A zero slope is horizontal or flat, and an undefined slope is vertical or straight up and down.
While this answer is in the textbook and notebook of anyone studying the subject, it is easy to get the two mixed up on test day. Try the following mnemonic (memory aid):
Look at the letters "z" for zero, and "u" for undefined. The top and bottom of the letter Z are horizontal lines. The two sides of the letter U are vertical lines.
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A line with an undefined slope is a vertical line. A line with a slope of zero is a horizontal line. If you use the formula for slope m = (y2-y1) divided by (x2-x1)... For an undefined slope you will get zero in the denominator, which you cannot have because you cannot divide by zero. For a slope equal to zero, you will get a zero in the numerator. Zero divided by any non-zero denominator, will give you a slope of zero.
The slope when you get a zero on top would be zero. Zero divided by any number is zero.
Flat slope is slope equals zero.
A line with a slope of zero is a flat, vertical line.
The parallel line would also have a slope of zero (both are horizontal lines).