Does line have to cross on graph to be an infinite number of solutions

Answer 1

For a system of linear equations to have an infinite number of solutions, the lines represented by these equations do not merely have to cross; they must be coincident, meaning they lie exactly on top of one another. This essentially means they represent the same line.

To explain further:

1. **Identical Lines:** If two lines are identical, every point on one line is also on the other line. This results in an infinite number of solutions because there are infinitely many points that satisfy both equations simultaneously. Mathematically, this happens when one equation is a scalar multiple of the other.

2. **Parallel Lines:** If two lines are parallel but not identical, they will never intersect, meaning there are no points (no solutions) that satisfy both equations at the same time. Parallel lines have the same slope but different y-intercepts.

3. **Intersecting Lines:** If two lines intersect at exactly one point, there is exactly one solution to the system of equations, which is the coordinates of the intersection point.

To summarize, for an infinite number of solutions:
- The equations must describe the same line.
- They need to be scaled versions of each other, confirming that every point on one line is also a point on the other.