To find out how many solutions there are in a system of linear equations, let’s look at three different cases:

1. No Solutions (Inconsistent):
   This happens when the lines are parallel.
   For example, if we have two equations like:

• (y = mx + b_1)
• (y = mx + b_2)
  Here, (b_1) and (b_2) are different numbers, so the lines never meet.

2. One Solution (Consistent and Independent):
   This is when the lines cross each other at one point.
   For instance:

• (y = m_1x + b_1)
• (y = m_2x + b_2)
  In this case, (m_1) and (m_2) are not the same, which means the lines intersect.

3. Infinite Solutions (Dependent):
   This occurs when the equations represent the same line.
   For example:

• (y = mx + b)
• (2y = 2mx + 2b)
  These two lines are actually the same line.

To sum it up:

• No Solutions: The lines are parallel.
• One Solution: The lines cross each other.
• Infinite Solutions: The lines are the same.