Which statement is true for any linear system of two equations in two variables?

A) The system is independent if the slopes of the two lines are equal.
B) The system is inconsistent if the y-intercepts of the two lines are equal.
C) The system is independent if the graphs of the two equations intersect once.
D) The system is dependent if it has no real solution.
E) The system is always consistent if the slopes of the two lines are equal.

I believe it is A. But I'm not sure.

To determine which statement is true for any linear system of two equations in two variables, we need to analyze each option and understand the concepts involved.

A) The statement says that the system is independent if the slopes of the two lines are equal. However, this is incorrect. The slopes of the two lines being equal merely indicates that the lines are parallel, not necessarily independent.

B) The statement states that the system is inconsistent if the y-intercepts of the two lines are equal. This is incorrect as well. The y-intercepts being equal does not guarantee that the lines do not intersect.

C) The statement claims that the system is independent if the graphs of the two equations intersect once. This statement is true. In a linear system of two equations, if the graphs of the equations intersect precisely at one point, it means that the system has a unique solution and therefore, is independent.

D) The statement suggests that the system is dependent if it has no real solution. This is incorrect. A linear system can be dependent, but it may still have a real solution.

E) The statement argues that the system is always consistent if the slopes of the two lines are equal. This is also incorrect. While equal slopes may imply consistency, it does not hold true in all cases. For instance, if the y-intercepts of the two lines are different, they will never intersect.

Therefore, the correct statement is option C: The system is independent if the graphs of the two equations intersect once.