Same Slope, Never Meet

Two lines are parallel when they never meet. Two lines with the same slope are parallel. They rise by the same amount for each step to the right, so the gap between them never changes. The lines $y = 2x-1$ and $y = 2x+3$ both have slope 2. At every $x$ the second line is 4 above the first, so they never meet.

The two lines have the same slope, so the gap between them stays the same.

Whether two lines are parallel depends only on the slope $m$. Two lines are parallel when they have the same slope and different intercepts. If two equations have the same slope and the same intercept, they are not parallel. They describe the same line, written twice.