<s>[INST] System of quadratic equations with three variables (generic form) Try solve a system of equation like this one.
\begin{cases} 
(O_x -A_x)^2+(O_y-A_y)^2+(O_z-Az)^2=x^2 \\
(O_x -B_x)^2+(O_y-B_y)^2+(O_z-Bz)^2=y^2 \\
(O_x -C_x)^2+(O_y-C_y)^2+(O_z-Cz)^2=z^2
\end{cases}
Is there any way to express $O_x, O_y, O_z$ in terms of $A_x, A_y, A_z, B_x, B_y, B_z,C_x,C_y,C_z, x, y, z$?
 [/INST] OK lets check this out I'll replace $x \  y \ z$ with $a\ b\ c$ to prevent name collision with function independent variables.
\begin{cases} 
(O_x -A_x)^2+(O_y-A_y)^2+(O_z-A_z)^2=a^2 \ ①\\
(O_x -B_x)^2+(O_y-B_y)^2+(O_z-B_z)^2=b^2 \ ②\\
(O_x -C_x)^2+(O_y-C_y)^2+(O_z-C_z)^2=c^2 \ ③
\end{cases}
$④=①-②$ and $⑤=②-③$ we get
\begin{cases} 
2(A_x-B_x)O_x+2(A_y-B_y)O_y+2(A_z-B_z)O_z-A_x^2-A_y^2-A_z^2+B_x^2+B_y^2+B_z^2+a^2-b^2=0;\ ④ \\
2(B_y-C_y)O_y+2(B_z-C_z)O_z+2(B_x-C_x)O_x-B_y^2-B_z^2-B_x^2+C_y^2+C_z^2+C_x^2+b^2-c^2=0;\ ⑤
\end{cases}
eliminate $O_y$ in $④$ and $O_z$ in $⑤$
\begin{cases} 
O_y=\frac{(B_z-C_z)(A_x^2+A_y^2+A_z^2-a^2)/2+(C_z-A_z)(B_x^2+B_y^2+B_z^2-b^2)/2+(A_z-B_z)(C_x^2+C_y^2+C_z^2-c^2)/2-(A_x(B_z-C_z)+B_x(C_z-A_z)+C_x(A_z-B_z))*O_x}{A_y(B_z-C_z)+B_y(C_z-A_z)+C_y(A_z-B_z)} \\ \\
O_z=\frac{(A_y-B_y)(C_z^2+C_x^2+C_y^2-c^2)/2+(B_y-C_y)(A_z^2+A_x^2+A_y^2-a^2)/2+(C_y-A_y)(B_z^2+B_x^2+B_y^2-b^2)/2-(C_x(A_y-B_y)+A_x*(B_y-C_y)+B_x(C_y-A_y))O_x}{C_z(A_y-B_y)+A_z(B_y-C_y)+B_z(C_y-A_y)}
\end{cases}
name
$A_\sharp =(A_x^2+A_y^2+A_z^2-a^2)/2\quad B_\sharp  =(B_x^2+B_y^2+B_z^2-b^2)/2\quad
C_\sharp  =(C_x^2+C_y^2+C_z^2-c^2)/2\\
DET_{xy}  =\begin{vmatrix}A_x&B_x&C_x\\1&1&1\\A_y& B_y&C_y\end{vmatrix}\quad
DET_{xz}  =\begin{vmatrix}A_x&B_x&C_x\\1&1&1\\A_z& B_z&C_z\end{vmatrix}\quad
DET_{yz}  =\begin{vmatrix}A_y&B_y&C_y\\1&1&1\\A_z& B_z&C_z\end{vmatrix}\\
DET_{\sharp x}  =\begin{vmatrix}A_\sharp&B_\sharp&C_\sharp\\1&1&1\\A_x& B_x&C_x\end{vmatrix}\quad
DET_{\sharp y}  =\begin{vmatrix}A_\sharp&B_\sharp&C_\sharp\\1&1&1\\A_y& B_y&C_y\end{vmatrix}\quad
DET_{\sharp z}  =\begin{vmatrix}A_\sharp&B_\sharp&C_\sharp\\1&1&1\\A_z& B_z&C_z\end{vmatrix}\quad$
$$ \bbox[,border:2px solid red]
{
\begin{aligned}
&\qquad  \text{ An interesting property to notice is}\\
& DET_{xz}DET_{\sharp y}-DET_{xy}DET_{\sharp z}=DET_{yz}DET{\sharp x}\\
&\qquad \quad also \quad DET_{xy}=-DET_{yx}
\end{aligned}
}$$
renaming $O_y$ and $O_z$ to
\begin{cases} 
O_y=\frac{(B_z-C_z)*A_\sharp+(C_z-A_z)*B_\sharp+(A_z-B_z)*C_\sharp+DET_{xz}*O_x}{-DET_{yz}} \\ \\
O_z=\frac{(A_y-B_y)*C_\sharp+(B_y-C_y)*A_\sharp-(C_y-A_y)*B_\sharp+DET_{xy}*O_x}{DET_{yz}}
\end{cases}
and further simplifies to
\begin{cases} 
O_y=\frac{\ DET_{\sharp z}-DET_{xz}*O_x}{DET_{yz}} \quad ⑥\\ 
O_z=\frac{-DET_{\sharp y}+DET_{xy}*O_x}{DET_{yz}} \quad ⑦
\end{cases}
Substitute $O_x$ and $O_y$ into $①$
$O_x^2-2A_xX+A_x^2-a^2
+\bigl[(\frac{DET_{\sharp z}-DET_{xz}O_x}{DET_{yz}})^2-2(\frac{DET_{\sharp z}-DET_{xz}O_x}{DET_{yz}})A_y+A_y^2\bigr]
+\bigl[(\frac{-DET_{\sharp y}+DET_{xy}O_x}{DET_{yz}})^2-2(\frac{-DET_{\sharp y}+DET_{xy}O_x}{DET_{yz}})A_z+A_z^2\bigr]=0$
and further reduce to
$(DET_{xy}^2+DET_{yz}^2+DET_{xz}^2)
-2(DET_{yz}^2A_x+DET_{xz}(DET_{\sharp z}-DET_{yz}Ay)+DET_{xy} (DET_{\sharp y}+DET_{yz}A_z))O_x
+DET_{yz}^2(A_x^2-a^2) +(DET_{\sharp z}-DET_{yz}Ay)^2+(DET_{\sharp y-DET_{yz}Az})^2=0$
then we can solve $O_x$ using "Quadratic formula"
$
Ox=\frac{DET_{yz}^2A_x+DET_{xz}(DET_{\sharp z}-DET_{yz}A_y)+DET_{xy}(DET_{\sharp y}+DET_{yz}A_z)
    \pm \sqrt{
-DET_{yz}^2(DET_{\sharp z}-DET_{yz}A_y)(DET_{\sharp z}-DET_{yz}A_y-2DET_{xz}A_x)  
-DET_{yz}^2(DET_{\sharp y}+DET_{yz}A_z)(DET_{\sharp y}+DET_{yz}A_z-2DET_{xy}A_x) 
-(DET_{xz}(DET_{\sharp y}+DET_{yz}A_z)-DET_{xy}(DET_{\sharp z}-DET_{yz}A_y))^2 
-DET_{yz}^2(DET_{xz}^2+DET_{xy}^2)A_x^2                         
+DET_{yz}^2(DET_{xy}^2+DET_{yz}^2+DET_{xz}^2)a^2}}              
{DET_{xy}^2+DET_{yz}^2+DET_{xz}^2} $
substitude $O_x$ into $⑥\ ⑦$ we get $O_y\ O_z$
or interestingly because $O_x O_y O_z$ are in symmetrical position all you have to do is to swap all $x \leftrightarrow y$ or  $x \leftrightarrow z$ to get O_y and O_z
$
Oy=\frac{DET_{xz}^2A_y+DET_{yz}(DET_{\sharp z}-DET_{xz}A_x)+DET_{yx}(DET_{\sharp x}+DET_{xz}A_z)
    \pm \sqrt{
-DET_{xz}^2(DET_{\sharp z}-DET_{xz}A_x)(DET_{\sharp z}-DET_{xz}A_x-2DET_{yz}A_y)  
-DET_{xz}^2(DET_{\sharp x}+DET_{xz}A_z)(DET_{\sharp x}+DET_{xz}A_z-2DET_{yx}A_y) 
-(DET_{yz}(DET_{\sharp x}+DET_{xz}A_z)-DET_{yx}(DET_{\sharp z}-DET_{xz}A_x))^2 
-DET_{xz}^2(DET_{yz}^2+DET_{yx}^2)A_y^2                         
+DET_{xz}^2(DET_{yx}^2+DET_{xz}^2+DET_{yz}^2)a^2}}              
{DET_{yx}^2+DET_{xz}^2+DET_{yz}^2} $
$
Oz=\frac{DET_{yx}^2A_z+DET_{zx}(DET_{\sharp x}-DET_{yx}A_y)+DET_{zy}(DET_{\sharp y}+DET_{yx}A_x)
    \pm \sqrt{
-DET_{yx}^2(DET_{\sharp x}-DET_{yx}A_y)(DET_{\sharp x}-DET_{yx}A_y-2DET_{zx}A_z)  
-DET_{yx}^2(DET_{\sharp y}+DET_{yx}A_x)(DET_{\sharp y}+DET_{yx}A_x-2DET_{zy}A_z) 
-(DET_{zx}(DET_{\sharp y}+DET_{yx}A_x)-DET_{zy}(DET_{\sharp x}-DET_{yx}A_y))^2 
-DET_{yx}^2(DET_{zx}^2+DET_{zy}^2)A_z^2                         
+DET_{yx}^2(DET_{zy}^2+DET_{yx}^2+DET_{zx}^2)a^2}}              
{DET_{zy}^2+DET_{yx}^2+DET_{zx}^2} $
</s>