To determine all solutions in positive integers for the equation 18x + 5y = 48 using a Diophantine equation, we can apply the method of Diophantine equations and explore possible combinations of values for x and y.

One approach is to use the Euclidean algorithm to find the greatest common divisor (GCD) of the coefficients 18 and 5. In this case, the GCD of 18 and 5 is 1, which means there is a unique solution in positive integers.

To find a specific solution, we can use the extended Euclidean algorithm. Starting with the equation:

18x + 5y = 48

We can express the GCD (1) as a linear combination of 18 and 5:

1 = 18(-2) + 5(7)

Now, we multiply the entire equation by 48 to get:

48 = 18(-96) + 5(336)

Therefore, a specific solution to the equation is x = -96 and y = 336.