How do I derive the demand function for a utility function of, say, $U(x,y)=\sqrt{11x+11y}$ for goods X and Y in terms of $P_x$, $P_y$, and income $I$, with basic mathematics (basic calculus, but no lagrange multiplier or partial differentiation)?
So far, I know that for a given value of U, $Y=\frac {U^2}{11}-x $ and $\frac {dy}{dx} = -\frac {MU_x}{MU_y} =-1$ and $\frac {MU_x}{MU_y}=\frac {P_x}{P_y} = 1$. I can't see how to get rid of $Y$ in $X = \frac{I - P_yY}{P_x} = \frac{I}{P_x}-Y$.
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$\begingroup$You don't need calculus for this, a simple figure suffices.
If I understand the problem correctly you want to maximize $U(x,y)$ under the constraints $$x\geq0,\quad y\geq 0,\quad P_x\>x+P_y\>y\leq I\ .\tag{1}$$ The constraints $(1)$ define a triangle $T$ in the first quadrant as set of feasible points.
On the other hand $U(x,y)$ is a monotonically increasing function of $x+y$. Therefore we have to pick the vertex of $T$ where $x+y$ is maximal, and this is either the vertex $\bigl({I\over P_x},0\bigr)$ or the vertex $\bigl(0,{I\over P_y} \bigr)$, depending on which of $P_x$ or $P_y$ is smaller.
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