In solving the system of equations $y = 7$ and $x^2+ y^2= 100,$ what is the sum of the solutions for $x?$
To find the solutions for $x$, I need to substitute $y = 7$ into the second equation and solve for $x$. That gives me $x^2 + 49 = 100,$ or $x^2 = 51.$ Taking the square root of both sides, I get $x = \pm \sqrt{51}.$ The sum of the solutions for $x$ is then $\sqrt{51} + (-\sqrt{51}) = 0.$ 