To find all the solutions in positive integers for the given system of equations, we can solve them one by one:
1. 2xy = 2:
Since 2xy = 2, we can divide both sides by 2 to simplify the equation: xy = 1. The only pair of positive integers that satisfies this equation is x = 1 and y = 1.
2. 3x - 4y = 0:
Rearranging the equation, we get 3x = 4y. Since both sides are divisible by 4, we can rewrite it as x = (4/3)y. For x to be a positive integer, y must be divisible by 3. The smallest positive integer solution is x = 4 and y = 3.
3. 7x + 15y = 51:
Rearranging the equation, we get 7x = 51 - 15y. Observing the right side, we see that for x to be a positive integer, 51 - 15y must be divisible by 7. We can test different values of y to find the solutions. Trying y = 1, 2, 3, we find that y = 3 is the only positive integer value that satisfies the equation. So, x = 6 for y = 3.
Therefore, the positive integer solutions to the system of equations are:
x = 1, y = 1
x = 4, y = 3
x = 6, y = 3