To determine the number of solutions to the system of linear equations given by:
- \( y = x + 3 \)
- \( y = x - 2 \)
we can set the right-hand sides of the equations equal to each other, since both equations equal \( y \):
\[ x + 3 = x - 2 \]
Now, we can simplify this equation. Subtract \( x \) from both sides:
\[ 3 = -2 \]
This statement is false. Since we reached a contradiction, it indicates that there are no values of \( x \) that satisfy both equations at the same time.
Thus, the system of equations has no solutions.
Therefore, the number of solutions to the system is:
\[ \boxed{0} \]