Do all lines with the same slope have the same line?

No, not necessarily. They could have different y intercepts.

For example, take one horizontal line, and another horizontal line.

These horizontal lines would both have the same slope, 0. But one could be higher up, or have a higher y intercept, than the other.

They would still have the same slope, but not be the same line.

Add the cents that add the left overs to get the first number