The point-slope equation of a line is useful for this purpose. It is often seen in the form
y -k = m(x -h)
for a line of slope m through the point (h, k).
You can add k to this equation to put it in a form more like slope-intercept form:
y = m(x -h) +k
Either of these can be used to write the equation of a parallel or perpendicular line through the point (h, k). The slope "m" of a parallel line will match that of the given line.
The slope "m" of a perpendicular line will be the negative reciprocal of the slope of the given line. That is, if you are given a line with slope "s", the slope of the perpendicular line will be m = -1/s.