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

No, not all lines with the same slope have the same line.

If they're y and/or x intercepts are different then they could have the same slope, but not have the same line.

Here's an example:

The equation for a linear line is y = mx + b

Where m = the slope and b = the y intercept.

If we have two lines y = 2x + 1 and y = 2x + 10, they both have the same slope, but they are different lines:

If they are the same line: I should be able to plug one x value into both equations and I would get the same value, but if they are different then I won't get the same y value for each x value.

ex: if x = 2

y = 2x + 1 = > y = 2 (2) + 1 = > y = 5

y = 2x + 10 = > y = 2 (2) + 10 = > y = 14

As you can see I plugged in the same x value into both equations (they have same slope) and I got different values.

Hope this answers your question!

-Jabba