 to get 5X - 4Y = 20. If an equation is a linear diophantine equation, this is always possible and multiplying through by a constant does not change the solutions to the equation.
Manipulate the equation into what is called "Y-intercept form." This is the form Y = mX + b, where m and b are integers. For example, if 5X - 4Y = 20, then 4Y = 5X - 20 or Y = (5/4)X - 5. This change is always possible. If aX + bY = c where a, b and c are numbers, then Y = (-a/b)X + c/b.
Choose proper values for X; there will be an infinite number of these. For each proper value of X, calculate the corresponding value of Y. These (X,Y) pairs are all integer solutions of the linear diophantine equation. The "proper" values of X are the values that make mX an integer. For example, for the equation Y = (5/4)X - 5, the proper values for X are the multiples of 4, that is 0, 4, 8, 12, 16, 20 and so on. For these values mX will be 0, 5, 10, 15, 20, 25 and so forth. For these values Y will have the values -5, 0, 5, 10, 15 and so on. The solutions to the linear diophantine equation are (0,-5), (4,0), (8,5), (12,10) and so forth.