In a dependent system of two equations, particularly when discussing linear equations in two variables, the equations represent the same line. This means that every point on the line is a solution to both of the equations, and since a line has infinitely many points, there are infinitely many solutions to the system.
Let's go through the options to understand why "There are infinitely many solutions" is the correct choice for a dependent system:
1. "One of the lines has a positive slope and the other has a negative slope." - This statement refers to two different lines that are neither the same nor dependent. This option would suggest an intersecting system where the lines meet at exactly one point, forming an X shape, which is not the case for a dependent system.
2. "The lines intersect at exactly one point." - This statement suggests that there are two distinct lines that cross each other at a single point. In a dependent system, the lines do not intersect at just one point; they coincide completely, meaning they lie on top of each other.
3. "There are infinitely many solutions." - This statement is true for a dependent system, where both equations describe the same line. Since a line has infinitely many points, there are infinitely many solutions – every point on the line is a solution.
4. "The lines are perpendicular." - This would mean the two lines intersect at a right angle (90 degrees), which would again suggest an intersecting system with exactly one point of intersection, which is not true for a dependent system.
Therefore, the correct answer is:
3. There are infinitely many solutions.
