Solve the non-linear system of equations.
\[
\begin{array}{l}
(x-5)^{2}+(y-1)^{2}=50 \\
2 x+y=-4
\end{array}
\]
Present your answer in the form of $(x, y)$. If there are more than one solutions, separate them by a comma. If there is no solution, enter DNE.

Step 1 ：The given system of equations is a combination of a circle equation and a linear equation. The first equation represents a circle with center at (5,1) and radius \(\sqrt{50}\), and the second equation represents a straight line. The solutions to the system are the points where the line intersects the circle.

Step 2 ：To solve this system, we can substitute y from the second equation into the first equation, and solve for x. Then we can substitute x into the second equation to solve for y.

Step 3 ：By substituting y from the second equation into the first, we get the equation \((-2*x - 5)^2 + (x - 5)^2 = 50\).

Step 4 ：Solving this equation gives us two solutions for x: -2 and 0.

Step 5 ：Substituting these values of x into the second equation, we get the corresponding values of y: 0 and -4.

Step 6 ：Therefore, the solutions to the system of equations are \(\boxed{(-2, 0)}\) and \(\boxed{(0, -4)}\).