Demand $z=x+y$ and $x^2/4 + y^2/5 + z^2/25 = 1$. What is the maximum value of $f(x,y,z) = x^2+y^2+z^2$? 
Demand $z=x+y$ and $x^2/4 + y^2/5 + z^2/25 = 1$. What is the maximum value of $f(x,y,z) = x^2+y^2+z^2$?

I've been attempting this with Lagrange multipliers in a few different ways. However, the resulting system of equations with two lagrangians has so many variables that it becomes very complicated. Can someone show how this is to be done manually?
I also attempted turning it into two unknowns by replacing $z$ with $x+y$. However, this also led nowhere.

If $3x=2y$, we get a value $10$.
We'll prove that it's a maximal value.
Indeed, we need to prove that
$$x^2+y^2+z^2\leq10\left(\frac{x^2}{4}+\frac{y^2}{5}+\frac{z^2}{25}\right)$$ or
$$x^2+y^2+(x+y)^2\leq10\left(\frac{x^2}{4}+\frac{y^2}{5}+\frac{(x+y)^2}{25}\right)$$ or
$$(3x-2y)^2\geq0$$ and we are done!
