all you need to do to find the slope here is put it into y = mx + b format.
5y + 6x - 2 = 0
the goal is to get the y alone, so...
5y = 2 - 6x
divide both sides by 5
y = 2/5 - (6/5)x
then, just to be clearer for y = mx + b ...... y = (-6/5)x + (2/5)
the slope should be (-6/5)