Solving a system of 3 variables

Question

How to solve or what is the algorithm to solve a system of equations like this: $$\eqalign{ (x +\phantom{3} z)^2 + (y +\phantom{3} w)^2 &= 52\cr (x + 3z)^2 + (y + 3w)^2 &= 296\cr (x -\phantom{3} z)^2 + (y -\phantom{3} w)^2 &= 8\cr (x -2z)^2 + (y - 2w)^2 &= 61? }$$

Answer 1

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:

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