Solve the following system of equations. Give the real solutions only. \[ \begin{array}{l} x^{2}+y^{2}=15 \\ 2 x=y^{2}-7 \end{array} \] Select ALL that apply.

Step 1 ：We are given the system of equations: \(x^{2}+y^{2}=15\) and \(2x=y^{2}-7\).

Step 2 ：We can solve this system of equations by substitution or elimination. Since the second equation is already solved for y, it would be easiest to substitute the expression for y from the second equation into the first equation. This will give us an equation in terms of x only, which we can solve.

Step 3 ：After substituting the expression for y from the second equation into the first equation, we get: \(x^{2} + 2x + 7 = 15\).

Step 4 ：Solving this equation, we find the solutions for x are -4 and 2.

Step 5 ：After finding the value(s) of x, we can substitute them back into the second equation to find the corresponding value(s) of y. The solutions for y include complex numbers, which are not real. Therefore, we need to discard the solutions that involve complex numbers.

Step 6 ：The real solutions for y are -\(\sqrt{11}\) and \(\sqrt{11}\) when x is 2. When x is -4, there are no real solutions for y.

Step 7 ：Therefore, the real solutions to the system of equations are \((2, -\sqrt{11})\), \((2, \sqrt{11})\).

Step 8 ：\(\boxed{\text{The real solutions to the system of equations are } (2, -\sqrt{11}), (2, \sqrt{11})}\)