$x$ and $y$ Problem

After simplification, we have 1/2=x/y

Putting values; x=1 and y=2 (minimum), we get the sum as 1+2=3

A Note was missing in the question...........

x and y are natural numbers

The given equation can be simplified as:

$\frac{1 + xy}{x} = 2 \times \frac{1 + xy}{y}$

Solving this we get:

y = 2x

We have to find minimum value of x+y i.e.

x + 2x = 3x

Thus substituting x=1(smallest natural number) we get

x+ y = 3.

We can solve for $x$ and $y$ by first combining each side into a single fraction: $\frac{1 + xy}{x} = 2 \frac{1 + xy}{y}.$ Since $x, y > 0$ , $1 + xy \neq0$ and we may divide both sides by this quantity to get $\frac{1}{x} = \frac{2}{y} \implies y = 2x$ , and therefore $x+ y = 3x$ . Over the real numbers, there is no minimum value, and the problem does not specify that our solution should be over the positive integers. With this restriction, however, the minimum is clearly achieved at $x = 1 \implies x + y = 3x = 3$ .