Explanation:
there are infinitely many values for "a" and "b" to make that happen.
so, let's start with the opposite cases.
for which "a" and "b" do they have infinitely many intersection points ?
in that case both lines and therefore equations must be identical - that includes also constant multiples of one equation to the other.
cx + dy = e
is identical to any
ncx + ndy = ne
in our case
ax - 2y = 1
6x - 4y = b
that means for a = 3, b = 2 we have exactly that situation
3x - 2y = 1
6x - 4y = 2
equation 2 is just equation 1 multiplied by 2. they are identical and have infinitely many intersection points, because they are on top of each other.
the next case :
for which "a" and "b" do they have no intersection point at all ?
for that the lines need to be parallel (the same slope) but not identical as before.
to be parallel we need a form
ncx + ndy = me
with m <> n.
in our case we need again a = 3, but b must be anything but 2 :
3x - 2y = 1
6x - 4y = b
b <> 2
so, to have exactly one intersection point,
a <> 3, b = anything from -infinity to +infinity
or
b is in (-infinity, +infinity)