If you have two variables, and changing one results in a the other changing as well, we say that those variables are proportional. If that change is just multiplying by a constant, then we call that constant the constant of proportionality.
If we look at the equation #y = 5/8x#, then we know #x# and #y# are proportional, because changing one results in the other changing as well. Furthermore, we can see that if we change #x#, then we are changing #y# by that amount multiplied by #5/8#. For example, it's easy to see that if #x=0#, then #y = 5/8*0 = 0#. Let's see what happens to #y# when we add #n# to #x#:
#y = 5/8*(0+n) = 5/8n = 0+5/8n#
So, when we increased #x# by #n#, we increased #y# by #5/8n#. We can use a similar process to show that it doesn't matter where we start. A change of #n# in #x# will result in a change of #5/8n# in #y#. This matches our description of a constant of proportionality.
You're also right about it being the slope. In a line, the slope is just another way of expressing the change in #y# with relation to the change in #x#. That is often the first way students are introduced to it:
#"slope" = "rise"/"run" = ("change in "y)/("change in "x)#
A very fast way to find the slope of a line given the equation is to put it into point-slope form.
If a line has the equation #y = mx + b#, where #m# and #b# are constants, then #m# is the slope of the line and #b# is the #y#-intercept (that is, the line passes through the point #(0, b)#).
Note that the given equation is actually in that form already, with #m = 5/8# and #b = 0#.
