-3x-4y+6=0 can be written 4y=6-3x and y=6/4-¾x.
The slope is -¾. This is the tangent of the angle of elevation of the given line.
If we make a right triangle out of the line by using a line segment as the hypotenuse (run) and the adjacent side as length 4 and the opposite side (rise) is length 3, then by Pythagoras, the sloping side has length √(32+42)=√(9+16)=√25=5. So rise/run=opposite/hypotenuse=⅗. However, the slope is negative so rise/run must also be negative and the true rise/run=-⅗.
Another way of looking at this is to note that:
slope=tangent=(rise/run)/(adjacent/run)=-¾=rise/adjacent=(-⅗)/(⅘).