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.