No, never.
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Horizontal lines always have a slope of zero. (i.e completely flat, level surfaces have a slope of zero). However a line does not have to have a slope of zero in order to be a line.
If a line on a graph is rising as it goes from left to right, it has a positive slope. If it is falling from left to right (or rising from right to left) it has a negative slope. If it is horizontal, it has a slope of zero.
The slope of the graph of a direct variation is always positive.
Yes
A horizontal line has a slope of zero. The equation of a horizontal line is y = a.