Question

the graph of a system of equations shows two lines that overlap completely. what does this tell you about the system? \bigcirc the system has one solution at the point where the lines cross. \bigcirc the system has exactly two solutions. \bigcirc the system has no solution because the lines never intersect. \bigcirc the system has infinitely many solutions because every point on the line satisfies both equations.

Explanation:

To determine the nature of the system of equations from the graph:

1. Recall the meaning of overlapping lines in a system of equations.
2. Analyze each option:

• Option 1: If lines overlap, they are not just intersecting at one point, so this is incorrect.
• Option 2: A system of linear equations can have 0, 1, or infinitely many solutions, not exactly two, so this is incorrect.
• Option 3: Overlapping lines do intersect (they are the same line), so this is incorrect.
• Option 4: When two lines overlap completely, they are the same line. Every point on one line is also on the other line, meaning every point on the line satisfies both equations, so there are infinitely many solutions. This is correct.

Answer:

D. The system has infinitely many solutions because every point on the line satisfies both equations.