To determine if a system of linear equations is consistent and independent, you need to consider the following:
- Consistent: The system has at least one solution.
- Independent: The two equations represent distinct lines that intersect at exactly one point.
Let's analyze the options given:
A) The x- and y-intercepts are the same: This would indicate that the lines are the same line (infinitely many solutions), which is not independent, and therefore this condition wouldn't work.
B) The slopes and y-intercepts are the same: If the slopes and y-intercepts are the same, it means the lines are identical (again, infinitely many solutions), which means they are not independent and inconsistent, so this condition wouldn't work either.
C) The two equations have two points in common: This suggests that the lines are either the same or intersecting at more than one point, which also indicates dependence or inconsistency if they are not the same line.
From this analysis, none of the provided options seem to accurately describe a scenario where the system is consistent and independent.
However, if we had to pick only from the given options based on the typical characteristics of linear independence and consistency, we would have to conclude none of them accurately meet the criteria.
In conclusion, the correct response, in line with standard interpretations of linear equations, would actually be none of the provided options.
If I must strictly adhere to your response options, I would say: None of the above suffice the criteria of being consistent and independent. But based on the closest potential interpretation, I would lean towards C assuming the context might suggest only one intersection point needed clarification.