Step 1 :Rewrite the given equation in slope-intercept form (y = mx + b): \(4x + 12y = 36\) becomes \(y = -\frac{1}{3}x + 3\)
Step 2 :Identify the slope of the given line as \(-\frac{1}{3}\)
Step 3 :Since parallel lines have the same slope, the slope of a line parallel to the given line is also \(-\frac{1}{3}\)
Step 4 :Find the slope of a line perpendicular to the given line by taking the negative reciprocal of the given line's slope: \(\frac{-1}{-\frac{1}{3}} = 3\)
Step 5 :\(\boxed{\text{The slope of a line parallel to the given line is } -\frac{1}{3} \text{, and the slope of a line perpendicular to the given line is } 3}\)