Hypothetical Question Help?

Question

Could x + y = x / y ? X and Y can be anything

Answer 1

Yes, as long as y does not equal either zero or 1:

x + y = x/y         (as y is on the denominator, it can't be zero)

y(x + y) = x

xy + y^2 = x

xy - x = -y^2

x(y-1) = -y^2

x = -y^2/(y-1)        (so y can't equal 1 as the denominator here would be zero)

You can now choose whatever values for y that you like and you can work out a value of x which will make your original statement true. (eg. if you pick y = 3, then x = -3^2/(3-1) = -9/2 and you can see that x + y = x/y if you put in these values).

Answer 2

With the obvious constraint that y is nonzero, a little algebra gives you:

y² +xy -x = 0

For each candidate value of x, this becomes a simple quadratic equation, which can have zero, one, or two real roots.  If you allow the values to be complex, each always has two roots.