Which of the following is always true of a dependent system of two equations?

• A. The system has no solution.
• B. The equations represent parallel lines with different intercepts.
• C. The system has exactly one solution.
• D. The system has infinitely many solutions.

Answer: (D)

Verified step by step guidance

1. Understand what a dependent system of two equations means: it occurs when the two equations represent the same line, meaning one equation can be derived from the other by multiplying by a constant.
2. Recall that if two equations represent the same line, every point on one line is also on the other, so they share infinitely many solutions.
3. Contrast this with independent systems, which have exactly one solution (the point where the lines intersect), and inconsistent systems, which have no solution (parallel lines with different intercepts).
4. Recognize that dependent systems cannot have no solution or exactly one solution because the lines coincide completely.
5. Therefore, the defining characteristic of a dependent system is that it has infinitely many solutions.