Question 77808
Let x and y represent the two integers and let us arbitrarily choose x>y.  The first statement "the difference between two positive integers is 52" can then be written as an equation:  x-y=52 (since x>y).
The second statement "one integer is 3 times as great as the other" must then be written as 3y=x (again, since x>y.  If 3x=y this would imply that y>x, prove this to yourself!).  So the two equations we have to work with are

1.  x-y=52
2.  3y=x

Since 3y=x, we can substitute 3y for x in equation (1) to get 3y-y=52 => 2y=52 => y=26.  Substitute this value of y=26 into equation (2) to get 3(26)=x => 78=x.

So the two positive integers are 78 and 26.  QED