Coordinates

The first step is to consider some triangles.

Once we find x, we can see that there are two congruent triangles with hypotenuse length \(10\sqrt{3}\).

One of the legs of this triangle will be y1 - y2, which is 14.

Using the pythagorean theorem, we can find the distance between the two x coordinates. 

\(\sqrt{(10\sqrt{3})^2 - 14^2}\)

This simplifies to:

\(\sqrt{104}\)

This simplifies to:

\(2\sqrt{26}\)

This means the x coordinate is \(2\sqrt{26}\) units away from the x coordinate of (4,7), and then we have these 2 possible values of x:

\(4 + 2\sqrt{26}\) and \(4 - 2\sqrt{26}\)

Since the question is looking for the sum of the possible values of x, then we just add the two possible values together:

\(4 + 2\sqrt{26} + 4 - 2\sqrt{26}\)

This simplifies to 8.

Therefore, the sum of all possible values of x is 8.