The lines represented by the equations \( y = -5x + 1 \) and \( y = -5x + 10 \) have the following characteristics:
- Both equations have the same slope of \(-5\).
- However, they have different y-intercepts (1 and 10, respectively).
Since they have the same slope but different y-intercepts, the lines are parallel and will never intersect.
The correct answer is: C. The linear equations have the same slope but different y-intercepts.