To determine the number of solutions to the given system of linear equations, we need to analyze the equations:
- \( y = x + 5 \)
- \( y = -x + 5 \)
We can set both equations equal to each other to find any intersection points:
\[ x + 5 = -x + 5 \]
Now, let's solve for \( x \):
\[ x + x = 5 - 5 \] \[ 2x = 0 \] \[ x = 0 \]
Now, substitute \( x = 0 \) back into either equation to find \( y \):
Using the first equation: \[ y = 0 + 5 = 5 \]
So, the intersection point (solution) is \( (0, 5) \).
Since the two lines intersect at a single point, the system of equations has exactly one solution.
Therefore, the number of solutions to the system is 1.