Determine the number of solutions to this system of linear equations:

To determine the number of solutions to the given system of linear equations, we can analyze the two equations:

1. \( y = x + 5 \)
2. \( y = -x + 5 \)

These equations represent two lines in the Cartesian plane.

1. The first equation \( y = x + 5 \) is a line with a slope of \( 1 \) and a y-intercept of \( 5 \).
2. The second equation \( y = -x + 5 \) is a line with a slope of \( -1 \) and a y-intercept of \( 5 \).

Next, we can find the point of intersection (where they meet) by setting the equations equal to each other:

\[ x + 5 = -x + 5 \]

Now, we'll solve for \( x \):

\[ x + x = 5 - 5 \] \[ 2x = 0 \] \[ x = 0 \]

Now that we have \( x = 0 \), we can substitute it back into either equation to find \( y \):

Using the first equation: \[ y = 0 + 5 = 5 \]

So the point of intersection is \( (0, 5) \).

Since the two lines intersect at exactly one point, the system of equations has exactly 1 solution.

Thus, the answer is \( 1 \).