so given five simultaneous equations,
$x+y+z=10$
$xy=4$
$2x=t+uy$
$2y=t+ux$
$2z=t$
How should I go about solving this? It's part of a question on Lagrange multipliers that requires me to find the minimum to the equation $f(x,y,z)$ = $x^2+y^2+z^2$. I've been banging my head trying to figure this one out. Every time I try to substitute one of the variable in the other equation, I run into the issue of t and u. Any help or insight would be appreciated.
$2x=2z+uy$,
$2y=2z+ux$,
$2x^2-2xz=2y^2-2yz$
$2x^2-2x(10-x-y)=2y^2-2y(10-x-y)$
$2x^2-20x+2x^2+2xy=2y^2-20y+2xy+2y^2$
$4x^2-20x=4y^2-20y={64\over x^2}-{80\over x}$
$4x^4-20x^3=64-80x$
$0=x^4-5x^3+20x-16=(x-1)(x^3-4x^2-4x+16)=(x-1)(x^2-4)(x-4)$