Solving Lagrange multipliers system

Question

I need help solving this system:

$$ \begin{cases} 2(x-1) = \lambda2x \\ 2(y-2) = \lambda2y \\ 2(z-2) = \lambda2z \\x^2 + y^2+z^2 = 1 \end{cases} $$

I can find $$ \lambda = (x-1)/x $$ but can't go further.

Any help?

Answer 1

First three equations lead to:

$$\lambda = \frac{x-1}{x} = \frac{y-2}{y} = \frac{z-2}{z}.$$

Now subtracting 1 on every term implies

$$\lambda-1 = -\frac{1}{x} = -\frac{2}{y} = -\frac{2}{z}.$$

Now you found a relation of $x,y$ and $z$: $$ 2x = y = z. $$ I believe you can take it from here.