To determine the number of solutions to the system of equations given by:
- \( y = 2x + 1 \)
- \( y = 3x + 1 \)
we can set the right-hand sides of the equations equal to each other since they both equal \( y \):
\[ 2x + 1 = 3x + 1 \]
Now, let's solve for \( x \):
- Subtract \( 2x \) from both sides:
\[ 1 = x + 1 \]
- Subtract 1 from both sides:
\[ 0 = x \]
Now we have found \( x = 0 \). We can substitute this value back into either equation to find the corresponding \( y \) value:
Using the first equation:
\[ y = 2(0) + 1 = 1 \]
Thus, the point of intersection (the solution to the system) is \( (0, 1) \).
Since we found exactly one solution to the system, we conclude that there is exactly one solution to the system of linear equations:
\[ \boxed{1} \]