Solving a system of 3 variables How to solve or what is the algorithm to solve a system of equations like this: 
$$\eqalign{
(x +\phantom{3} z)^2 + (y +\phantom{3} w)^2 &amp;= 52\cr
(x + 3z)^2 + (y + 3w)^2 &amp;= 296\cr
(x -\phantom{3} z)^2 + (y -\phantom{3} w)^2 &amp;= 8\cr
(x -2z)^2 + (y - 2w)^2 &amp;= 61?
}$$
By trying to solve this equation (adding and subtracting first and third equations, putting results to fourth equation) I eventually got these:

$$\begin{cases}z^2+w^2=25\\
  x^2+y^2=5\\
 zx+wy=11\end{cases}$$

All of the given equations are the results of the equations I have provided. (You can check it).
And I found that the only integers which can solve the equation are:

$$x=2, y=1, z=4, w=3\quad\text{or}\quad
x=-2, y=-1, z=-4, w=-3$$
By trying to solve this equation (adding and subtracting first and third equations, putting results to fourth equation) I eventually got these:

$$\begin{cases}z^2+w^2=25\\
  x^2+y^2=5\\
 zx+wy=11\end{cases}$$

All of the given equations are the results of the equations I have provided. (You can check it).
And I found that the only integers which can solve the equation are:

$$x=2, y=1, z=4, w=3\quad\text{or}\quad
x=-2, y=-1, z=-4, w=-3$$