Derivation of a Demand Equation A consumer's utility function is $U(x,y)=\sqrt x + y$. Assuming we have an interior solution, I need to show that the demand for $x$ does not depend on income.

I know that the consumer consumes where the marginal utilities per dollar of $x$ and $y$ are equal, so I have found the following two equations:


$\frac 1 {2P_x\sqrt x}$ = $\frac 1 {P_y}$
${P_x}x + {P_y}y = I$


Then, I am stuck. I do not see how I can get rid of both $y$ and $I$ in this case? And how does knowing that we have an interior solution help?
hint:

Solve for optimal $x^*,y^*$ then substitute the solutions into budget constraint and then solve the budget constraint for $x^*,y^*$ respectively and you will get the demands as a function of the parameters and of course conditional on the optimal consumption of the other variable.

Knowing that we have an interior solution helps because that means we know demand(s) is(are) not zero.