Ross' argument with equations says that, if you choose a Cartesian coordinate system in the plane, every straight line can be written as
$$
y = ax + b \ , \qquad \text{or} \qquad x = 0 \ ,
$$
for some real numbers $a, b$. Now, choosing a direction means picking up some $a \in \mathbb{R}$. Everyone of your finite (or countable) number of points $(x_n, y_n)_{n=1,2, \dots}$ determines a straight line: you just solve the equation for $b$ in terms of $y_n, a, x_n$
$$
y_n = a x_n + b \qquad \Longleftrightarrow \qquad b = y_n - ax_n \ .
$$
And then you just need to choose any $b \neq y_n - ax_n, \ n = 1,2,\dots$, which you can do, since there is just a finite (or countable) number of them, but an uncountable quantity of real numbers.
Alternatively, you could fix the height $b$ instead of the slope $a$; for instance, $b=0$, and the points $(x_n, y_n)_{n=1,2,\dots}$ would give you a finite (or countable) number of slopes of straight lines from the origin $(0,0)$ to each of your points, using the same equation as above:
$$
a = \dfrac{y_n}{x_n} \ .
$$
(In case you have $x_n = 0$, you just have the straight line $x=0$.)
Now you pick up any $a\neq\dfrac{y_n}{x_n} $, which you can do for the same reason as before, and you are done.