Diophantine Equation Solution in Plain Text
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This time, I want to edit the solution's description in "vi".
The equation is:

  3       5
 ---  +  --- = 4
  x       y

What we need are all pairs of integers (x,y) that satisfy the equation. The
equation can be solved by finding y in terms of xi or x in terms of y. So,
let us start by multiplying both sides by xy. Remember that neither x nor y
can be zero:

    3y + 5x = 4xy ==>

==>  5x = 4xy - 3y = (4x-3)y ==>

           5x       4x - 3 + x + 3          x + 3
==>  y = -------- = ---------------- = 1 + --------
         4x - 3         4x - 3             4x - 3

And y is an integer, which means that the denominator divides the numerator.
Or 4x - 3 divides x + 3, which means that 4x - 3 divides 4(x+3)=4x+12. And of
course
4x - 3 divides itself, thus it divides the difference between itself and
4x+12.
So, 4x - 3 divides 4x + 12 - (4x - 3) = 15
Let tablens to find values of x and y if 4x - 3 divides 15:

 +========+=====+==============+
 | 4x - 3 |   x  |       y     |
 +========+======+=============+
 |     1  |  1   |           5 |
 +--------+-----+--------------+
 |    -1  |  0.5 |  Never mind |
 +--------+------+-------------+
 |     3  |  1.5 |  Never mind |
 +--------+------+-------------+
 |    -3  |  0   |  Invalid    |
 +--------+------+-------------+
 |     5  |  2   |           2 |
 +--------+------+-------------+
 |    -5  | -0.5 |  Never mind |
 +--------+------+-------------+
 |    15  |  4.5 |  Never mind |
 +--------+------+-------------+
 |   -15  | -3   |           1 |
 +========+======+=============+

From the table above, we get that the solution set for (x,y) is:

 {(1, 5), (2, 2),(-3, 1)}

Another method to find pairs satisfying the equation is to find a bounded set
from which to check integer values of x:
If 4x - 3 divides x + 3, it means that either

 x + 3 = 0

or

 | 4x - 3 | <= | x + 3 |

We can solve the absolute value inequality by squaring both sides. And then
you will have less values from which to select.