What makes two lines parallels is the slope. If the slope is the same, the lines cannot intersect. Let's try to see it algebraically.
We have the first line with the equation
#y=mx+a#
and the second line has the same slope #m# so has equation
#y=mx+b#
If the lines have an intersection it means that the #x# and the #y# are the same. So we can, for example, equate the #y# and say
#mx+a = mx+b# because they both are equal to the same #y#.
But we can remove the #mx# from both equations because it is identical and we obtain
#a=b#.
What does it mean? Clearly if #a=b# the two lines are the same line, and this is the only case when two parallel line can have an intersection, when they intersects everywhere (and are then the same line).
So if #a# is not equal to #b# the lines cannot intersect.
Now that we know how to describe a parallel line, we have
#y=3x+6#.
All the line with the form
#y=3x+a# are parallel. So #y=3x+2# is parallel, #y=3x-18# is parallel etc.
There are infinite parallels, one for each chose of #a#. We are searching for one in particular, it is the line that passes from the point #(-10, 2.5)#.
We then know that when #x=-10# we must have #y=2.5#. We use this information in our general equation of the parallel #y=3x+a#.
#y=3x+a# and #x=-10, y=2.5#, then
#2.5=3*(-10)+a#
#2.5=-30+a# and we can solve for #a#
#a=32.5#.
This fix one parallel that is
#y=3x+32.5#.
