Find all the points that lie $ 3$ units from each of the points $ (2,0,0), (0,2,0), \text{ and }(0,0,2)$ 
I calculated the result $$ \{(x,y,z) \in\mathbb{R^3}: x=y=z\}.$$
I'm wondering whether I did this problem correctly and if I did how to draw the set of solutions. 
I used the euclidean distance formula with the square root (the euclidean norm) and set up three equations for each point $(2,0,0)$, $(0,2,0)$ and $(0,0,2)$ that made sure the distance from arbitrary $(x,y,z)$ to these three points equal to  $3$ and arrived at the solution set. –

To avoid forgetting solutions it is best to keep the whole system together.
$\begin{cases}
(x-2)^2+y^2+z^2=9\\
x^2+(y-2)^2+z^2=9\\
x^2+y^2+(z-2)^2=9\\
\end{cases}\iff\begin{cases}
(-4x+4)+x^2+y^2+z^2=9\\
(-4y+4)+x^2+y^2+z^2=9\\
(-4z+4)+x^2+y^2+z^2=9\\
\end{cases}$
Now replacing row $(2)$ by $(2)-(1)$ you get $4x=4y$ and replacing row $(3)$ by $(3)-(1)$ you get $4x=4z$.
But you still have to carry equation $(1)$ for the system to stay complete. This is probably what you forget to do in your solution.
Or maybe you also replaced $(1)$ by a combination of rows, and lost information. In Gauss elimination at least one row should stay untouched at each step.
$\begin{cases} (-4x+4)+x^2+y^2+z^2=9\\ x=y\\ y=z\\ \end{cases} \iff 
\begin{cases} (-4x+4)+3x^2=9\\ x=y\\ y=z\\ \end{cases}\iff
\begin{cases} 3x^2-4x-5=0\\ x=y\\ y=z\\ \end{cases}$
Which solves to $\displaystyle x=y=z=\frac{2\pm\sqrt{19}}3$.
