System of 3 simultaneous equation where one is non-linear Hi I have a problem where I need to solve the following set of equations:
$$
v = U u 
$$
$$
u = 1 -Uv
$$
$$
U^2 = u^2 + v^2
$$
I have tried subbing $u$ and $v$ into the expression for $U^2$ but it seems to get very messy very quickly.
Any help solving for $u$,$v$ and $U$ would be greatly appreciated.

$v = Uu, u = 1-Uv, U^2 = u^2 + v^2$
I solve the first for $U = v/u$ and substitute into both of the others:
$$u = 1-\frac{v^2}{u}, \frac{v^2}{u^2} = u^2 + v^2$$
Then the first becomes
$$ u^2 = u - v^2$$ and the second becomes
$$v^2 = u^4 + u^2v^2$$
The former rearranges into $u = v^2 + u^2$ and putting that into the original third equation, I find
$U^2 = u$.  So
$$\frac{v^2}{u^2} = U^2 = u,$$ so $v^2 = u^3$ or $v = u^{3/2}$
Finally, $u = 1-Uv$ becomes $u = 1-u^2$, which quadratic equation solves to
$$u = {{-1\pm\sqrt{5}}\over {2}}$$
