Natural number solutions of bivariate linear equation 2x + 3Y = 15?

Natural number solutions of bivariate linear equation 2x + 3Y = 15?

2x+3y=15
The solution of natural number is as follows
x=0,y=5
x=3,y=3
x=6,y=1
Supplementary process:
2x=15-3y
x=(15-3y)/2
Because X and y are natural numbers,
So x ≥ 0, and 15-3y ≥ 0, 15-3y is even
So x ≥ 0, y ≤ 5, y is odd
So:
When x = 0, y = 5
When x = 3, y = 3
When x = 6, y = 1

Finding all natural number solutions of quadratic equation 2x + 3Y = 15

(1)x=0 y=5 (2)x=3 y=3 (3)x=6 y=1
There should be only three

The quadratic equation 2x + 3Y = 15 has natural number solutions____ Group, positive integer solution___ Group, integer solution___ Group?

The quadratic equation 2x + 3Y = 15 has natural number solutions__ 3_ Group, positive integer solution_ 2_ Group, integer solution_ Infinity_ Group?
The binary linear equation 2x + 3Y = 15 is transformed into y = 5-x + X / 3
Y is an integer, so x / 3 is an integer, let t = x / 3
Then: x = 3T, y = 5-2t
When t is an arbitrary integer, a set of integer solutions can be obtained. Therefore, the quadratic equation 2x + 3Y = 15 has infinite sets of integer solutions
When x > 0, Y > 0, t > 0, t > 0

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