Distance between a point $X$ and a line $2x-y = 1$ I am asked to state a condition for a point $X$ such that the point $X$ is a distance of 10 units from the line $2x-y = 1$, and then find the Cartesian equation of the set of all points $X$ that are a fixed distance $10$ from that line.
I am not sure about the first condition required, but for the cartesian equation would it be
$$
  10\sqrt{5} = |2x-y-1|
$$ 
which would give me two equations, one being $y = 2x -1 - 10\sqrt{5}$ and the other being $y = 2x -1 + 10\sqrt{5}$ I am not sure which one is the Cartesian equation of all points. I am also unsure of this "condition" they speak of

The "condition" here is the constraint on $X$ that it's distance from $2x-y=1$ is $10$ units. By imposing the condition, you get the locus of $X$.
In this case, the locus is a pair of parallel straight lines parallel to the given line, which you have found out. 
The reason why there are two lines is that $X$ can lie on either side of $2x-y=1$, and each line takes into account one side.
