How to find coordinates of reflected point? How can I find the coordinates of a point reflected over a line that may not necessarily be any of the axis? 
Example Question: 

If P is a reflection (image) of point (3, -3) in the line $2y = x+1$, find the coordinates of Point P. 

I know the answer is $(-1,5)$ by drawing a graph but other than that, I cannot provide any prior workings because I don't know how to start...

Similar answer to @Vrisk, but a bit faster

Consider the line $L=Ax+By+C=0$ and  find image of point $(u,v)$  assuming $Au + Bv + C \neq 0 $

If we extend the space which the equation exists into $R^3$ , we will find that the equation $Ax+By+C=0$ denotes a plane with a unit normal vector as : $$ \hat{n} = \frac{<A,B>}{\sqrt{A^2 +B^2} } $$
Now, consider the unsigned perpendicular distance of this point from the line/plane :
$$ d =  \frac{Au +Bv +C}{\sqrt{A^2 +B^2}}$$
Now, depending on sign of the above quantity we can see which part of the plane/ line which the point is (see here).
If, from our point we were translate in twice the direction of unit normal scaled up by the distance from line to point, we would reach the image point. Hence, the image coordinates$(I)$ are given by the vector:
$$I =<u,v>- \frac{2(Au+Bv+c)}{A^2 +B^2} <A,B>$$
Or more concisely:
$$I= <u,v> -2d \hat{n}$$
Here is a useful diagram:

