Based on some algebraic considerations solutions can be found:
When we let we get and .
When we let we get the inverse relationship where and .
When we let k=4 we get and
18.805... = 18.805...
When we let we get the inverse scenario where and
There are some restriction on . The functions for and are undefined when since you it results in and and since we can't divide by zero this is undefined. Interestingly though is defined as and therefore . Although this is outside our solution set since no value of seems to produce this result. Other cases that are undefined include the case where since it again results in a division by zero. The solution is also outside this solution set since there is no value of that produces one for and one for . You will find that k can also not be a negative number since it results in negative roots and that is an operation that is undefined.
When we let we get and
For a similar reason k can not be a complex number. Complex numbers also produce undefined results using this method. Taking a complex root is undefined.
If we look at a graph of the equation for x and y we can see this clearer. is in blue and is in green. As we can see and have the same domain for but they have different ranges. The one sided limit for as k approaches zero appears to be one. That means as k gets smaller and smaller y will get closer to one. On the other hand if we consider what happens to x it seems to behave in the opposite manner. It apporaches infinity as k approaches zero. This explains the behavior we saw earlier with the inverses of k. As k changes x and y have to swap places. If y starts out larger than x and we decrease y eventually y will equal x. We find at k=1.643 x=y =3.556. As y decreases x will increase to maintain the relationship. Unfortunately the fact that x and y do not have the same range means that y can not be less than one but x can which is not exactly true because we know we can just flip x and y and the equation will still look the same. So we know simply by looking at the graph of x and y that this does not include the full solution set.